diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzhnhw" "b/data_all_eng_slimpj/shuffled/split2/finalzzhnhw" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzhnhw" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nIn this note we consider the Dirichlet problem for the complex Monge-Amp\\`ere equation in a strictly pseudoconvex domain $\\Omega\\subset \\mathbb{C} ^n.$\nLet $\\psi$ be a continuous function on the boundary of $\\Omega.$\n We look for the solution to the equation:\n\\[\\label{eq:dirichlet-prob}\n\\begin{aligned}\n&\tu\\in PSH(\\Omega) \\cap C^0(\\bar\\Omega), \\\\\n&\t(dd^c u)^n = d\\mu, \\\\\n&\tu = \\psi \\quad \\mbox{on } \\d\\Omega.\n\\end{aligned}\\]\n\nIt was shown\nin \\cite{ko96} that for the measures satisfying certain bound in terms of the Bedford-Taylor capacity \\cite{BT2}\n the Dirichlet problem has a (unique) solution. The precise statement is as follows.\n\nLet $h : \\mathbb R_+ \\rightarrow (0, \\infty ) $ be an increasing function such that\n\\[\\notag\n\\label{eq:admissible}\n\t\\int_1^\\infty \\frac{1}{x [h(x) ]^{\\frac{1}{n}} } \\, dx < +\\infty.\n\\]\nWe call such a function \n{\\em admissible}. If $h$ is admissible, then so is $A h$ for any number $A >0$.\nDefine\n\\[\\notag\n\tF_h(x) = \\frac{x}{h(x^{-\\frac{1}{n}})}.\n\\]\nSuppose that for such a function $F_h (x)$ a Borel measure $\\mu $ satisfies\n\n\\begin{equation}\n\\label{eq:magrowth}\n\t\\int_E d\\mu \\leq F_h( cap(E)),\n\\end{equation}\nfor any Borel set $E \\subset \\Omega$. Then, by \\cite{ko96} the Dirichlet problem \\eqref{eq:dirichlet-prob} has a solution.\n\nThis statement is useful as long as we can verify the condition \\eqref{eq:magrowth}.\nIn particular if $\\mu$ has density with respect to the Lebesgue measure in $L^p$, $p>1$\nthen this bound is satisfied \\cite{ko96}. By the recent results in\n \\cite{Ng17, Ng18} if $\\mu$ is bounded by the Monge-Amp\\`ere measure of a H\\\"older continuous plurisubharmonic function $\\varphi$:\n\\[\\notag\n\\label{eq:sub}\n\t\\mu \\leq (dd^c\\varphi)^n \\quad \\mbox{in }\\Omega ,\n\\]\nthen \\eqref{eq:magrowth} holds for a specific $h$, and consequently, the Dirichlet problem \\eqref{eq:dirichlet-prob} is solvable with H\\\"older continuous solution.\nOur result in this paper says that we can considerably weaken the assumption on $\\varphi$ and still get a continuous solution of the equation.\n\nLet $\\varpi (t):= \\varpi(t;\\varphi,\\bar\\Omega)$ denote the modulus of continuity of $\\varphi$ on $\\bar\\Omega$, i.e, \n\\[\\notag\n\t\\varpi(t)= \\sup \\left\\{|\\varphi(z) -\\varphi(w)| : z,w \\in \\bar\\Omega, \\quad |z-w| \\leq t\\right\\}.\n\\]\nThus\n$\n\t|\\varphi(z) - \\varphi(w)| \\leq \\varpi (|z-w|)\n$ for every $ z,w\\in \\bar\\Omega$.\nLet us state the first result.\n\n\\begin{thm} \n\\label{thm:sub}\nLet $\\varphi \\in PSH(\\Omega) \\cap C^{0}(\\bar\\Omega), $ $\\varphi =0$ on $\\partial\\Omega$. Assume that its modulus of continuity satisfies the Dini type condition\n\\[\\label{eq:modulus-ass}\n\t\\int_0^{1} \\frac{[\\varpi(t)]^\\frac{1}{n}}{t |\\log t|} dt <+\\infty.\n\\]\nIf the measure $\\mu$ satisfies $\\mu \\leq (dd^c\\varphi)^n$ in $\\Omega$, then the Dirichlet problem \\eqref{eq:dirichlet-prob} admits a unique solution.\n\n\\end{thm}\n\nLet us mention in this context that it is still an open problem if a continuous subsolution $\\varphi$ implies the\nsolvability of \\eqref{eq:dirichlet-prob}.\n\n\nThe modulus of continuity of solution to the Dirichlet problem~\\eqref{eq:dirichlet-prob} was obtained in \\cite{BT76} for \n$\\mu = fdV_{2n}$ with $f(x)$ being continuous on $\\bar\\Omega$. We also wish to study this problem for the measures which satisfy the inequality \\eqref{eq:magrowth}. For simplicity we restrict ourselves to measures belonging to $\\mathcal{H}(\\alpha,\\Omega)$. In other words, we take the function $h(x) = C x^{n\\alpha}$ for positive constants $C,\\alpha>0$ in the inequality \\eqref{eq:magrowth}.\n\n We introduce the following notion, which generalizes the one in \\cite{ko94}.\nConsider a continuous increasing function $F_0:[0,\\infty) \\to [0,\\infty)$ with $F(0)=0$. \n\n\\begin{defn}\nThe measure $\\mu$ is called uniformly locally dominated by capacity with respect to $F_0$ if for every cube $I(z,r)=:I \\subset B_I:= B(z, 2r) \\subset \\subset \\Omega$ and for every set $E\\subset I$,\n\\[\\label{eq:u-vol-cap}\n\t\\mu(E) \\leq \\mu(I) F_0\\left(cap (E, B_I) \\right).\n\\]\n\\end{defn}\n\nAccording to \\cite{ACKPZ} the Lebesgue measure $dV_{2n}$ satisfies this property with $F_0 = C_\\alpha \\exp (-\\alpha\/ x^{-1\/n})$ for every $0< \\alpha < 2n$. The case $F_0(x)= C x$ was considered in \\cite{ko94}. We refer the reader to \\cite{BJZ} for more examples of measures satisfying this property.\nHere is our second result.\n\n\\begin{thm} \n\\label{thm:modulus}\nAssume $\\mu \\in \\mathcal{H}(\\alpha,\\Omega)$ with compact support and satisfying the condition \\eqref{eq:u-vol-cap} for some $F_0$. Then, the modulus of continuity of the solution $u$ of the Dirichlet problem~\\eqref{eq:dirichlet-prob} satisfies for $0< \\delta < R_0$ and $2R_0 = \\mbox{\\rm dist} (\\mbox{supp }\\mu, \\d\\Omega)>0$,\n\\[\\notag\n\t\\varpi(\\delta;u,\\Omega) \\leq \\varpi(\\delta;\\psi,\\d\\Omega) + C \\left[ \\left(\\log \\frac{R_0}{\\delta}\\right)^{-\\frac{1}{2}} + F_0 \\left(\\frac{C_0}{[\\log (R_0\/\\delta)]^\\frac{1}{2}}\\right)\\right]^{\\alpha_1},\n\\]\nwhere the constants $C, \\alpha_1$ depend only on $\\alpha, \\mu, \\Omega$.\n\\end{thm}\n\n\n\\bigskip\n\n{\\bf Acknowledgement.} The first author was partially supported by NCN grant \n2017\/27\/B\/ST1\/01145. The second author was supported by the NRF Grant 2011-0030044 (SRC-GAIA) of The Republic of Korea. He also would like to thank Kang-Tae Kim for encouragement and support. \n\n\n\\section{Proof of Theorem~\\ref{thm:sub}}\n\nIn this section we will prove Theorem~\\ref{thm:sub}. We need the following lemma. The proof of this lemma is based on a similar idea as the one in \\cite[Lemma~3.1]{KN18a} where the complex Hessian equation is considered. The difference is that we have much stronger volume-capacity inequality for the Monge-Amp\\`ere equation.\n\n\\begin{lem} \n\\label{lem:vol-cap-key}\nAssume the measure $\\mu$ is compactly supported. Fix $0< \\alpha < 2n$ and $ \\tau = \\alpha\/(2n+1)$. There exists a uniform constant $C$ such that for every compact set $K \\subset \\Omega$,\n\\[\\label{eq:vol-cap-key}\n\t\\mu(K) \\leq C \\left\\{ \\varpi\\left( \\exp \\left(\\frac{-\\tau}{2[cap(K)]^\\frac{1}{n}}\\right) \\right) + \\exp\\left(\\frac{2n\\tau -\\alpha}{2[cap(K)]^\\frac{1}{n}} \\right)\\right\\} \\cdot cap(K)\n\\]\nwhere $cap(K):= cap(K,\\Omega).$\n\\end{lem}\n\n\\begin{proof} Fix a compact subset $K\\subset\\subset \\Omega$. Without loss of generality we may assume that $K$ is regular (in the sense that its relative extremal function \\cite{BT2}\nis continuous) as $\\mu$ is a Radon measure. Denote by $\\varphi_\\varepsilon$ the standard regularization of $\\varphi$. We choose $\\varepsilon>0$ so small that \n\\[\\notag\n\t\\mbox{supp } \\mu \\subset \\Omega'' \\subset\\subset \\Omega' \\subset \\Omega_\\varepsilon \\subset \\Omega,\n\\]\nwhere $\\Omega_\\varepsilon = \\{z\\in \\Omega: dist(z, \\d\\Omega) > \\varepsilon\\}$. Since for every $K \\subset \\Omega''$ we have\n\\[\\notag\n\tcap(K, \\Omega') \\sim cap(K,\\Omega)\n\\]\n(up to a constant depending only on $\\Omega, \\Omega'$) in what follows we will write $cap (K)$ for either one of these capacities.\nWe have \n\\[\\notag\n\t0\\leq \\varphi_\\varepsilon - \\varphi \\leq \\varpi(\\varepsilon) := \\delta \\quad\n\t\\mbox{on } \\Omega'.\n\\]\nLet $u_K$ the relative extremal function for $K$ with respect to $\\Omega'$. \nConsider the set $K' = \\{ 3\\delta u_K + \\varphi_\\varepsilon < \\varphi - 2\\delta\\}$. Then, \n\\[\\label{eq:supp}\n\tK \\subset K' \\subset \\left\\{u_K < -\\frac{1}{2} \\right\\} \\subset \\Omega'.\n\\]\nHence, by the comparison principle \\cite{BT2},\n\\[\\label{eq:compare}\n\tcap(K') \\leq 2^n cap(K). \n\\]\nNote that \n\\[\\label{eq:bound-ep}\n\tdd^c \\varphi_\\varepsilon \\leq \\frac{C}{\\varepsilon^2} \\; dd^c |z|^2, \\quad\n\t\\|\\varphi_\\varepsilon + u_K\\|_\\infty=:M \\leq \\|\\varphi\\|_\\infty +1.\n\\]\nThe comparison principle, the bounds \\eqref{eq:bound-ep} and the volume-capacity inequality from \\cite{ACKPZ} (in the last inequality below) give us that \n\\[\\begin{aligned}\n\t\\int_{K'} (dd^c \\varphi )^n \n&\\leq \t\\int_{K'} (dd^c (3\\delta u_K + \\varphi_\\varepsilon) )^n \\\\\n&\\leq\t\t3\\delta\\int_{K'} \\left[dd^c (u_K + \\varphi_\\varepsilon)\\right]^n + \\int_{K'} (dd^c \\varphi_\\varepsilon )^n \\\\\n&\\leq\t\t3\\delta M^n cap(K') + C(\\alpha) \\varepsilon^{-2n} \\exp\\left(\\frac{-\\alpha}{[cap(K')]^\\frac{1}{n}} \\right) cap(K').\n\\end{aligned}\\]\nChoose $$\\varepsilon = \\exp\\left(\\frac{-\\tau}{[cap(K')]^\\frac{1}{n}} \\right)$$ (we assume that $\\varepsilon$ is so small that it satisfies \\eqref{eq:supp}, otherwise the inequality \\eqref{eq:vol-cap-key} holds true by increasing the constant) and plug in the formula for $\\delta$ we get that\n\\[\\notag\n\\begin{aligned}\n\t\\mu(K) \n&\\leq \t\\int_{K'} (dd^c (\\varphi) )^n \\\\\n&\\leq \t3 M^n \\varpi\\left( \\exp \\left(\\frac{-\\tau}{[cap(K')]^\\frac{1}{n}}\\right) \\right) \\cdot cap(K') \\\\\n&\\quad + C \\exp\\left(\\frac{2n\\tau -\\alpha}{[cap(K')]^\\frac{1}{n}} \\right).\n\\end{aligned}\\]\nThis combined with \\eqref{eq:compare} gives the desired inequality.\n\\end{proof}\n\nWe are ready to finish the proof of the theorem. \nIt follows from Lemma~\\ref{lem:vol-cap-key} that a suitable function $h$ for the measure $\\mu$ which satisfies \\eqref{eq:magrowth} is\n\\[\\notag\n\th (x)= \\frac{1}{C \\varpi(\\exp (-\\tau x))}\n\\]\nonce we had \n\\[\\notag\n\t\\int_1^\\infty \\frac{1}{x [h(x) ]^{\\frac{1}{n}} } \\, dx < +\\infty.\n\\]\nBy changing the variable $s= 1\/x$, and then $t = e^{-\\tau\/s}$, this is equivalent to \n$$\n\t\\int_0^{e^{-\\tau}} \\frac{\\left[\\varpi(t) \\right]^\\frac{1}{n}}{t |\\log t|} dt <+\\infty.\n$$\nThe finiteness is guaranteed by \\eqref{eq:modulus-ass}.\nThus, our assumption on the modulus of continuity $\\varpi(t)$ implies that $h$ is admissible in the case of $\\mu$ with compact support.\nThen, by \\cite{ko96} the Dirichlet problem~\\eqref{eq:dirichlet-prob} has a unique solution.\n\nTo deal with the general case consider the exhaustion of $\\Omega$ by \n$$\nE _j =\\{ \\varphi \\leq -1\/j \\}\n$$\nand define $\\mu _j $ to be the restriction of $\\mu$ to $E_j$. Denote by $u_j$ the solution\nof \\eqref{eq:dirichlet-prob} with $\\mu$ replaced by $\\mu _j$. By the comparison principle\n$$\nu_j + \\max (\\varphi , -1\/j ) \\leq u \\leq u_j ,\n$$\nand so the sequence $u_j$ tends to $u=\\lim u_j$ uniformly which gives the continuity of $u$.\nThe proof is completed. \n\n\n\n\n\n\\section{the modulus of continuity of solutions} \n\nIn this section we study the modulus of continuity of the solution of the Dirichlet problem with the right hand side in the class $\\mathcal{H}(\\alpha,\\Omega)$ (definition below)\nunder the additional condition that a given measure is locally dominated by capacity.\n\n\n\nRecall that a positive Borel measure $\\mu$ belongs to $\\mathcal{H}(\\alpha,\\Omega)$, $\\alpha>0$, if there exists a uniform constant $C>0$ such that for every Borel set $E \\subset \\Omega$,\n\\[\\notag\n\t\\mu(E) \\leq C \\left[cap (E,\\Omega) \\right]^{1+ \\alpha}.\n\\]\nThe following result \\cite[Lemma~2]{ko94} will be used in what follows.\n\n\\begin{lem}\n\\label{lem:ko94}\n Suppose $0< 3r< R$ and\n$\n\tB(z,r) \\subset B(z, R) \\subset \\subset \\Omega.\n$\nLet $v\\in PSH(\\Omega)$ be such that $-1 \\leq v \\leq 0$. Denote \n\\[\\notag\n\tE(\\varepsilon, v, B(z,r)) := \\{z \\in B(z, r) : (1-\\varepsilon)v \\leq \\sup_{B(z,r)} v\\},\n\\]\nwhere $\\varepsilon \\in (0,1)$. Then, there exists $C_0$ depending only on $n$ such that\n\\[\\notag\n\tcap(E, B(z,2r)) \\leq \\frac{C_0}{\\varepsilon \\log (R\/r)}.\n\\]\n\\end{lem}\n\n\\begin{proof} See Appendix.\n\\end{proof}\n\n\nLet us proceed with the proof of Theorem~\\ref{thm:modulus}.\nSince $\\mu \\in \\mathcal{H}(\\alpha,\\Omega)$, according to \\cite{ko96} we can solve the Dirichlet problem~\\eqref{eq:dirichlet-prob} to obtain a unique continuous solution $u$.\nDefine for $\\delta>0$ small\n\\[\\notag\n\t\\Omega_\\delta := \\left\\{z \\in \\Omega : dist(z, \\d \\Omega) > \\delta\\right\\};\n\\]\nand for $z\\in \\Omega_{\\delta}$ we define\n\\[\\notag\n\tu_\\delta(z) := \\sup_{|\\zeta| \\leq \\delta} u(z+ \\zeta).\n\\]\nThanks to the arguments in \\cite[Lemma~2.11]{Ng17} it is easy to see that there exists $\\delta_0>0$ such that\n\\[\\label{eq:boundary-est-b}\n\tu_\\delta(z) \\leq u(z) + \\varpi(\\delta;\\psi,\\d\\Omega)\n\\]\nfor every $z \\in \\d\\Omega_\\delta$ and $0< \\delta<\\delta_0$.\nHere we used the result of Bedford and Taylor \\cite[Theorem~6.2]{BT76} (with minor modifications) to extend $\\psi$ plurisubharmonically onto $\\Omega$\nso that its modulus of continuity on $\\bar\\Omega$ is controlled by the one on the boundary. \nTherefore, for a suitable extension of $u_\\delta$ to $\\Omega$, using the stability estimate for measure in $\\mathcal{H}(\\alpha,\\Omega)$ as in \\cite[Theorem~1.1]{GKZ08} (see also \\cite[Proposition~2.10]{Ng17}) we get \n\n\\begin{lem}\n\\label{lem:stability}\n There are uniform constants $C, \\alpha_1$ depending only on $\\Omega, \\alpha, \\mu$ such that\n\\[\\notag\t\\sup_{\\Omega_\\delta} (u_\\delta - u) \\leq \\varpi(\\delta; \\psi, \\d\\Omega) + C \\left(\\int_{\\Omega_\\delta} (u_\\delta -u) d\\mu \\right)^{\\alpha_1}\n\\]\nfor every $0<\\delta<\\delta_0$.\n\\end{lem}\nThanks to this lemma we know that the right hand side tends to zero as $\\delta$ decreases to zero. We will use the property \"locally dominated by capacity\" to obtain a quantitative bound via Lemma~\\ref{lem:ko94}.\n\n\\begin{proof}[End of Proof of Theorem~\\ref{thm:modulus}]\nLet us denote the support of $\\mu$ by $K$. Since $\\|u\\|_\\infty$ is controlled by a contant $C = C(\\alpha,\\Omega, \\mu)$, without loss of generality we may assume that \n\\[\\notag\n\t-1 \\leq u \\leq 0.\n\\]\nThen for every $0<\\varepsilon<1$\n\\[\\label{eq:divide}\n\\begin{aligned}\n\t\\int_{\\Omega_\\delta} (u_\\delta -u) d\\mu \\leq \\varepsilon \\; \\mu(\\Omega) + \\int_{\\{u < u_\\delta - \\varepsilon\\} \\cap K} d\\mu\n\\end{aligned}\\]\nWe shall now estimate the second term on the right hand side.\n\nLet us fix the notation that will be used later on. \nWe may assume that $\\Omega \\subset\\subset [0,1]^{2n}$. Let us write $z = (x^1, ...,x^{2n}) \\in \\mathbb{R}^{2n}$ and denote the semi open cube centered at a point $z_0$ of diameter $2r$ by\n\\[\\notag\n\tI(z_0,r):= \\{z = (x^1, ..., x^{2n})\\in \\mathbb{C}^n : -r \\leq x^i - x_0^i< r \\; \\forall i = 1,...,2n\\}.\n\\]\nThen, by the assumption $\\mu$ satisfies for every cube $$I(z,r)=:I \\subset B_I := B(z,2r) \\subset\\subset \\Omega$$ and for every set $E \\subset I$, \n\\[ \\label{eq:local-dominate}\n\t\\mu (E) \\leq \\mu (I(z,r)) F_0\\left( cap (E, B_I) \\right),\n\\]\nwhere $F_0: [0,\\infty] \\to [0,\\infty]$ is an increasing continuous function and $F_0 (0) =0$.\n\n\nConsider the semi-open cube decomposition of $\\Omega \\subset\\subset I_0:=[0,1)^{2n} \\subset \\mathbb{R}^{2n}$ into $3^{2ns}$ congruent cubes of diameter $3^{-s} = 2\\delta$, where $ s \\in \\mathbb{N}$. \nThen \n\\[ \\label{eq:inclusion}\n\\{u < u_\\delta -\\varepsilon\\} \\cap I_s \\subset \\{z\\in B_{I_s} : u< \\sup_{B_{I_s}} u -\\varepsilon\\},\n\\]\nwhere $I_s = I(z_s, \\delta)$ and $B_{I_s} = B(z_s, 2\\delta)$ for some $z_s\\in I_0$.\nHence\n\\[\\notag\n\t\\int_{\\{u < u_\\delta - \\varepsilon\\}} d\\mu \\leq \\sum_{I_s \\cap K \\neq \\emptyset }\\int_{\\{u< u_\\delta -\\varepsilon \\} \\cap I_s} d\\mu.\n\\]\nUsing \\eqref{eq:local-dominate}, \\eqref{eq:inclusion}, and then applying Lemma~\\ref{lem:ko94} for $r=2\\delta$ and $R= 2R_0$, we have for $B_s : = B(z_s, 4\\delta)$ corresponding to each cube $I_s$:\n\\[\\label{eq:key-bound-u}\n\\begin{aligned}\n\t\\int_{\\{u< u_\\delta - \\varepsilon\\} \\cap I_s} d\\mu \n&\\leq \t \\mu(I_s) F_0(cap(E(\\varepsilon, u, B_{I_s}), B_s)) \\\\\n&\\leq\t\t \\mu(I_s) \\; F_0 \\left(\\frac{C_0}{\\varepsilon \\log (R_0\/\\delta)}\\right),\t\n\\end{aligned}\\]\nwhere $2R_0 = \\mbox{dist} (K, \\d\\Omega)$.\nTherefore, combining the above inequalities, we get that\n\\[\\notag\n\t\\int_{\\{u < u_\\delta - \\varepsilon\\}} d\\mu \\leq \\mu(\\Omega) F_0 \\left(\\frac{C_0}{\\varepsilon \\log (R_0\/\\delta)}\\right).\n\\]\nWe conclude from this and Lemma~\\ref{lem:stability} that\n\\[\\notag\n\t\\omega(\\delta;u, \\bar\\Omega) \\leq \\sup_{\\Omega_\\delta} (u_\\delta - u) \\leq \\varpi(\\delta;\\psi,\\d\\Omega) + C \\left[\\varepsilon + F_0 \\left(\\frac{C_0}{\\varepsilon \\log (R_0\/\\delta)}\\right)\\right]^{\\alpha_1}.\n\\]\nIf we choose $\\varepsilon = (\\log R_0\/\\delta)^{-1\/2}$ then Theorem~\\ref{thm:modulus} follows. \n\\end{proof}\n\n\n\n\\section{Appendix}\n\nFor the reader's convenience we give the details of the proof of Lemma~\\ref{lem:ko94}.\nThe following inequality is due to Alexander and Taylor \\cite[Lemma~3.3]{AT84}.\n\n\\begin{lem} \n\\label{lem:AT}\nLet $B' = \\{|z-z_0| 1$, $R_0>0$ such that\n$\n\tB_M \\subset \\subset\\Omega,\n$\nand given $v\\in PSH(\\Omega)$ such that $-1 \\delta t^{-1} > E_g^{(0) }$, where $\\Lambda$ is the ultra-violet (UV) cut-off and $E_g^{(0)}$ is the mass gap of the initial pre-quench state. In this case universal scaling emerges from the fixed point at the UV. This scaling has been observed in free theories, as well as using holography. See the dissertation \\cite{damianthesis} and references therein. The slow regime is when $\\delta t^{-1} < E_g^{(0)} < \\Lambda $. In this case too one may break adiabaticty. Obviously, there is no escape if one crosses a critical point during the quench, but adiabaticity also breaks if one comes \\textit{close enough} to one. The criteria for adiabatic breakdown can be obtained by calculating the time dependent gap using \\textit{adiabatic perturbation theory}. This is a calculation in a derivative expansion, where one assumes, $g > \\dot{g} > \\ddot {g} \\dots$. It results in $E_g(t) = E_g^{(0)}(t) + E_g^{(1)}(t) + \\dots$, where $E_g^{(1)}$ is the leading correction to the gap and further corrections are denoted by the $\\dots$. This expansion breaks down when $E_g^{(1)} \\sim E_g^{(0)}$, which happens at a particular time, dubbed as the Kibble-Zurek (KZ) time, $t_{KZ}$. $t_{KZ}$ is the time when the system fails to respond to the quench, and all correlators freeze at this scale. The KZ time is generically a function of the rate, $t_{KZ} = t_{KZ}(\\delta t)$. Operator expectations are dictated by this scale : $\\vev{\\CO_\\Delta}\\sim t_{KZ}^{-\\Delta}$ \\cite{Kibble:1976sj, Zurek:1985qw}. For a nice review including experimental results, see \\cite{Polkovnikov:2010yn}. Both the \\textit{fast} as well as the KZ scalings have extensively been investigated in explicitly solvable models either in the continuum or on the lattice. See \\cite{Das:2016eao, Das:2014hqa, Das:2014jna, Das:2015jka, Das:2016lla} for studies in the continuum and \\cite{Mondal2010, Divakaran2010, Das:2017sgp} for some lattice examples. In all of the field theoretic examples exhibiting the KZ scalings, investigations have been limited to Hamiltonia which can be quadraticized, else in scenarios involving one ( or more ) critical point(s), and thereby in certain cases using the CFT technology. It is a much wider and unknown area to explore quenches in interacting quantum field theories away from criticality. Recently, in \\cite{Goykhman:2018iaz} the authors considered a $\\phi^4$ theory in $4-\\epsilon$ dimensions and addressed the quench of the quartic coupling constant. Perturbatively (in the quartic coupling) they were able to show how renormalization group effects via time-dependent counterterms, induce a quench in the running mass. In the quartic fermionic model we also find a induced time dependent gap resulting from the quartic coupling quench. Our calculation however is non-perturbative in the coupling, as we rely on the largeness of the number ($N$) of fermionic flavours, which allows us to focus on a special class of diagrams that can be summed. \\\\\nLarge $N$ methods have previously been used to study quenches in interacting theories \\cite{Boyanovsky:1997cr, Boyanovsky:1996rw, Sotiriadis:2010si, Das2012, Gemsheim:2019qed}. As a result of the quench the system often exhibits an effective thermal behaviour, however there has not been any explicit KZ scaling analysis in this setting. In our work we present such a scenario where a KZ scale emerges. We find that the scaling is \\textit{tied} to restoration of a dynamically broken symmetry of the system. It is also known that this broken symmetry gets restored at finite temperatures \\cite{Jacobs:1974ys, Harrington:1974tf}. We show that the restoration during the quench can be understood as an \\textit{effective thermalization} at a temperature which takes the system to the symmetric, disordered phase. While one is guaranteed to break adiabaticity while crossing a critical point, the adiabatic expansion can still breakdown \\textit{close} to the critical point. In our set-up too we shall stay close to the $g=0$, UV fixed point, close enough in order to get the emergent KZ scaling.\\\\\n\\textbf{Organization :} In \\S\\ref{sec:GN} we describe the model and briefly state its equilibrium properties. Then we set up the problem in the Schwinger-Keldysh contour which allows us to conveniently study the quench in large $N$. In \\S\\ref{sec:dynsad} we focus on the dynamical gap equation using derivative expansion. In \\S\\ref{sec:num} we solve it numerically and state the results which show that quenches lead towards restoration of a discrete symmetry. In \\S\\ref{efftherm} we argue how the broken-restoration transition during quench can be interpreted as an effective thermalization. Next we investigate KZ in \\S\\ref{sec:KZ} and analytically argue for the scaling of the symmetry restoration time. In the subsection \\S\\ref{sec:liou} we show that the order parameter dynamics has a natural double scaling limit wherein the symmetry restored configuration can be understood via Liouville quantum mechanics. We outline some future directions and end with some discussions in \\S\\ref{sec:conc}. Appendix \\S\\ref{app:eps} contains detail of the derivative expansion. Numerics relevant for \\S\\ref{sec:KZ} is relegated to appendix \\S\\ref{app:tanhnum}. A full-fledged analysis of a similar model is carried out in \\S\\ref{app:njl} which also exhibits restoration of symmetry, albeit a continuous one.\n\n\n\\section{Setting-up the quench in the Gross-Neveu model}\\label{sec:GN}\nThe focus of our study is the theory of two dimensional massless fermions with quartic interactions, introduced by Gross and Neveu \\cite{PhysRevD.10.3235}. The Gross-Neveu model is a renormalizable field theory admitting a $1\/N$ expansion (where $N$ is the number of fermion flavours) and displays asymptotic freedom and dynamical symmetry breakdown. The discrete chiral symmetry $\\mathbb{Z}_2$ of the model is dynamically broken via the generation of a $\\langle\\bar{\\psi} \\psi \\rangle$ mass term, which is non-perturbative in the quartic coupling. It is also known that the symmetry gets restored at finite temperature \\cite{Jacobs:1974ys, Harrington:1974tf} and also in presence of finite curvature \\cite{buchkiri89}. We implement the smooth quench by promoting the constant quartic coupling $g$ to $g(t\/\\delta t)$. A global quench can be thought of as turning on a time-dependent metric. Furthermore, it is also expected that a strongly interacting system will eventually thermalize. Therefore it may be expected that under a quench the symmetry may get restored eventually. \nAt equilibrium, the Gross-Neveu (GN) action is given by,\n\\begin{equation}\nS =\\int d^2 x \\{ \\bar{\\psi}_{i} i \\slashed{\\partial} \\psi_{i} + \\frac{1}{2N}g^2 (\\bar{\\psi}_{i} \\psi_{i})^2\\} \\label{eq:a}\n\\end{equation}\nwhere coupling $ g^2 $ is kept fixed as $ N \\rightarrow \\infty $. The large number $N$ here, is the number of fermionic species, $i \\in \\{1, N\\}$. In two dimensions the bare fermion mass dimension is $\\tfrac{1}{2}$, hence the coupling is dimensionless and consequently gets only logarithmic corrections $\\implies$ the theory is renormalizable. $\\gamma^0$ and $\\gamma^1$ are $2\\times 2$ gamma matrices in 2 dimensions, which we take to be in the Weyl basis (\\S \\ref{free}). A mass term is a priori excluded by the discrete chiral symmetry,\n$$\n\\psi_i \\rightarrow \\gamma^5 \\psi_i, \\,\\, \\bar{\\psi}_i \\rightarrow - \\bar{\\psi}_i \\gamma^5. $$ It is this symmetry that suffers from a dynamic breakdown. To solve the theory in large $N$ one introduces an auxiliary scalar field $\\sigma$, \n\\begin{equation}\nS = \\int d^2 x \\{ \\bar{\\psi} i \\slashed{\\partial} \\psi -\\frac{N}{2 g^2} \\sigma^2 + \\sigma\n\\bar{\\psi} \\psi \\} .\\label{eq:b}\n\\end{equation}\nIntegrating over $\\sigma$ gives back eq\\eqref{eq:a}. The discrete chiral symmetry acts as a simple $\\mathbb{Z}_2$, $\\sigma \\rightarrow - \\sigma$. In equilibrium one can regard $\\sigma(x,t) = \\sigma$ to be a spacetime constant. Integrating out the fermions and then evaluating the saddle in the $\\sigma$ functional integral yields, \\begin{equation}\\label{equilibrium} \\sigma = 2\\Lambda \\frac{e^{\\pi\/g^2}}{e^{2\\pi\/g^2}-1} \\sim 2\\Lambda e^{- \\pi\/g^2 }, \\end{equation} where $\\Lambda$ is the cut-off scale. Note that the last $\\sim$ approximation holds for $g\\rightarrow 0$ only. In our analysis we do not make any small $g$ approximation. This non-zero value of $\\sigma = \\tfrac{g^2}{N} \\bar{\\psi}\\psi$ signals the breakdown of the $\\mathbb{Z}_2$ symmetry, and results in a mass for the fermions. For the time-dependent $g$, all amplitudes now need to be calculated using the Schwinger-Keldysh contour. Also as we shall see, that we can no longer choose $\\sigma(x,t)$ as a spacetime constant. The partition function evaluated using the Schwinger-Keldysh contour is given by, \n\\begin{eqnarray}\n\\mathcal{Z} = \\int \\mathcal{D} \\bar{\\psi}_{\\pm} \\mathcal{D}\\psi_{\\pm} \\mathcal{D}\\sigma_{\\pm} \\exp \\left[ i \\{S(\\bar{\\psi}_{+},\\psi_{+}, \\sigma_{+}) - S(\\bar{\\psi}_{-},\\psi_{-}, \\sigma_{-}) \\} \\right] \\label{eq:e}\n\\end{eqnarray}\nwhere,\n\\begin{align}\nS(\\bar{\\psi}_{+},\\psi_{+}, \\sigma_{+}) - S(\\bar{\\psi}_{-},\\psi_{-}, \\sigma_{-}) &= \\int d^2 x [\\bar{\\psi}_{+} ( i \\slashed{\\partial} + \\sigma_{+} ) \\psi_{+} - \\bar{\\psi}_{-} ( i \\slashed{\\partial} + \\sigma_{-} ) \\psi_{-} \\nonumber \\\\&- \\frac{N}{2 g^2(t)} (\\sigma^2_{+} - \\sigma^2_{-}) ] \\label{eq:f}.\n\\end{align}\nNext, we integrate out $ \\bar{\\psi}_{\\pm}, \\psi_{\\pm}$ to obtain the effective action,\n\\begin{align}\n&S_{\\text{eff}}(\\sigma) = -N \\, {{\\rm Tr}} \\log D - \\frac{N}{2g^2(t)} \\int d^2 x (\\sigma_{+}^2 - \\sigma_{-}^2) \\label{eq:g}, \\nonumber \\\\&\\text{ with}\n\\,\\,\\, D= \\left( \\begin{matrix}\ni\\slashed{\\partial} + \\sigma_{+} && 0 \\\\\n0 && - i\\slashed{\\partial} - \\sigma_{-}\n\\end{matrix} \\right).\n\\end{align}\nAt large-$N$, $S_{\\text{eff}}(\\sigma)$ is dominated by the saddle point configuration which is given by, \n \\begin{equation}\\label{saddle}\n\\frac{\\sigma_{\\pm}}{g^2(t)} = - {{\\rm Tr}}\\left[\\frac{1}{ i \\slashed{\\partial} + \\sigma_{\\pm}} \\right] = - \\int_{-\\infty}^\\infty dk\\, \\langle \\bar{\\chi}_k(t) \\chi\n_k(t) \\rangle. \n\\end{equation}\nNote, that in general due to the boundary condition at the turning point of the Schwinger-Keldysh contour, there are non-trivial correlations between the $+$ and the $-$ fields, however the classical saddle remains unaffected \\cite{Kamenev_2009}. The first equality in eq\\eqref{saddle} shows that $\\sigma$ can no longer be constant in time. Therefore in the second equality, $ \\chi$ is a free massive fermion in two spacetime dimensions with a time dependent mass $m(t)= -\\sigma(t)$, whose correlator we seek. The fermionic field $\\chi$ is used as an auxiliary field employed to calculate the trace. This is the problem we turn to next.\n\\subsection{Free fermion with time-dependent mass}\\label{free}\nThe relevant Dirac equation that we solve for is,\n\\begin{equation}\n\\left( i \\gamma^0 \\partial_0 + i \\gamma^1 \\partial_1 - m(t) \\right) \\chi(x,t) = 0. \n\\end{equation}\nWe follow the conventions used in \\cite{PhysRevD.97.125012}, and work in the Weyl-basis where the $\\gamma$ matrices take the form,\n\\begin{align} \\label{weylbasis}\n\\gamma^0 &= \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix}, \\,\\,\\,\\,\\,\\gamma^1 = \\begin{pmatrix} 0 & 1 \\\\ -1 & 0 \\end{pmatrix}. \n\\end{align}\nNext, we decompose the Dirac field into momenta modes,\n$\\chi(x,t) = \\sum_k e^{i k x} \\chi_k (t) . $\nThe equation satisfied by $\\chi_k(t)$ is, \n$\n\\left( i \\gamma^0 \\partial_0 -k \\gamma^1 - m(t) \\right) \\chi_k(t) = 0. \n$\nWe choose the spinor basis of solutions as,\n\\begin{align}\nu_k(t) &= \\frac{ e^{ikx} }{\\sqrt{2\\pi } } \\begin{pmatrix} h_k^I(t) \\\\ -h_k^{II}(t) \\end{pmatrix} , \\,\\,\\,\nv_k(t) = \\frac{ e^{-ikx} }{\\sqrt{2\\pi } } \\begin{pmatrix} h_{-k}^{II*}(t) \\\\ h_{-k}^{I*}(t) \\end{pmatrix}.\n\\end{align}\nThe equation of motion in momentum space translates into the following coupled equations of motion, \n\\begin{align} \\label{h1h2}\n\\dot{h}_k^I - i k h_k^I - i m h_k^{II} &= 0, \\,\\,\\, \n\\dot{h}_k^{II} + i k h_k^{II} - i m h_k^{I} = 0,\n\\end{align}\nalong with the standard normalization condition, \n$\n| h_{\\pm k}^I |^2 + | h_{\\pm k}^{II} |^2 = 1. \n$\nNow we can write down the second-quantized mode expansion for the Dirac field as,\n$\n\\chi(x,t ) = \\int\\, dk \\,\\, [ B_k^{ \\vphantom{\\dagger}} u_k(t,x) + D^\\dagger_k v_k(t,x) ],\n$\nwhere, \n $\\{ B_k^{ \\vphantom{\\dagger}} , B_{k'}^\\dagger\\} = \\delta(k-k')$ and $\\{ D_k^{ \\vphantom{\\dagger}} , D_{k'}^\\dagger\\} = \\delta(k-k')$. \nWith these relations one can check that, \n$ \\{ \\chi^{ \\vphantom{\\dagger}}(x,t) , \\chi^\\dagger(y,t) \\}$ $= \\delta(x-y) \\mathbb{I}_{2\\times 2}. \n$\nA straightforward algebra gives, \n$$\n\\int dk\\, \\langle \\bar{\\chi}_k(t) \\chi_k(t) \\rangle = \\int_{-\\infty}^\\infty \\frac{dk}{2\\pi} \\left( h_{-k}^I h_{-k}^{II*} + h_{-k}^{II} h_{-k}^{I*} \\right\n).$$ To proceed we propose the following \\textit{ansatz}, \n\\begin{eqnarray}\nh_k^I &=& \\sqrt{ \\frac{\\omega_k(t) - k }{2\\omega_k(t) }} e^{-i \\int^t \\omega_k(t') dt' },\\,\\,\\,\n h_k^{II} = -\\sqrt{ \\frac{\\omega_k(t) + k }{2\\omega_k(t) } }e^{-i \\int^t \\omega_k(t') dt' }. \n \\label{ansatz}\n \\end{eqnarray}\nThis is inspired from equations \\eqref{h1h2}\\footnote{ In equilibrium, the ansatz automatically satisfies, $h_k^{II} = \\tfrac{1}{im} \\dot{h}_k^I - \\tfrac{k}{m} h_k^I$, and maintains $\n| h_{\\pm k}^I |^2 + | h_{\\pm k}^{II} |^2 = 1. \n$ for all times.}. Note, that this is quite different from the ansatz used in the bosonic cases \\cite{Gemsheim:2019qed, Das2012}, wherein there is only a single mode to solve for, and the prefactor is also considerably simpler. \n\n\\section{Dynamic saddle : symmetry restoration}\\label{sec:dynsad}\nWith the ansatz \\eqref{ansatz}, the gap equation \\eqref{saddle} simplifies to, \n\\begin{equation}\n\\frac{\\sigma(t) }{g^2(t)} = \\int_0^\\infty \\frac{dk}{\\pi } \\mathcal{A}(k,t),\\,\\,\\text{where,}\\, \\mathcal{A}(k,t)= \\frac{ \\text{Re} \\bigg[ \\sqrt{\\omega_k(t) - k } \\bigg( \\sqrt{ \\omega_k(t) + k }\\bigg)^* \\bigg]}{ | \\omega_k(t) | }. \\label{saddle2}\n\\end{equation}\nNote, that in the static equilibrium case, the above reproduces the correct gap equation, \n\\begin{equation}\n\\frac{1}{g^2} = \\int_0^\\infty \\frac{dk}{\\pi } \\frac{1}{\\sqrt{ k^2 + \\sigma^2 } },\n\\end{equation}\nwhich has the equilibrium solution, given by \n\\begin{equation} \\label{equi}\n\\sigma = m_0= 2 \\Lambda \\frac{ e^{\\pi \/ g^2}}{ e^{2\\pi \/ g^2 } -1 }.\n\\end{equation}\n One can now use the derivative expansion to solve the for $\\omega_k(t)$. The differential equation satisfied by $\\omega_k(t)$ can be obtained from eq\\eqref{h1h2} by plugging in the ansatz eq\\eqref{ansatz} (eq\\eqref{ode1}). See \\S Appendix \\ref{app:eps} for further details. The integrand in eq\\eqref{saddle2} takes the form given by equation \\eqref{Akt}. Integrating this over the momentum (with cut-off $= \\Lambda$) we obtain a second order differential equation for the saddle $\\sigma(t)$,\n\\begin{align}\\label{ode2}\n&\\frac{24 \\pi \\sigma^4 }{g^2}= 6 ( 4\\sigma^4 - \\dot\\sigma^2 +\\sigma \\ddot\\sigma ) \\log \\bigg( \\Lambda + \\sqrt{\\Lambda^2 + \\sigma^2 } \\bigg) + \\frac{3 \\sigma^2 ( \\dot\\sigma^2 + \\sigma \\ddot\\sigma )}{\\Lambda^2 + \\sigma^2 } \\nonumber \\\\ &+ \\frac{2\\Lambda \\sigma^2 ( 2\\dot\\sigma^2 + \\sigma \\ddot\\sigma )}{( \\Lambda^2 + \\sigma^2 )^{3\/2} }+ \\Lambda \\frac{5\\dot \\sigma^2 - 8 \\sigma \\ddot \\sigma }{\\sqrt{\\Lambda^2 + \\sigma^2 } } - 3( \\dot \\sigma^2 - \\sigma \\ddot \\sigma ) \\log \\bigg( \\Lambda^2 + \\sigma^2 \\bigg) - 3\\sigma^4 \\dot\\sigma^2 \\nonumber \\\\ &\\bigg( \\frac{\\Lambda}{(\\Lambda^2 +\\sigma^2 )^{5\/2} } + \\frac{1}{(\\Lambda^2 +\\sigma^2 )^2 } \\bigg)\n- 3 \\sigma \\ddot\\sigma (1 + 4\\log \\sigma ) + 12 \\dot\\sigma^2 \\log \\sigma - 24 \\sigma^4 \\log \\sigma. \n \\end{align}\nWe solve this equation numerically (without any approximation) with initial equilibrium conditions that depend upon the quench protocol. Fig.\\ref{b5der} justifies the derivative expansion, in that the solution $\\sigma(t)$ is always larger than its time derivatives. \n\n\\subsection{Numerical results}\\label{sec:num}\n\nWe use $\\texttt{Mathematica}^{\\tiny{\\textregistered}}$ to numerically integrate the differential equation \\eqref{ode2} for the smooth standard quench profile asymptoting between two constant values, the $\\tanh t\/\\delta t$ function. Working within the regime of validity of the approximation, we find that the order parameter quickly settles to zero, signalling the approach towards \\textit{restoration} of the dynamically broken symmetry. We choose equilibrium initial conditions at early times, {\\it i.e.,}\\ equation \\eqref{equi} along with $\\dot\\sigma(-\\infty) = 0$. \n\n\\subsubsection*{Tanh quench}\nMore explicitly we choose, \n\\begin{equation}\\label{tanhprof}g(t) = \\frac{ g_i + g_f}{2} + \\frac{g_f - g_i}{2} \\tanh \\frac{t}{\\delta t}. \\end{equation}\nThe plot in Fig. \\ref{tanh1} shows how the order parameter quickly settles to vanishing value. \n\\begin{figure}[h!]\n \\centering\n \\includegraphics[scale=0.52]{tanh1.pdf} \n \\caption{ Plot of $\\sigma(t)$ as a function of time. We have chosen cut-off, $\\Lambda = 3$ with $g_i = 1.10, g_f = 1.1 + 10^{-4.5}, \\delta t = 10$, the red inset curve shows $g(t)$ vs. $t$. In the zoomed inset, the orange line corresponds to the equilibrium $\\sigma$ for $g=g_i$ and the green corresponds to $\\sigma$ for $g=g_f$. }\n \\label{tanh1}\n\\end{figure}\n\n\n\\FloatBarrier\n\n\\subsubsection{Breaking-Restoring transition}\\label{brt}\nHere we numerically investigate for the \\textit{tanh} quench protocol, the transition from the broken phase to the restored phase as we vary the quench amplitude for a fixed $\\delta t$. We notice from Fig.(\\ref{tanhsmall}) that if $g_i$ and $g_f$ are very close to each other, then the quench does not restore the broken symmetry. In fact, there is a transition at a particular quench amplitude. In Fig.(\\ref{tanhsmallbig}) we show the evidence for the transition. In this figure, the green and the red quenches do not restore the broken symmetry, while the orange and blue quenches do\\footnote{One can once again check the self-consistency of the numerical solutions by comparing with the time derivatives as carried out for Fig.(\\ref{b5der}).}. The green and red corresponds to the case, $g_f-g_i < 10^{-7.18}$ while the orange and the blue curves corresponds to $g_f-g_i > 10^{-7.18}$. The critical amplitude depends on the rate $\\delta t$. A slower quench requires larger quench amplitude to restore the broken symmetry. It is the amplitude of the quench at a fixed rate which causes the transition. One could have equivalently chosen $g_i > g_f$ and also discussed this transition.\\\\\nIn the next section we have shown that this dynamic transition for a fixed rate as a function of the quench amplitude can be phrased in terms of thermalization. Later we have also focussed on the dependence of the transition time on the rate of the quench at a fixed amplitude. It is this latter case that can be associated to a Kibble-Zurek scaling. \n\\begin{figure}[!htbp]\n \\centering\n \\subfigure[Small amplitude Tanh quench]{\n \\includegraphics[scale=0.4]{tanhsmall.pdf} \\label{tanhsmall}}\n \\subfigure[Transition from broken to restoration]{\n \\includegraphics[scale=0.47]{tanhsmallbig.pdf} \\label{tanhsmallbig}}\n \\caption{(a) Plot of $\\sigma(t)$ as a function of time. We have chosen, $g_i = 1.1, g_f = 1.1+10^{-7.184}, \\delta t = 10, \\Lambda = 3$, the red inset curve shows $g(t)$ vs. $t$. The orange line corresponds to the equilibrium $\\sigma$ for $g=g_i$ and the green corresponds to $\\sigma$ for $g=g_f$. In (b) we plot the various $\\sigma$ profiles for different quench amplitudes. }\n\\end{figure}\n\n\\FloatBarrier\n\n\\subsubsection{Effective thermalization}\\label{efftherm} In this subsection we show how the symmetry broken-restoration transition during the quench can be understood as an effective thermalization. It is useful firstly to define an \\textit{instantaneous gap}, $\\text{m}(t) = 2\\Lambda \\tfrac{e^{\\pi \/g^2(t)}}{e^{2 \\pi \/g^2(t)} -1 }$ which becomes the gap in equilibrium, $m_0$, eq\\eqref{equi}. Unlike the non-linear bosonic O(N)${}_3$ model where the thermal phase transition can be analytically investigated \\cite{sachdev_2011}, the GN${}_2$ thermal transition can only be fully understood numerically \\cite{Jacobs:1974ys, Harrington:1974tf}. In equilibrium as the temperature is increased the $\\sigma =0$ point becomes the true minima instead of $m_0$ and the chiral symmetry gets restored. The critical inverse temperature is found to be, $\\beta_{\\text{crit}} = \\frac{\\pi e^{-\\gamma} }{m_0}$. In the quench context we therefore naturally define a critical time-dependent $\\beta_{\\text{crit}}(t) = \\frac{\\pi e^{-\\gamma} }{m(t)}$. In equilibrium, one can also define a temperature-dependent fermionic mass by extremizing the finite-temperature effective potential\\cite{Jacobs:1974ys}. This is given by, \n\\begin{equation}\\label{temp}\n 2 f\\left( \\beta^2 m_\\beta^2 \\right) = \\gamma + \\log \\frac{m_0 \\beta}{\\pi}. \n\\end{equation}\nIn the above equation $$f(a) = \\displaystyle \\sum_{n=0}^\\infty \\frac{1}{2n+1}\\left\\{ 1 - \\left( 1 + \\frac{a}{(2n+1)^2\\pi^2}\\right)^{-1\/2} \\right\\},$$ and $\\gamma= 0.5772156649$ is the Euler-Mascheroni constant. Unfortunately, eq\\eqref{temp} cannot be solved to obtain, $m_\\beta$ as a function of $\\beta$ or vice-versa in a closed form. However from the time-dependent analog of eq\\eqref{temp} we can numerically extract an \\textit{effective} $\\beta(t)$. The equation we solve is, \n\\begin{equation}\\label{temp2} 2 f\\left(\\beta^2(t) \\sigma^2(t)\\right) = \\gamma + \\log \\tfrac{m(t) \\beta(t) }{\\pi}. \\end{equation}\nWe plot the results in Fig.\\ref{tanh-beta}. We choose parameters and colours as in \\S\\ref{brt}. We notice that both the symmetry restoring quenches (orange, blue points) cross $\\beta_{\\text{crit}}(t)$ (red curve) while the symmetry broken quenches (red and the green points) do not!\n\n\\begin{figure}[!htbp]\n \\centering\n \\includegraphics[scale=0.45]{tanh-beta.pdf} \n \\caption{ Plot of effective temperature as a function of time for various $tanh$ quench profiles. We have chosen, $\\Lambda =3$, $g_i = 1.1$, $\\delta t = 10$. The blue, orange, red and green corresponds to choosing $g_f = \\{1.1+10^{-7.16},1.1+10^{-7.172},1.1+10^{-7.184},1.1+10^{-7.188} \\}$, respectively. The insets are zoomed in views of different parts. }\n \\label{tanh-beta}\n\\end{figure}\n\n\\FloatBarrier\n\n\n\\section{Emergence of Kibble-Zurek scaling} \\label{sec:KZ}\nIn this section, we show that the time at which the order parameter goes to zero, can be identified with the Kibble-Zurek (KZ) time scale in the problem. The Kibble-Zurek time is the moment in time when the adiabatic corrections to the time-dependent gap\\footnote{which depends non-trivially on the order parameter and its derivatives.}, become of the same order as the leading answer. This emergent time-scale dictates all correlations in the system, which freezes at the KZ scale. As in \\S\\ref{efftherm}, it is useful to think of the quench as a change in the dimensionful parameter, $m(t)$, which is identified with the mass gap at equilibrium. We let $m(t) = \\frac{ m_i + m_f}{2} + \\frac{m_f - m_i}{2} \\tanh \\frac{t}{\\delta t}$. This is achieved practically by making $g(t)$ a function of time. Next, we plot the dynamic order parameter $\\sigma(t)$ during the quench for different $\\delta t$ values while keeping the amplitude of the quench fixed. The results are consistent with the expectation that for slower quenches the order parameter takes longer time to vanish. See appendix \\S\\ref{app:tanhnum} for further details. We have taken zero to be some fixed small number ($\\epsilon$) and find the set of times when $\\sigma = \\epsilon$ for the different $\\delta t$'s. We make sure to be in the regime, $\\Lambda > m_f, \\, m_i > \\delta t^{-1}$, where one expects KZ. However as emphasised in the introduction, we do not cross criticality ({\\it i.e.,}\\ $\\,$vanishing $m(t)$). Now, we find analytically the scaling of $t_{KZ}$ with $\\delta t$ and show that the zero times extracted from the numerics exhibit the same KZ scaling. \\\\\nThe analytical analysis is simpler in the $\\Lambda\/\\sigma \\gg 1$ limit of \\eqref{ode2} which simplifies now to, \n\\begin{equation}\\label{limit}\n\\frac{\\pi}{g^2} = \\log \\frac{\\Lambda}{\\sigma} - \\frac{ \\dot\\sigma^2}{2\\sigma^4} \\log \\frac{\\Lambda}{\\sigma} + \\frac{\\ddot \\sigma}{2\\sigma^3} \\log \\frac{\\Lambda}{\\sigma}.\n\\end{equation}\nIn the derivative expansion this equation can now be inverted to give (till quadratic order),\n\\begin{equation}\n\\sigma = m \\left( 1 + \\frac{\\pi}{2 m^2 g^2 }\\left( \\frac{\\ddot{m}}{m} - \\frac{\\dot{m}^2 }{ m^2 } \\right) + \\dots \\right).\n\\end{equation}\nThe KZ scale in the problem, arises from the condition of adiabatic breakdown near criticality when the equilibrium mass $m(t) \\rightarrow 0$. Mathematically, this occurs when the correction term is of the leading order. Near this region for slow rates, assuming $m(t) \\sim t\/\\delta t$, we find the adiabaticity breakdown condition to be, \n$ \\log t_{KZ}\/\\delta t \\simeq t_{KZ}^4\/\\delta t^2$, this can be solved for $t_{KZ}$ in terms of the Lambert function to give the scaling: \n\\begin{equation}\\label{scaling}\nt_{KZ} \\sim \\sqrt{\\delta t} \\,\\, W(4\\delta t^2)^{1\/4}. \n\\end{equation}\nThe units are made out of the masses (which in turn depend on $\\Lambda$) to make the above equation dimensionally consistent. Next, we find that the same scaling as in eq \\eqref{scaling} is exhibited for the restoration time, see Fig. \\ref{tkz}. \n \\begin{figure}[!htbp]\n\\centering\n\\includegraphics[scale=0.5]{tkz} \n\\caption{ A fit of the zero times extracted for various ranges as in Fig.\\ref{tzero} with the scaling function given by eq\\eqref{scaling}. We use parameters as listed in eq\\eqref{params}.}\\label{tkz}\n \\end{figure}\n \n\\subsection{Saddle dynamics as Liouville quantum mechanics}\\label{sec:liou} \nWe start with the order parameter equation in the limit, $\\Lambda\/\\sigma \\rightarrow \\infty$ {\\it i.e.,}\\ , eq\\eqref{limit}.\nUnder the change of variables to $\\sigma = \\Lambda e^{-y}$\n, the first derivative cancels, and the equation becomes\n\\begin{equation}\n\\ddot y + 2 \\Lambda^2 e^{-2y} \\frac{\\pi - y g^2 }{ y g^2} = 0.\n\\end{equation}\nNow if we look at the limit, $y\\rightarrow \\infty$ (keeping $\\sigma = \\Lambda e^{-y}$ fixed, such that $g^2 y \\gg \\pi$), quite surprizingly the above equation becomes independent of $g^2$! $\\ddot{y} - 2\\Lambda^2 e^{-2y} =0.$ Redefining, $\\phi = -2y $ and analytically continuing, $\\tau = i t$, the resulting equation of motion follows from the Euclidean action, \\begin{equation}\\label{liouac} S = \\int d\\tau\\, ( \\frac{\\dot{\\phi}^2}{8} + \\Lambda^2 e^{2 \\phi } ). \\end{equation} This is the action of Liouville quantum mechanics, which results under the double scaling limit, $\\Lambda \\rightarrow \\infty, \\phi \\rightarrow -\\infty$, with $\\sigma = \\Lambda e^{ \\phi\/2}$ fixed. \n Liouville quantum mechanics captures the dynamics of the zero mode of the Liouville field theory which describes 2D induced gravity in the conformal gauge \\cite{Ginsparg:1993is}. Note that the Liouville field $\\phi$, being real, we focus only on positive $\\sigma$. Using the conventions of \\cite{Bagrets_2016}, we look at the wavefunction $\\Psi[\\phi(\\sigma)] = \\vev{\\phi(\\sigma) | k }$, which satisfies the Schr\\\"odinger equation, derived from the Hamiltonian corresponding to the action \\eqref{liouac}:\n \\begin{equation}\n\\label{schrodinger}\n-2 \\partial_\\phi^2 \\Psi_k + \\Lambda^2 e^{\\phi} \\Psi_k = 2k^2 \\Psi_k. \n{\\cal E}\nThis has a continuous spectra labelled by $k>0$. The wavefunction which is normalized such that at $\\phi \\rightarrow -\\infty$, the state : $\\vev{k|k'} = 2\\pi \\delta(k-k')$ is given by $\\Psi_k = \\tfrac{2}{\\Gamma(2 i k)} K_{2ik}\\left(\\sqrt{2}\\sigma \\right)$.\\footnote{ We also demand, $\\Psi[\\sigma] \\rightarrow 0$, for $\\sigma \\gg \\Lambda$. } Close to $k\\rightarrow 0$, $|\\Psi_k|^2 = 16 K_0(\\sqrt{2}\\sigma)^2 k^2 + \\mathcal{O}(k^3)$. $K_0(x)$ is known to satisfy the inequality: $ \\frac{\\sqrt{\\pi} e^{-x}}{\\sqrt{2(x +a ) }} < K\n_0(x) <\\frac{\\sqrt{\\pi} e^{-x}}{\\sqrt{2 x }}$, for $x>0$ and $a\\geq 1\/4$,\\cite{besselk0}. This implies that the $\\sigma \\rightarrow 0$ configuration dominates at low energies. \nThis provides a likely explanation in the \\textit{double scaling limit} for the restoration of chiral symmetry. It is interesting to note that the Liouville quantum mechanics also describes the long time Schwarzian dynamics in the SYK model \\cite{Bagrets_2016}. \n\\vspace{.5cm}\n\n\n\\section{Discussion}\\label{sec:conc}\nIn this work we have implemented smooth global quench of the quartic coupling in the Gross-Neveu model\\footnote{and also in the Nambu and Jona-Lasinio model, (see appendix \\S\\ref{app:njl}) in 2 dimensions.} at zero temperature and in absence of any chemical potential. We find that as a result of the quench, the order parameter, which is the dynamical symmetry breaking fermionic mass, becomes time dependent. Our main finding is that the quench drives the order parameter asymptotically to zero, thus restoring the symmetry. Additionally, we have shown that the time of the symmetry restoration exhibits Kibble-Zurek scaling. Very interestingly, the quench dynamics echoes Liouvillian dynamics which raises further questions. In \\S\\ref{efftherm} we have also interpreted the dynamical symmetry broken-restoration transition as an \\textit{effective thermalization}. In smooth quenches in the non-linear $O(N)$ model \\cite{Gemsheim:2019qed} and in the SYK model\\cite{Eberlein:2017wah}, effective thermalization was also observed. The interplay of smooth quenches and thermalization was also studied holographically in \\cite{Bhaseen:2012gg}. We have not studied extensively the dependence of the temperature and the thermalization rate on the quench amplitude or the quench rate, which we leave for future work. \\\\ \nNow, it is known that the thermal GN model (in $2+1$ dimensions) can be mapped to the non-critical M-theory \\cite{Petkou:2005se, Horava:2005tt}. The non-critical M-theory contains the $c=1$ Matrix model in its solution space which in the double scaling limit is described by the Liouville conformal field theory \\cite{Ginsparg:1993is}. Since we start from a $1+1$ dimensional GN theory, it may not be surprizing that we end up with the Liouville quantum mechanics. It may be insightful to follow this series of connections and understand effective thermalization following a quench, from this perspective. Another situation where similar physics seems to play a role is in \\cite{PhysRevB.91.054306},\\footnote{We thank A. Polkovnikov for bringing this work to our attention.} wherein one once again solves driven self-consistency equations similar to eq\\eqref{saddle2}. Interestingly, the authors find an effective \\textit{heating} near the critical point and a KZ scale emerge. Similar ideas of symmetry restoration upon driving also seem to be deep-seated in \\cite{Kofman_2004}. Apart from a more extensive understanding of these connections, we also leave a few technicalities for future work.\\\\\n\\textbf{Fast ? } In this work we were always in the KZ regime, such that $\\delta t^{-1} < m_i, m_f$. Fast scalings emerge when the rate is larger than the mass scales \\cite{Das:2014hqa}. Presently our derivative expansion does not allow us to access this regime. A full numerical treatment of the problem by solving directly equation \\eqref{ode1} will be necessary to study fast quenches.\\\\\n\\textbf{1\/N ?} It will also be challenging but useful to investigate the $1\/N$ corrections to the saddle point equations. There will be two sources, one coming from the Schwinger-Keldysh propagator's off-diagonal entries, and the other from the traditional $1\/N$ fluctuations around the semi-classical saddle. If the quench really thermalizes the system, then the expectation is that the fluctuations will be suppressed. It is to be noted, that using Liouville quantum mechanics \\cite{Bagrets_2016} in the double scaling regime, one can estimate the fall-off $\\vev{\\sigma(t) \\sigma(0)} \\sim t^{-3\/2}$. It will also be interesting to check if the suppression of matrix elements during the quench is consistent with the Eigenstate Thermalization Hypothesis. The ETH gives a likely criteria for thermalization \\cite{DAlessio:2016rwt}, note however \\cite{Mori:2017dip}. \\\\\n\\textbf{Holography ? } A UV completion of the zero temperature Gross-Neveu theory in the string theoretic setting has been formulated in terms of intersecting D4-D6 branes in \\cite{Antonyan:2006qy, Basu:2006eb}. In the weak string coupling limit, the 1+1-D Gross-Neveu theory emerges in this set-up where the role of the four-fermion coupling is played by the inverse of the separation between the D6 branes. Thus a global quench would amount to studying the effective D-brane dynamics under a drive that moves the D6 branes around. It will be interesting to investigate how KZ arises here and what happens in the strong string coupling regime which is when the D-branes are too close to each other. It will also be interesting if the near horizon geometry in this set-up has any semblance of AdS${}_2$ which may give a holographic explanation for the emergence of the Liouville quantum mechanics description. \\\\ \n\\textbf{Chaos ?} Another interesting computation will be that of the out of time ordered correlator \\cite{1969JETP...28.1200L, Maldacena:2015waa}, both in the field theory as well as in holography. It will be interesting to compare the KZ time regime with the scrambling time scale. The Lyapunov exponent was also defined recently in the context of evolution of chiral condensates \\cite{Hashimoto:2016wme}, it will be interesting to also explore this in the NJL${}_2$ context. \\\\\n\\textbf{Quenching in extended phases ?} The Gross-Neveu model also has a rich phase diagram at finite temperature and charge density exhibiting both second as well as a first order transition \\cite{Wolff:1985av}. An obvious generalization to the present study is to understand the quenches between phases in this rich set-up. Additionally both the zero charge \/ temperature as well as the finite case is known to have integrability structures \\cite{Basar:2010mu}, which had been used to find spacetime dependent condensate solutions to the gap equation. Understanding how such solutions dynamically contribute during a quench may lead to insights towards understanding the KZ scaling. \\\\ \nTo conclude : There are a lot of interesting non-equilibrium phenomena for which the Gross-Neveu model provides an excellent setting for fruitful investigations. \\\\\n\n\\section*{Acknowledgement}\nIt is a pleasure to thank Joydeep Chakrabortty, Sumit R. Das, Amit Dutta, Arijit Kundu, R. Loganayagam, Sridip Pal, Anatoli Polkovnikov, and Spenta R. Wadia for discussions and useful comments on the manuscript. The authors will like to acknowledge the support provided by the Max Planck Partner Group grant MAXPLA\/PHY\/2018577. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{introduction}\\label{intro}\n\nThis paper is devoted to the stability analysis of a unique (up to translation) traveling wave solution to a thermo-diffusive model of flame propagation with stepwise temperature kinetics and first-order reaction (see \\cite{BGKS15}) at high Lewis numbers, namely $\\Le>1$. The problem reads in one spatial dimension:\n\\begin{eqnarray}\\label{problem-1}\n\\left\\{\n\\begin{array}{l}\n\\displaystyle\\frac{\\partial\\Theta}{\\partial t}=\\frac{\\partial^2\\Theta}{\\partial x^2}+W(\\Theta,\\Phi), \\\\[2mm]\n\\displaystyle\\frac{\\partial\\Phi}{\\partial t}={\\Le}^{-1}\\frac{\\partial^2\\Phi}{\\partial x^2}-W(\\Theta,\\Phi).\n\\end{array}\n\\right.\n\\end{eqnarray}\nHere, $\\Theta$ and $\\Phi$ are appropriately normalized temperature and concentration of deficient reactant,\n$x\\in \\R$ denotes the spatial coordinate, $t>0$ the time. The nonlinear term $W(\\Theta,\\Phi)$ is a scaled reaction rate given by (see \\cite[Section 2, formula (3)]{BGKS15}):\n\\begin{eqnarray}\\label{problem-2}\nW(\\Theta,\\Phi)=\n\\left\\{\n\\begin{array}{lllll}\nA\\Phi, & \\mbox{if} & \\Theta\\ge \\Theta_i, \\\\[2mm]\n0, & \\mbox{if} &\\Theta<\\Theta_i.\n\\end{array}\n\\right.\n\\end{eqnarray}\nIn \\eqref{problem-2}, $0<\\Theta_i<1$ is the reduced ignition temperature, $A>0$ is a normalized factor depending on $\\Theta_i$ and $\\Le$, to be determined hereafter for the purpose of ensuring that the speed of traveling wave is set at unity. Moreover, the following boundary conditions hold at $\\pm \\infty$:\n\\begin{eqnarray}\n\\label{boundary condition-12}\n\\begin{matrix}\n\\Theta(t, -\\infty)=1, & & \\Theta(t,\\infty)=0,\\\\\n\\Phi(t, -\\infty)=0, & & \\Phi(t,\\infty)=1.\n\\end{matrix}\n\\end{eqnarray}\n\nIn this first-order stepwise kinetics model, $\\Phi$ does not vanish except as $t$ tends to $-\\infty$. Thus, problem \\eqref{problem-1}-\\eqref{boundary condition-12} belongs to the class of parabolic Partial Differential Equations with discontinuous nonlinearities. Models in combustion theory and other fields (see, e.g. \\cite[Section 1]{AF82}) involving discontinuous reaction terms have been used by physicists and engineers for long because of their manageability; as a result, elliptic and parabolic PDEs with discontinuous nonlinearities, and related Free Boundary Problems, have received a close attention from the mathematical community (see \\cite[Section 1]{ABLZ18} and references therein). We quote in particular the paper \\cite{C80}, by K.-C. Chang, which contains a systematical study of elliptic PDEs with discontinuous nonlinearities (DNDE).\n\t\nIn this paper, we consider the case of a free \\textsl{ignition interface} $g(t)$ defined by\n\\begin{equation}\n\\label{ignition}\n\\Theta(t,g(t))=\\Theta_i,\n\\end{equation}\nsuch that $\\Theta(t,x)>\\Theta_i$ for $x>g(t)$ and $\\Theta(t,x)<\\Theta_i$ for $x0$ and $x \\in \\R, x \\neq g(t)$:\n\\begin{eqnarray}\n\\label{system-1}\n&&\\left\\{\\begin{aligned}\n&\\frac{\\partial\\Theta}{\\partial t}=\\frac{\\partial^2\\Theta}{\\partial x^2}+A\\Phi,&xg(t),\\\\\n&\\frac{\\partial\\Phi}{\\partial t}={\\rm \\Le}^{-1}\\frac{\\partial^2\\Phi}{\\partial x^2},\\quad &x>g(t).\n\\end{aligned}\\right.\n\\end{eqnarray}\nAt the free interface $x=g(t)$, the following continuity conditions hold:\n\\begin{equation}\\label{system 1-2}\n[\\Theta]=[\\Phi]=0, \\qquad\\;\\, \\bigg [\\frac{\\partial\\Theta}{\\partial x}\\bigg ]=\\bigg [\\frac{\\partial\\Phi}{\\partial x}\\bigg ]=0,\n\\end{equation}\nwhere we denote by $[f]$ the jump of a function $f$ at a point $x_0$, i.e., the difference $f(x_0^+)-f(x_0^-)$.\n\nThe system above admits a unique (up to translation) traveling wave solution $\\pmb U=(\\Theta^0,\\Phi^0)$ which propagates with constant positive velocity $V$. In the moving frame coordinate $z=x-Vt$, by choosing\n\\begin{equation}\n\\label{eqn:A}\nA=\\frac{\\Theta_i}{1-\\Theta_i}\\bigg(1+\\frac{\\Theta_i}{\\Le(1-\\Theta_i)}\\bigg),\n\\end{equation}\nto have $V=1$ and, hence, $z=x-t$, the traveling wave solution is explicitly given by the following formulae:\n\\begin{eqnarray*}\n&\\Theta^0(z)&=\n\\left\\{\\begin{aligned}\n&1-(1-\\Theta_i)e^{\\frac{\\Theta_i}{1-\\Theta_i}z},& \\ \\ &z<0,&\\\\\n&\\Theta_ie^{-z},& \\ \\ &z>0,&\n\\end{aligned}\\right.\\\\[2mm]\n&\\Phi^0(z)&=\n\\left\\{\\begin{aligned}\n&\\frac{\\Theta_i}{A(1-\\Theta_i)}e^{\\frac{\\Theta_i}{1-\\Theta_i}z},& \\ \\ &z<0,&\\\\\n&1+\\left (\\frac{\\Theta_i}{A(1-\\Theta_i)}-1\\right )e^{-\\Le z},& \\ \\ &z>0.&\n\\end{aligned}\\right.\n\\end{eqnarray*}\n\nThe goal of this paper is the analysis of the stability of the traveling wave solution $\\pmb U$ in the case of high Lewis numbers ($\\Le>1$). Here, stability refers to orbital stability with asymptotic phase, because of the translation invariance of the traveling wave. It is known (see \\cite[Section 3.2]{BGKS15})\nthat large enough Lewis numbers give rise to \\textit{pulsating instabilities}, i.e., oscillatory behavior of the flame. This is\nvery unlike \\textit{cellular instabilities} for relatively small Lewis number ($\\Le<1$), that is pattern formation; in the latter case, a paradigm for the evolution of the disturbed flame front is the Kuramoto-Sivashinsky equation (see \\cite{MS79, S80}, and also \\cite{BHL13,BHL09, BHL11, BHLS10, BLSX10}).\n\nThe paper is organized as follows: In Section \\ref{linear operator}, we first transform the free interface problem to a system of parabolic equations on a fixed domain. Then, in the spirit of \\cite{BHL00,Lorenzi02,Lorenzi02-b}, the perturbation $\\pmb u$ of the traveling wave $\\pmb U$ is split as $\\displaystyle\\pmb u= s\\frac{d\\pmb U}{d\\xi} +\\pmb v$ (``ansatz 1''), in which $s$ is the perturbation of the front $g$. The largest part of the section is devoted to a thorough study of the linearization at $0$ of the elliptic part of the parabolic system in a weighted space $\\bm{\\mathcal W}$ where its realization $L$ is sectorial (see Subsection \\ref{linearized subsect} for further details about the use of a weighted space). Furthermore, we determine the spectrum of $L$ which contains $(-\\infty,-\\frac{1}{4}]$, a parabola and its interior, the roots of the so-called dispersion relation, and the eigenvalue $0$.\nThereafter, an important point is getting rid of the eigenvalue 0 which, as it has been already stressed, is generated by translation invariance. In Section \\ref{sect-3}, we use a spectral projection $P$ as well as ``ansatz 2'' and then derive the fully nonlinear problem (see, e.g. \\cite{Lunardi96}) for $\\pmb w$:\n\\begin{equation*}\n\\frac{\\partial\\pmb w}{\\partial\\tau} = (I-P)L\\pmb w + {F}(\\pmb w).\n\\end{equation*}\n\nNext, in Sections \\ref{stability} and \\ref{sect-5} we use the bifurcation parameter $m$ defined by\n\\begin{eqnarray*}\nm:=\\frac{\\Theta_i}{1-\\Theta_i}\n\\end{eqnarray*}\nto investigate the stability of the traveling wave. Simultaneously, as one already noted that pulsating instability is likely to occur at large Lewis number, it is natural to introduce a small perturbation parameter $\\varepsilon >0$ (dimensionless diffusion coefficient) defined by $\\varepsilon:=\\Le^{-1}$, so that \\eqref{eqn:A} reads $A =m+\\varep m^2$.\nThe simplest situation arises in the asymptotic case of gasless combustion when $\\Le =\\infty$, as in \\cite{GJ09}. As it is easily seen, as $\\varepsilon \\to 0$, problem \\eqref{system-1}-\\eqref{system-2} converges formally to:\n\\begin{eqnarray}\n\\label{system-1-limit}\n&&\\left\\{\\begin{aligned}\n&\\frac{\\partial\\Theta}{\\partial t}=\\frac{\\partial^2\\Theta}{\\partial x^2}+A\\Phi,\\quad &xg(t),\\\\\n&\\Phi\\equiv 1,&x>g(t),\n\\end{aligned}\\right.\n\\end{eqnarray}\nwith conditions $[\\Theta]= [\\Phi] = 0$, $\\displaystyle\\left [\\frac{\\partial\\Theta}{\\partial x}\\right ]=0$ at the free interface $x=g(t)$. However,\nthe limit free interface system \\eqref{system-1-limit}-\\eqref{system-2-limit} is only partly parabolic.\n\nAt the outset, we fix $m$ in Section \\ref{stability} and let $\\varep$ tend to $0$, which allows to apply the classical Hurwitz Theorem in complex analysis to the \\textit{dispersion relation} $D_{\\varep}(\\lambda,m)$. Our first main result, Theorem \\ref{stability theorem TW}, states that, for $2m^c=6$, it is unstable. To give a broad picture, we take advantage of the regular convergence of the point spectrum as $\\varep \\to 0$.\n\nSection \\ref{sect-5} is devoted to the proof of Hopf bifurcation in a neighborhood of the critical value $m^c=6$. The difficulty is twofold: first, the framework is that of a fully nonlinear problem; second, $m$ is not fixed in the sequence of parameterized analytic functions $D_{\\varep}(\\lambda,m)$ which prevents us from using Hurwitz Theorem directly. The trick is to find a proper approach to combining $m$ with $\\varep$: to this end we construct a sequence of critical values $m^c(\\varep)$ such that $m^c(0)=m^c$ and apply Hurwitz Theorem to $D_{\\varep}(\\lambda,m^c(\\varep))$. Proposition \\ref{give critical value} and Theorem \\ref{Hopf bifurcation theorem} are crucial to prove Hopf bifurcation at $m^c(\\varep)$ for $\\varep$ small enough. Finally, in three appendices, we collect some formulae and results that we use to prove our main results.\n\n\n\\section{The linearized operator}\\label{linear operator}\nIn this section, we first derive the governing equations for the perturbations of the traveling wave solution. As usual, it is convenient to transform the free interface problem to a system on a fixed domain. More specifically, we use the general method of \\cite{BHL00} that converts free interface problems to fully nonlinear problems with\ntransmission conditions at a fixed interface (see \\cite{ABLZ18}). Then, we are going to focus on the linearized system.\n\n\\subsection{The system with fixed interface}\\label{fixed}\nTo begin with, we rewrite problem \\eqref{system-1}-\\eqref{system 1-2} in a new system of coordinates that fixes the position of the ignition interface at the origin:\n\\begin{eqnarray*}\n\\tau=t,\\ \\\n\\xi=x-g(\\tau).\n\\end{eqnarray*}\nHereafter, we are going to use, whenever it is convenient, the superdot to denote differentiation with respect to time and the prime to denote partial differentiation with respect to the space variable.\n\nThen, the system for ${\\pmb X}=(\\Theta,\\Phi)$ and $g$ reads:\n\\begin{eqnarray}\n\\label{perturbation T}\n&&\\left\\{\\begin{aligned}\n\\frac{\\partial\\Theta}{\\partial\\tau}-\\dot{g}\\frac{\\partial\\Theta}{\\partial\\xi}=&\\frac{\\partial^2\\Theta}{\\partial\\xi^2}+A\\Phi, \\ \\ &\\xi<0,\\\\\n\\frac{\\partial\\Phi}{\\partial\\tau}-\\dot{g}\\frac{\\partial\\Phi}{\\partial\\xi}=&\\Le^{-1}\\frac{\\partial^2\\Phi}{\\partial\\xi^2}-A\\Phi, \\ \\ &\\xi<0, \\end{aligned}\\right.\\\\[1mm]\n\\label{perturbation W}\n&&\\left\\{\\begin{aligned}\n\\frac{\\partial\\Theta}{\\partial\\tau}-\\dot{g}\\frac{\\partial\\Theta}{\\partial\\xi}=&\\frac{\\partial^2\\Theta}{\\partial\\xi^2}, \\ \\ &\\xi>0,\\\\\n\\frac{\\partial\\Phi}{\\partial\\tau}-\\dot{g}\\frac{\\partial\\Phi}{\\partial\\xi}=&\\Le^{-1}\\frac{\\partial^2\\Phi}{\\partial\\xi^2}, \\ \\ &\\xi>0. \\end{aligned}\\right.\n\\end{eqnarray}\nMoreover, $\\Theta$, $\\Phi$ and their first-order space derivatives are continuous at the fixed interface $\\xi=0$, thus\n\\begin{equation}\n\\Theta(\\cdot,0)=\\Theta_i, \\qquad\\;\\, [\\Theta]=[\\Phi]=0, \\qquad\\;\\, \\bigg [\\frac{\\partial\\Theta}{\\partial\\xi}\\bigg ]=\\bigg [\\frac{\\partial\\Phi}{\\partial\\xi}\\bigg ]=0.\n\\label{interface-Theta-Phi}\n\\end{equation}\nIn addition, at $\\xi=\\pm \\infty$, $\\Theta$ and $\\Phi$ satisfy \\eqref{boundary condition-12}.\n\nNext, we introduce the small perturbations $\\pmb u=(u_1,u_2)$ and $s$, respectively of the traveling wave $\\pmb U$ and of the front $g$, more precisely,\n\\begin{align*}\n&u_1(\\tau,\\xi)=\\Theta(\\tau, \\xi)-\\Theta^0(\\xi),\\\\\n&u_2(\\tau,\\xi)=\\Phi(\\tau, \\xi)-\\Phi^0(\\xi),\\\\\n&s(\\tau)=g(\\tau)-\\tau.\n\\end{align*}\nIt then follows that the perturbations $\\pmb u$ and $s$ verify the system\n\\begin{eqnarray}\n\\label{u_1}\n&&\\left\\{\\begin{aligned}\n\\frac{\\partial u_1}{\\partial \\tau}&=\\frac{\\partial^2 u_1}{\\partial \\xi^2}+\\frac{\\partial u_1}{\\partial \\xi}+Au_2+\\dot{s}\\frac{d\\Theta^0}{d\\xi}+\\dot{s}\\frac{\\partial u_1}{\\partial \\xi}, \\ &\\xi<0,&\\\\\n\\frac{\\partial u_2}{\\partial \\tau}&=\\Le^{-1}\\frac{\\partial^2 u_2}{\\partial \\xi^2}+\\frac{\\partial u_2}{\\partial \\xi}-Au_2+\\dot{s}\\frac{d\\Phi^0}{d\\xi}+\\dot{s}\\frac{\\partial u_2}{\\partial \\xi}, \\ &\\xi<0,&\n\\end{aligned}\\right.\\\\[1mm]\n\\label{u_2}\n&&\\left\\{\\begin{aligned}\n\\frac{\\partial u_1}{\\partial \\tau}&=\\frac{\\partial^2 u_1}{\\partial \\xi^2}+\\frac{\\partial u_1}{\\partial \\xi}+\\dot{s}\\frac{d\\Theta^0}{d\\xi}+\\dot{s}\\frac{\\partial u_1}{\\partial \\xi}, \\ &\\xi>0,&\\\\\n\\frac{\\partial u_2}{\\partial \\tau}&=\\Le^{-1}\\frac{\\partial^2 u_2}{\\partial \\xi^2}+\\frac{\\partial u_2}{\\partial \\xi}+\\dot{s}\\frac{d\\Phi^0}{d\\xi}+\\dot{s}\\frac{\\partial u_2}{\\partial \\xi}, \\ &\\xi>0,&\n\\end{aligned}\\right.\n\\end{eqnarray}\nand the corresponding interface conditions obtained from \\eqref{interface-Theta-Phi} are:\n\\begin{equation}\\label{interface-u}\nu_1(\\tau,0)=0,\\qquad\\;\\, [u_1]=[u_2]=\\bigg [\\frac{\\partial u_1}{\\partial \\xi}\\bigg ]=\\bigg [\\frac{\\partial u_2}{\\partial \\xi}\\bigg ]=0.\n\\end{equation}\n\n\\subsection{Ansatz 1}\n\\label{subsect-2.2}\nIn the spirit of \\cite{BHL00,Lorenzi02}, we introduce the following splitting or ansatz:\n\\begin{equation}\\label{ansatz1}\n\\begin{aligned}\nu_1(\\tau,\\xi)=&s(\\tau)\\frac{d\\Theta^0}{d\\xi}(\\xi)+v_1(\\tau,\\xi),\\\\\nu_2(\\tau,\\xi)=&s(\\tau)\\frac{d\\Phi^0}{d\\xi}(\\xi)+v_2(\\tau,\\xi),\n\\end{aligned}\n\\end{equation}\nin which $v_1$, $v_2$ are new unknown functions. In a more abstract setting, the ansatz reads\n\\begin{equation*}\n\\pmb u(\\tau,\\xi)= s(\\tau)\\frac{d\\pmb U}{d\\xi} +\\pmb v(\\tau,\\xi), \\qquad\\;\\, \\pmb v=(v_1,v_2).\n\\end{equation*}\nSubstituting \\eqref{ansatz1} into \\eqref{u_1}-\\eqref{u_2}, we get the system for $\\pmb u$ and $s$:\n\\begin{eqnarray}\n\\label{v1}\n&&\\left\\{\\begin{aligned}\n\\frac{\\partial v_1}{\\partial \\tau}&=\\frac{\\partial^2 v_1}{\\partial \\xi^2}+\\frac{\\partial v_1}{\\partial \\xi}+Av_2+\\dot{s}\\left (s\\frac{d^2\\Theta^0}{d\\xi^2}+\\frac{\\partial v_1}{\\partial \\xi}\\right ), \\ \\ &\\xi<0&,\\\\\n\\frac{\\partial v_2}{\\partial \\tau}&=\\Le^{-1}\\frac{\\partial^2 v_2}{\\partial \\xi^2}+\\frac{\\partial v_2}{\\partial \\xi}-Av_2+\\dot{s}\\left (s\\frac{d^2\\Phi^0}{d\\xi^2}+\\frac{\\partial v_2}{\\partial \\xi}\\right ), \\ \\ &\\xi<0&,\n\\end{aligned}\\right.\\\\[1mm]\n\\label{v2}\n&&\\left\\{\\begin{aligned}\n\\frac{\\partial v_1}{\\partial \\tau}&=\\frac{\\partial^2 v_1}{\\partial \\xi^2}+\\frac{\\partial v_1}{\\partial \\xi}+\\dot{s}\\left (s\\frac{d^2\\Theta^0}{d\\xi^2}+\\frac{\\partial v_1}{\\partial \\xi}\\right ), \\ \\ &\\xi>0&,\\\\\n\\frac{\\partial v_2}{\\partial \\tau}&=\\Le^{-1}\\frac{\\partial^2 v_2}{\\partial \\xi^2}+\\frac{\\partial v_2}{\\partial \\xi}+\\dot{s}\\left (s\\frac{d^2\\Phi^0}{d\\xi^2}+\\frac{\\partial v_2}{\\partial \\xi}\\right ), \\ \\ &\\xi>0&.\n\\end{aligned}\\right.\n\\end{eqnarray}\nAt $\\xi=0$, it is easy to see that the new interface conditions are:\n\\begin{eqnarray*}\n[v_1]=[v_2]=0,\\qquad\\,\\,\\bigg [\\frac{\\partial v_1}{\\partial \\xi}\\bigg ]=-s\\bigg [\\frac{d^2\\Theta^0}{d\\xi^2}\\bigg ],\\qquad\\,\\, \\bigg [\\frac{\\partial v_2}{\\partial \\xi}\\bigg ]=-s\\bigg [\\frac{d^2\\Phi^0}{d\\xi^2}\\bigg ],\\qquad\\,\\,v_1(\\tau,0)=-s\\frac{\\partial\\Theta^0}{\\partial\\xi}(0).\n\\end{eqnarray*}\nTaking advantage of the conditions\n\\begin{eqnarray*}\n\\frac{d\\Theta^0}{d\\xi}(0)=-\\Theta_i,\\quad\\bigg [\\frac{d^2\\Theta^0}{d\\xi^2}\\bigg ]=\\frac{\\Theta_i}{1-\\Theta_i},\\quad \\bigg [\\frac{d^2\\Phi^0}{d\\xi^2}\\bigg ]=-\\frac{\\Le\\Theta_i}{1-\\Theta_i},\n\\end{eqnarray*}\nwhere we used (\\ref{eqn:A}) to derive the last condition, it follows that\n\\begin{equation}\\label{transm}\ns(\\tau)=\\frac{v_1(\\tau,0)}{\\Theta_i},\\qquad\\;\\, \\bigg [\\frac{\\partial v_1}{\\partial \\xi}\\bigg ]=-\\frac{v_1(\\tau,0)}{1-\\Theta_i},\\qquad\\;\\, \\bigg [\\frac{\\partial v_2}{\\partial \\xi}\\bigg ]=\\frac{v_1(\\tau,0)\\Le}{1-\\Theta_i}.\n\\end{equation}\n\nSummarizing, the free interface problem \\eqref{system-1}-\\eqref{system-2} has been converted to ($\\ref{u_1}$)-($\\ref{u_2}$), which constitutes a nonlinear system for $v_1$, $v_2$ and $s$, with transmission conditions \\eqref{transm} at $\\xi=0$. The next subsections are devoted to the study of the linearized problem (at zero) in an abstract setting, with simplified notation $\\pmb u=(u,v)$ for convenience.\n\n\\subsection{The linearized problem}\\label{linearized subsect}\nNow, we consider the linearization at $0$ of the system \\eqref{v1}-\\eqref{transm}, which reads as follows:\n\\begin{eqnarray}\n\\label{linear pb-u}\n&&\\left\\{\\begin{aligned}\n\\frac{\\partial u}{\\partial \\tau}&=\\frac{\\partial^2u}{\\partial\\xi^2}+\\frac{\\partial u}{\\partial\\xi}+Av, &\\xi<0,\\\\\n\\frac{\\partial v}{\\partial\\tau}&=\\Le^{-1}\\frac{\\partial^2v}{\\partial \\xi^2}+\\frac{\\partial v}{\\partial\\xi}-Av,\\quad &\\xi<0,\\\\\n\\end{aligned}\\right.\\\\[1mm]\n\\label{linear pb-v}\n&&\\left\\{\\begin{aligned}\n\\frac{\\partial u}{\\partial \\tau}&=\\frac{\\partial^2u}{\\partial\\xi^2}+\\frac{\\partial u}{\\partial\\xi}, & \\xi>0,\\\\\n\\frac{\\partial v}{\\partial\\tau}&=\\Le^{-1}\\frac{\\partial^2v}{\\partial\\xi^2}+\\frac{\\partial v}{\\partial\\xi}, &\\quad \\xi>0,\n\\end{aligned}\\right.\n\\end{eqnarray}\nwith the interface conditions\n\\begin{equation}\n\\label{linear pb-interface}\n[u]=[v]=0,\\qquad\\;\\, \\bigg [\\frac{\\partial u}{\\partial\\xi}\\bigg ]=-\\frac{u(\\tau,0)}{1-\\Theta_i},\\qquad\\;\\,\n\\bigg [\\frac{\\partial v}{\\partial\\xi}\\bigg ]=\\frac{u(\\tau,0)\\Le}{1-\\Theta_i}.\n\\end{equation}\n\nProblem \\eqref{linear pb-u}-\\eqref{linear pb-v} can be written in the more compact form\n$\\displaystyle\\frac{\\partial\\pmb u}{\\partial\\tau}={\\mathcal L}\\pmb u$, where $\\pmb u=(u,v)$,\n\\begin{eqnarray*}\n\\mathcal{L}=\\left(\\begin{matrix}\n\\displaystyle\\frac{\\partial^2}{\\partial\\xi^2}+\\frac{\\partial}{\\partial\\xi}& &A\\chi_{-}\\\\\n0& &\\displaystyle\\Le^{-1}\\frac{\\partial^2}{\\partial\\xi^2}+\\frac{\\partial}{\\partial \\xi}-A\\chi_{-}\n\\end{matrix}\\right)\n\\end{eqnarray*}\nand $\\chi_-$ denotes the characteristic function of the set $(-\\infty,0)$.\n\nWe now introduce the weighted space $\\bm{\\mathcal W}$ where we analyze the system \\eqref{linear pb-u}-\\eqref{linear pb-interface}.\nAs a matter of fact, the introduction of exponentially weighted spaces for proving stability of traveling waves has been a standard tool since the pioneering work of Sattinger (see \\cite{S76}), its role being to shift the continuous spectrum to the left and, thus, creating a gap with the imaginary axis which simplifies the analysis.\n\n\\begin{definition}\nThe exponentially weighted Banach space $\\bm{\\mathcal W}$ is defined by\n\\begin{align*}\n\\bm{\\mathcal W}=\\Big\\{&\\pmb u:\ne^{\\frac{1}{2}\\xi}u, e^{\\frac{1}{2}\\xi}v\\in C_b((-\\infty,0);\\mathbb C),\\\ne^{\\frac{1}{2}\\xi}u, e^{\\frac{\\Le}{2}\\xi}v\\in C_b((0,\\infty);\\mathbb C), \\lim_{\\xi\\to 0^{\\pm}}u(\\xi),\\ \\lim_{\\xi\\to 0^{\\pm}}v(\\xi)\\in\\mathbb R\\Big\\},\n\\end{align*}\nequipped with the norm:\n\\begin{align*}\n\\|\\pmb u\\|_{\\bm{\\mathcal W}}=&\\sup_{\\xi <0}|e^{\\frac{1}{2}\\xi}u(\\xi)|+\\sup_{\\xi>0}|e^{\\frac{1}{2}\\xi}u(\\xi)|+\\sup_{\\xi <0}|e^{\\frac{1}{2}\\xi}v(\\xi)|+\\sup_{\\xi>0}|e^{\\frac{\\Le}{2}\\xi}v(\\xi)|.\n\\end{align*}\t\n\\end{definition}\nIn the above definition, $C_b(I;\\mathbb C)$ denotes the space of bounded and continuous functions from $I$ to $\\mathbb{C}$, $I$ being either the interval $(-\\infty,0)$ or $(0,\\infty)$. We finally introduce the realization $L$ of the operator ${\\mathcal L}$ in $\\bm{\\mathcal W}$ defined by\n\\begin{align*}\n&D(L)=\\bigg\\{\\pmb u\\in \\bm{\\mathcal W}: \\frac{\\partial \\pmb u}{\\partial\\xi},\\frac{\\partial^2\\pmb u}{\\partial\\xi^2}\\in \\bm{\\mathcal W},\\ [u]=[v]=0,\\ \\bigg [\\frac{\\partial u}{\\partial\\xi}\\bigg ]=-\\frac{u(0)}{1-\\Theta_i},\\ \\bigg [\\frac{\\partial v}{\\partial\\xi}\\bigg ]=\\frac{\\Le\\ u(0)}{1-\\Theta_i}\\bigg\\},\\\\[1mm]\n&L\\pmb u=\\mathcal{L}\\pmb u,\\qquad\\;\\, \\pmb u\\in \\bm{\\mathcal W}.\n\\end{align*}\n\n\\begin{remark}\n\\label{rem-simple}\n{\\rm We observe that, for any Lewis number, the pair\n$\\displaystyle\\frac{d\\pmb U}{d\\xi}=\\left (\\frac{d\\Theta^0}{d\\xi},\\frac{d\\Phi^0}{d\\xi}\\right )$ verifies System \\eqref{linear pb-u}, \\eqref{linear pb-v}, and it belongs to the space $\\bm{\\mathcal W}$. In other words, $\\displaystyle\\frac{d\\pmb U}{d\\xi}$ is an eigenfunction of the operator $L$ associated with the eigenvalue 0.}\n\\end{remark}\nThe above remark gives a first justification for the choice of the exponential weights in the definition of $\\bm{\\mathcal W}$.\nWe also stress that, following the same strategy as in the proof of the forthcoming Theorem \\ref{thm-2.3} it can be easily checked that the spectrum of the realization of the operator ${\\mathcal L}$ in the nonweighted space of pairs $(u,v)$ such that $u$, $v$ are bounded and continuous in $(-\\infty,0)\\cup (0,\\infty)$, contains a parabola which is tangent at $0$ to the imaginary axis.\n\n\\subsection{Analysis of the operator $L$}\n\\label{Resolvent operator}\nNext theorem is devoted to a deep study of the operator $L$. For simplicity of notation, for $j=1,2$ we set\n\\begin{align}\nH_{1,\\lambda}=\\sqrt{1+4\\lambda},\\qquad\\;\\,H_{2,\\lambda}=\\sqrt{\\Le^2+4\\Le(A+\\lambda)},\\qquad\\;\\,H_{3,\\lambda}=\\sqrt{\\Le^2+4\\Le\\lambda}\n\\label{formula-1}\n\\end{align}\nand\n\\begin{align}\n&k_{j,\\lambda}=\\frac{-1+(-1)^{j+1}H_{1,\\lambda}}{2},\\qquad k_{2+j,\\lambda}=\\frac{-\\Le+(-1)^{j+1}H_{2,\\lambda}}{2},\\qquad k_{4+j,\\lambda}=\\frac{-\\Le+(-1)^{j+1}H_{3,\\lambda}}{2}.\n\\label{formula-3}\n\\end{align}\n\n\\begin{theorem}\n\\label{thm-2.3}\nThe operator $L$ is sectorial and therefore generates an analytic semigroup. Moreover, its spectrum has components:\n\\begin{enumerate}[\\rm (1)]\n\\item\n$(-\\infty,-1\/4]\\cup \\mathcal{P}$, where $\\mathcal{P}=\\{\\lambda\\in\\mathbb{C}:a\\Re\\lambda+b(\\Im\\lambda)^2+c\\le 0\\}$ with\n\\begin{align*}\na=\\bigg (1-\\frac{1}{\\Le}\\bigg )^2,\\qquad\\;\\,\nb=\\frac{1}{\\Le},\\qquad\\;\\,\nc=\\frac{2A+1}{2}+\\frac{8A-5}{4\\Le}+\\frac{1+A}{\\Le^2}-\\frac{1}{4\\Le^3};\n\\end{align*}\n\\item\nthe simple isolated eigenvalue $0$, the kernel of $L$ being spanned by $\\displaystyle\\frac{d\\pmb U}{d\\xi}$;\n\\item\nadditional eigenvalues given by the solution of the dispersion relation\n\\begin{equation}\n\\label{d,r,Le}\nD(\\lambda; \\Theta_i, \\Le):=(k_{6,\\lambda}-k_{3,\\lambda})(k_{3,\\lambda}-k_{2,\\lambda})\\big [1-(1-\\Theta_i)\\sqrt{1+4\\lambda}\\big ]+A\\Le,\n\\end{equation}\nwhere $A$ is given by \\eqref{eqn:A}.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nSince the proof is rather lengthy, we split it into four steps.\nIn the first two steps, we prove properties (1) and (3). Step 3 is devoted to the proof of property (2).\nFinally, in Step 4, we prove that the operator $L$ is sectorial in $\\bm{\\mathcal W}$.\n\nFor notational convenience, throughout the proof, we set\n\\begin{align*}\n&{\\mathscr I}_1:=\\int_{0}^{\\infty}f_1(s)e^{-k_1s}ds,&\n&{\\mathscr I}_2:=\\int_{-\\infty}^{0}f_1(s)e^{-k_2s}ds,&\n&{\\mathscr I}_3:=\\int_{-\\infty}^{0}f_2(s)e^{-k_2s}ds,\\\\\n&{\\mathscr I}_4:=\\int_{-\\infty}^{0}f_2(s)e^{-k_4s}ds, &\n&{\\mathscr I}_5:=\\int_{0}^{\\infty}f_2(s)e^{-k_5s}ds,&\n\\end{align*}\nfor any fixed $\\pmb f=(f_1,f_2)\\in\\bm{\\mathcal W}$, where, here and Step 1 to 3, we simply write $k_j$ instead of $k_{j,\\lambda}$ to enlighten the notation.\n\n\\vskip 1mm\n{\\em Step 1}. To begin with, we prove that the interval $(-\\infty,-1\/4]$ belongs to the point spectrum of $L$. We first assume that $\\lambda\\le-\\Le\/4$ (recall that $\\Le>1$). In such a case, $\\Re(k_1)=\\Re(k_2)=-1\/2$, $\\Re(k_5)=\\Re(k_6)=-\\Le\/2$ and the function $\\pmb u$ defined by\n\\begin{equation}\nu(\\xi)=\n\\left\\{\n\\begin{array}{ll}\nc_1 e^{k_1\\xi}+c_2 e^{k_2 \\xi}, & \\xi<0,\\\\\nc_5 e^{k_1\\xi}+c_6 e^{k_2 \\xi}, & \\xi\\ge 0,\n\\end{array}\n\\right.\n\\qquad\\;\\,\nv(\\xi)=\n\\left\\{\n\\begin{array}{ll}\n0, &\\xi<0,\\\\\nc_7 e^{k_5\\xi}+c_8 e^{k_6\\xi}, &\\xi\\ge 0,\n\\end{array}\n\\right.\n\\label{eigenvalue-1\/4}\n\\end{equation}\nbelongs to $\\bm{\\mathcal W}$ and solves the equation $\\lambda \\pmb u-{\\mathcal L}\\pmb u={\\bf 0}$ for any choice of the complex parameters $c_1$, $c_2$, $c_5$, $c_6$, $c_7$ and $c_8$.\nSince there are only four boundary conditions to impose to guarantee that $\\pmb u\\in D(L)$,\nthe resolvent equation $\\lambda \\pmb u-{\\mathcal L}\\pmb u={\\bf 0}$ is not uniquely solvable in $\\bm{\\mathcal W}$. Thus, $\\lambda$ belongs to the point spectrum of $L$.\n\nNext, we consider the case when $\\lambda\\in(-\\Le\/4, -1\/4]$. In this situation, $\\Re(k_1)=\\Re(k_2)=-1\/2$, however, $\\Re(k_5)+\\Le\/2>0$, $\\Re(k_6)+\\Le\/2<0$.\nThanks to the fact that $e^{\\frac{\\Le}{2}\\xi}v(\\xi)$ should be bounded in $(0,\\infty)$, the constant $c_7$ in \\eqref{eigenvalue-1\/4} is zero, whereas the constants $c_1$, $c_2$, $c_5$, $c_6$ $c_8$ are arbitrary. As above, the resolvent equation $\\lambda\\pmb u-L\\pmb u={\\bf 0}$ cannot be solved uniquely. Consequently, we conclude that $(-\\infty,-1\/4]$ belongs to the point spectrum of the operator $L$.\n\nFrom now on, we consider the case when $\\lambda\\notin (-\\infty,-1\/4]$. Then,\n$\\Re(k_1)+1\/2>0$, $\\Re(k_2)+1\/2<0$, $\\Re(k_5)+\\Le\/2>0$ and $\\Re(k_6)+\\Le\/2<0$.\nSimilarly to the previous procedure, using the formulae \\eqref{rs-u-positive}, \\eqref{rs-v-positive} and \\eqref{resolvent-u} as well as the fact that the functions $\\xi\\mapsto e^{\\frac{1}{2}\\xi}u(\\xi)$ and $\\xi\\mapsto e^{\\frac{\\Le}{2}\\xi}v(\\xi)$ should be bounded in $\\mathbb{R}$ and in $(0,\\infty)$ respectively, the constants $c_2$, $c_5$, $c_7$ can be determined explicitly and they are given by\n \\begin{align*}\n&c_2=\\frac{1}{H_{1,\\lambda}}\\int_{-\\infty}^{0}(Av(s)+f_1(s))e^{-k_2s}ds,\\qquad\\;\\, c_5=\\frac{1}{H_{1,\\lambda}}{\\mathscr I}_1,\\qquad\\;\\,c_7=\\frac{\\Le}{H_{3,\\lambda}}{\\mathscr I}_5.\n\\end{align*}\n\nWe now consider formula (\\ref{resolvent-v}). Since $\\Le>1$, it follows that $\\Re(k_4)+1\/2<0$. Moreover, we observe that\nthe inequality $\\Re(k_3)+{1}\/{2}\\le 0$ is satisfied if and only if $\\lambda\\in {\\mathcal P}$. Indeed, fix any $\\lambda\\in \\stackrel{\\circ}{{\\mathcal P}}$, the interior of ${\\mathcal P}$, so that $\\Re(k_3)+1\/2<0$, and take\n\\begin{equation*}\nf_1(\\xi)=\n\\left\\{\n\\begin{array}{ll}\ne^{-\\frac{1}{2}\\xi}, &\\xi<0,\\\\\n0, &\\xi\\ge 0,\n\\end{array}\n\\right.\n\\qquad\\;\\,\nf_2\\equiv 0 \\ \\text{in} \\ \\mathbb{R}.\n\\end{equation*} \t\nIn such a case, the more general solution, $\\pmb u\\in\\bm{\\mathcal W}$, to the equation\n$\\lambda \\pmb u-{\\mathcal L}\\pmb u=\\pmb f$ is given\nby $u(\\xi)=c_6e^{k_2\\xi}$ and $v(\\xi)=c_8e^{k_6\\xi}$ for $\\xi\\ge 0$, whereas $v\\equiv 0$ in $(-\\infty,0)$ and\n$u(\\xi)=c_1e^{k_1\\xi}+2H_{1,\\lambda}^{-2}(2e^{-\\frac{1}{2}\\xi}-e^{k_1\\xi})$\nfor $\\xi<0$. Note that $k_1\\neq k_3$ for $\\lambda\\in\\stackrel{\\circ}{\\mathcal P}$.\nImposing the boundary conditions, we deduce that $c_6=c_8=0$,\n$c_1=-2H_{1,\\lambda}^{-2}$ and $k_1c_1=2H_{1,\\lambda}^{-2}k_2$,\nwhich is clearly a contradiction. We conclude that the domain $\\stackrel{\\circ}{{\\mathcal P}}$ and, consequently, its closure belong to the continuous spectrum of $L$.\t\nSummarizing, property (1) in the statement of the theorem is established.\n\n\\vskip 1mm\n{\\em Step 2}. Here, we consider the equation $\\lambda \\pmb u-{\\mathcal L}\\pmb u=\\pmb f$ for $\\pmb f\\in\\bm{\\mathcal W}$ and values of $\\lambda$ which are not in $(-\\infty,-1\/4]\\cup{\\mathcal P}$. For such $\\lambda$'s and $j=1,2$ it holds that\n\\begin{equation}\n\\Re(k_{2j-1})+\\frac{1}{2}>0,\\qquad\\;\\, \\Re(k_{2j})+\\frac{1}{2}<0,\\qquad\\;\\, \\Re(k_5)+\\displaystyle\\frac{\\Le}{2}>0,\\qquad\\;\\, \\Re(k_6)+\\displaystyle\\frac{\\Le}{2}<0.\n\\label{cond-1}\n\\end{equation}\nWe first assume that $k_1\\neq k_3$. Imposing that the function $\\pmb u$ defined by \\eqref{rs-u-positive}-\\eqref{resolvent-v} belongs to $\\bm{\\mathcal W}$, we can\nuniquely determine the constants $c_2$, $c_4$, $c_5$ and $c_7$ and we get\n\\begin{align}\n\\label{re-u-negative}\nu(\\xi)\n=&c_1e^{k_1\\xi}+\\frac{e^{k_1\\xi}}{H_{1,\\lambda}}\\int_{\\xi}^0 f_1(s)e^{-k_1s}ds\n+\\frac{e^{k_2\\xi}}{H_{1,\\lambda}}\\int_{-\\infty}^{\\xi} f_1(s)e^{-k_2s}ds \\nonumber\\\\\n&+\\frac{A}{H_{1,\\lambda}}\\bigg\\{\\bigg (\\frac{e^{k_3\\xi}}{k_3-k_2}-\\frac{e^{k_3\\xi}-e^{k_1\\xi}}{k_3-k_1}\\bigg )c_3\n+\\frac{\\Le}{H_{2,\\lambda}}\\bigg[\\bigg (\\frac{e^{k_1\\xi}-e^{k_3\\xi}}{k_3-k_1}\n-\\frac{e^{k_3\\xi}}{k_3-k_2}\\bigg )\\int_{\\xi}^0f_2(s)e^{-k_3s}ds\\notag\\\\\n&\\phantom{-\\frac{A}{H_{1,\\lambda}}\\bigg\\{\\;\\,}+\\frac{e^{k_1\\xi}}{k_3-k_1}\\int_\\xi^0 f_2(s)e^{-k_1s}ds\n+\\bigg (\\frac{e^{k_1\\xi}-e^{k_4\\xi}}{k_4-k_1}+\\frac{e^{k_4\\xi}}{k_4-k_2}\\bigg )\\int_{-\\infty}^\\xi f_2(s)e^{-k_4s}ds\\nonumber\\\\\n&\\phantom{-\\frac{A}{H_{1,\\lambda}}\\bigg\\{\\;\\,}+\\frac{e^{k_1\\xi}}{k_4-k_1}\\int_\\xi^0 f_2(s)(e^{-k_4s}\\!-\\!e^{-k_1s})ds\\!+\\! \\frac{(k_4-k_3)e^{k_2\\xi}}{(k_3-k_2)(k_4-k_2)}\\int_{-\\infty}^{\\xi}f_2(s)e^{-k_2s}ds\\bigg]\\bigg\\},\n\\\\[1mm]\n\\label{re-v-negative}\nv(\\xi)&=\\bigg (c_3+\\frac{\\Le}{H_{2,\\lambda}}\\int_{\\xi}^0f_2(s)e^{-k_3s}ds\\bigg )e^{k_3\\xi}\n+\\frac{\\Le\\,e^{k_4\\xi}}{H_{2,\\lambda}}\\int_{-\\infty}^{\\xi}f_2(s)e^{-k_4s}ds,\n\\end{align}\nfor $\\xi<0$. Note that $k_2-k_3\\neq 0$ (see Appendix \\ref{appendix-A}). For $\\xi>0$, we get\n\\begin{align}\n\\label{re-u-positive}\nu(\\xi)&=\\frac{e^{k_1\\xi}}{H_{1,\\lambda}}\\int_{\\xi}^{\\infty}f_1(s)e^{-k_1s}ds+\\bigg (c_6+{\\frac{1}{H_{1,\\lambda}}\\int_0^{\\xi}f_1(s)e^{-k_2s}ds}\\bigg )e^{k_2\\xi},\\\\[1mm]\n\\label{re-v-positive}\nv(\\xi)&=\\frac{\\Le\\,e^{k_5\\xi}}{H_{3,\\lambda}}\\int_{\\xi}^{\\infty}f_2(s)e^{-k_5s}ds+\\bigg (c_8+\\frac{\\Le}{H_{3,\\lambda}}\\int_0^{\\xi}f_2(s)e^{-k_6s}ds\\bigg )e^{k_6\\xi}.\n\\end{align}\n\nImposing the boundary conditions, we obtain the following linear system for the unknowns $c_1$, $c_3$, $c_6$ and $c_8$:\n\\begin{equation}\n\\begin{pmatrix}\n1 &\\frac{A}{(k_3-k_2)H_{1,\\lambda}} & -1 & 0\\\\\n0 & 1 & 0 & -1\\\\\nk_1 & \\frac{Ak_2}{(k_3-k_2)H_{1,\\lambda}} & \\frac{1}{\\Theta_i-1}-k_2 & 0\\\\\n0 & k_3 & \\frac{\\Le}{1-\\Theta_i} & -k_6\n\\end{pmatrix}\n\\begin{pmatrix}\nc_1\\\\\nc_3\\\\\nc_6\\\\\nc_8\n\\end{pmatrix}\n=\\begin{pmatrix}\nF_1\\\\\nF_2\\\\\nF_3\\\\\nF_4\n\\end{pmatrix},\n\\label{matrix}\n\\end{equation}\nwhere\n\\begin{align*}\nF_1=&-\\frac{A\\Le}{(k_4-k_2)H_{1,\\lambda}H_{2,\\lambda}}{\\mathscr I}_4-\\frac{1}{H_{1,\\lambda}}{\\mathscr I}_2+\\frac{1}{H_{1,\\lambda}}{\\mathscr I}_1\n-\\frac{A\\Le(k_4-k_3)}{(k_3-k_2)(k_4-k_2)H_{1,\\lambda}H_{2,\\lambda}}{\\mathscr I}_3;\\\\[1mm]\nF_2=&\\frac{\\Le}{H_{3,\\lambda}}{\\mathscr I}_5-\\frac{\\Le}{H_{2,\\lambda}}{\\mathscr I}_4;\\\\[1mm]\nF_3=&-\\frac{A\\Le k_2}{(k_4-k_2)H_{1,\\lambda}H_{2,\\lambda}}{\\mathscr I}_4\\!-\\!\\frac{k_2}{H_{1,\\lambda}}{\\mathscr I}_2\\!+\\!\\frac{1}{H_{1,\\lambda}}\\bigg (k_1\\!+\\!\\frac{1}{1-\\Theta_i}\\bigg ){\\mathscr I}_1\\!+\\!\\frac{A\\Le k_2}{(k_3-k_2)(k_4-k_2)H_{1,\\lambda}}{\\mathscr I}_3;\\\\[1mm]\nF_4=&\\frac{\\Le k_5}{H_{3,\\lambda}}{\\mathscr I}_5-\\frac{\\Le k_4}{H_{2,\\lambda}}{\\mathscr I}_4-\\frac{\\Le}{(1-\\Theta_i)H_{4,\\lambda}}{\\mathscr I}_1.\n\\end{align*}\nThis system is uniquely solvable if and only if $\\overline{D}(\\lambda;\\Theta_i,\\Le)=[\\Le (k_2-k_3)]^{-1}D(\\lambda; \\Theta_i, \\Le)$, the\ndeterminant of the matrix in left-hand side of \\eqref{matrix}, does not vanish, where\n$D(\\lambda; \\Theta_i, \\Le)$ is defined in \\eqref{d,r,Le}.\nHence, the solutions to the equation $D(\\lambda; \\Theta_i, \\Le)=0$ are elements of the point spectrum of $L$. Property (3) is proved.\nOn the other hand, as it is easily seen, if $\\lambda\\notin (-\\infty,-1\/4]\\cup {\\mathcal P}$ is not a root of the dispersion relation, then\nit is easy to check that the function $\\pmb u$ given by \\eqref{re-u-negative}-\\eqref{matrix} belongs to $D(L)$, so that $\\lambda$ is an element of the resolvent set of operator $L$.\n\nFinally, we consider the case when $k_3=k_1$, which gives $\\lambda=\\lambda_{\\pm}:=-\\frac{A\\Le}{\\Le-1}\\pm \\frac{i\\sqrt{A\\Le(\\Le-1)}}{\\Le-1}$ (see Appendices \\ref{appendix-A} and \\ref{appendix-B}).\nIt is easy to check that this pair of conjugate complex numbers does not belong to ${\\mathcal P}$.\nIt thus follows that $u$ for $\\xi\\ge 0$ and $v$ for $\\xi\\in\\R$ are still given by \\eqref{re-v-negative}, \\eqref{re-u-positive} and \\eqref{re-v-positive}.\nOn the other hand, for $\\xi<0$, $u$ is given by\n\\begin{align*}\nu(\\xi)=&c_1e^{k_1\\xi}-\\frac{Ac_3}{H_{1,\\lambda}}\\xi e^{k_1\\xi}+\\frac{e^{k_1\\xi}}{H_{1,\\lambda}}\\int_{\\xi}^0f_1(s)e^{-k_1s}ds+\\frac{e^{k_2\\xi}}{H_{1,\\lambda}}\\int_{-\\infty}^{\\xi}f_1(s)e^{-k_2s}ds\\\\\n&+\\frac{A\\Le\\, e^{k_1\\xi}}{H_{1,\\lambda}H_{2,\\lambda}}\\int_{\\xi}^0(s-\\xi)f_2(s)ds-\\frac{A\\Le\\, e^{k_1\\xi}}{H_{1,\\lambda}H_{2,\\lambda}^2}\\int_{-\\infty}^0f_2(s)e^{-k_4s}ds\\\\\n&+\\frac{A\\Le\\ e^{k_1\\xi}}{H_{1,\\lambda}H_{2,\\lambda}^2}\\int_{\\xi}^0f_2(s)e^{-k_1s}ds+\\frac{A\\Le\\, e^{k_4\\xi}}{H_{1,\\lambda}H_{2,\\lambda}^2}\\int_{-\\infty}^{\\xi}f_2(s)e^{-k_4s}ds\\\\\n&+\\frac{A}{H_{1,\\lambda}}\\bigg\\{\n\\frac{e^{k_1\\xi}}{k_1-k_2}c_3+\\frac{\\Le}{H_{2,\\lambda}}\\bigg[\n\\frac{e^{k_4\\xi}}{k_4-k_2}\\int_{-\\infty}^{\\xi}f_2(s)e^{-k_4s}ds-\\frac{e^{k_1\\xi}}{k_1-k_2}\\int_{\\xi}^0 f_2(s)e^{-k_1s}ds \\nonumber\\\\\n&\\phantom{+\\frac{A}{H_{1,\\lambda}}\\bigg\\{\\frac{e^{k_1\\xi}}{k_1-k_2}c_3+\\frac{\\Le}{H_{2,\\lambda}}\\bigg[\\;\\,}+ \\frac{(k_4-k_1)e^{k_2\\xi}}{(k_1-k_2)(k_4-k_2)}\\int_{-\\infty}^{\\xi}f_2(s)e^{-k_2s}ds\\bigg]\\bigg\\}.\n\\end{align*}\nNotice that $\\sup_{\\xi<0}e^{\\frac{1}{2}\\xi}|u(\\xi)|<\\infty$; therefore, $\\pmb u$ belongs to $\\bm{\\mathcal W}$. Imposing the boundary conditions, we get a linear system for the unknowns $(c_1,c_3,c_6,c_8)$, whose matrix is the same as in\n\\eqref{matrix}. Since the determinant is not zero when $\\lambda=\\lambda_{\\pm}$ (see Appendix \\ref{appendix-B}) and the first- and second-order derivatives of\n $\\pmb u$ belong to $\\pmb {\\mathcal W}$, we conclude that $\\lambda_{\\pm}$ are in the resolvent set of operator $L$.\n\n\\vskip 1mm\n{\\em Step 3}.\nNow, we proceed to show that $0$ is an isolated simple eigenvalue of the operator $L$.\nIn view of the previous steps, in a neighborhood of $\\lambda=0$ the solution $\\pmb u=R(\\lambda,L)\\pmb f$ of the equation $\\lambda \\pmb u-L\\pmb u=\\pmb f$ is given by\n\\eqref{re-u-negative}-\\eqref{re-v-positive} for any $\\pmb f\\in\\bm{\\mathcal W}$, where\n\\begin{align*}\nc_1=&\\frac{\\Le (k_2\\!-\\!k_3)}{D(\\lambda;\\Theta_i,\\Le)}\\bigg\\{\n\\bigg [\\frac{(k_6\\!-\\!k_3)(1-\\Theta_i)}{\\Le}\\!-\\!\\frac{A}{(k_3\\!-\\!k_2)H_{1,\\lambda}}\\bigg]{\\mathscr I}_1\\!+\\!\\frac{k_6\\!-\\!k_3}{\\Le H_{1,\\lambda}}{\\mathscr I}_2\n\\!-\\!\\frac{A(k_6\\!-\\!k_3)}{(k_3\\!-\\!k_2)(k_4\\!-\\!k_2)H_{1,\\lambda}}{\\mathscr I}_3\\\\\n&\\phantom{\\frac{\\Le (k_2\\!-\\!k_3)}{D(\\lambda;\\Theta_i,\\Le)}\\bigg\\{\\,}\n+\\frac{A}{H_{1,\\lambda}H_{2,\\lambda}}\\bigg(\\frac{k_6-k_3}{k_4-k_2} -\\frac{k_6-k_4}{k_3-k_2} \\bigg){\\mathscr I}_4-\\frac{A}{(k_3-k_2)H_{1,\\lambda}}{\\mathscr I}_5\\bigg\\},\\\\[1mm]\nc_3=&\\frac{\\Le (k_2\\!-\\!k_3)}{D(\\lambda;\\Theta_i,\\Le)}\\bigg\\{{\\mathscr I}_1+{\\mathscr I}_2-\\frac{A\\Le}{(k_4-k_2)(k_3-k_2)}{\\mathscr I}_3\\\\\n&\\phantom{\\frac{\\Le (k_2\\!-\\!k_3)}{D(\\lambda;\\Theta_i,\\Le)}\\bigg\\{\\;\\,}\n+\\frac{1}{H_{2,\\lambda}}\\bigg [(k_6\\!-\\!k_4)\\big[1-H_{1,\\lambda}(1-\\Theta_i)\\big]\\!+\\!\\frac{A\\Le }{k_4-k_2} \\bigg]{\\mathscr I}_4\n\\!+\\!\\big[1-H_{1,\\lambda}(1-\\Theta_i)\\big]{\\mathscr I}_5\\bigg\\},\\\\[1mm]\nc_6=&\\frac{\\Le (k_2\\!-\\!k_3)}{D(\\lambda;\\Theta_i,\\Le)}\\bigg\\{\\frac{1}{H_{1,\\lambda}}\\bigg (\\frac{A}{k_3-k_2}+\\frac{k_6-k_3}{\\Le}\\bigg ){\\mathscr I}_1\n\\!+\\!\\frac{(k_6\\!-\\!k_3)(1\\!-\\!\\Theta_i)}{\\Le}{\\mathscr I}_2\\!-\\!\\frac{A(k_6\\!-\\!k_3)(1\\!-\\!\\Theta_i)}{(k_3\\!-\\!k_2)(k_4\\!-\\!k_2)}{\\mathscr I}_3\n\\\\\n&\\phantom{\\frac{\\Le (k_2-k_3)}{D(\\lambda;\\Theta_i,\\Le)}\\bigg\\{\\;\\,}\n\\!+\\!\\frac{A(1\\!-\\!\\Theta_i)}{H_{2,\\lambda}}\\bigg(\\frac{k_6\\!-\\!k_3}{k_4\\!-\\!k_2}\\!-\\!\\frac{k_6\\!-\\!k_4}{k_3\\!-\\!k_2}\\bigg){\\mathscr I}_4\n-\\frac{A(1\\!-\\!\\Theta_i)}{k_3\\!-\\!k_2}{\\mathscr I}_5\n\\bigg\\},\\\\[1mm]\nc_8=&\\frac{\\Le (k_2\\!-\\!k_3)}{D(\\lambda;\\Theta_i,\\Le)}\\bigg\\{{\\mathscr I}_1\\!+\\!{\\mathscr I}_2\\!-\\!\\frac{A\\Le}{(k_3\\!-\\!k_2)(k_4\\!-\\!k_2)}{\\mathscr I}_3+\\bigg[1\\!-\\!H_{1,\\lambda}(1\\!-\\!\\Theta_i)\\!+\\!\\frac{A\\Le}{(k_3\\!-\\!k_2)(k_4\\!-\\!k_2)}\\bigg]{\\mathscr I}_4\n\\\\\n&\\phantom{\\frac{\\Le (k_2\\!-\\!k_3)}{D(\\lambda;\\Theta_i,\\Le)}\\bigg\\{}\n+\\bigg[\\frac{A\\Le}{(k_3-k_2)H_{3,\\lambda}}\n+[1-H_{1,\\lambda}(1-\\Theta_i)]\\bigg (1+\\frac{k_6-k_3}{H_{3,\\lambda}}\\bigg )\\bigg]{\\mathscr I}_5\\bigg\\}.\n\\end{align*}\nAs it is immediately seen, the function $D(\\cdot;\\Theta_i,\\Le)$ is analytic in a neighborhood of $\\lambda=0$, which is simple zero of such a function, and the other functions appearing in \\eqref{re-u-negative}-\\eqref{re-v-positive} are holomorphic in a neighborhood of $\\lambda=0$. Hence, we conclude that zero is a simple pole of the resolvent operator $R(\\lambda,L)$.\nSince $\\displaystyle\\frac{d\\pmb U}{d\\xi}$ belongs to the kernel of $L$ (see Remark \\ref{rem-simple}) and the matrix in \\eqref{matrix} has rank three at $\\lambda=0$, this function\ngenerates the kernel, so that the geometric multiplicity of the eigenvalue $\\lambda=0$ is one. This is enough to conclude that\n$\\lambda=0$ is a simple eigenvalue of $L$. Property (2) is established and the spectrum of $L$ is completely characterized.\n\n\\vskip 1mm\n{\\em Step 4}. In order to prove that $L$ is sectorial, it is sufficient to show that there exist two positive constants $C$ and $M$ such that\n\\begin{align}\n\\label{resolvent estimate}\n\\|R(\\lambda,L)\\|_{L(\\bm{\\mathcal W})}\\le\nC|\\lambda|^{-1},\\qquad\\;\\,\\Re{\\lambda}\\ge M.\n\\end{align}\nWithout loss of generality, we can assume that $k_{1,\\lambda}\\neq k_{3,\\lambda}$ and the conditions in \\eqref{cond-1} are all satisfied if $\\Re\\lambda\\ge M$.\nThroughout this step, $C_j$ denotes a positive constant, independent of $\\lambda$ and $\\pmb f\\in\\bm{\\mathcal W}$.\n\nWe begin by estimating the terms $H_{j,\\lambda}$ ($j=1,2,3$). As it is easily seen,\n\\begin{align}\n|H_{2,\\lambda}|\\ge\\Re(H_{2,\\lambda})=\\sqrt{\\frac{|{\\Le}^2+4\\Le(A+\\lambda)|+{\\Le}^2+4\\Le(A+\\Re\\lambda)}{2}}\\ge\\sqrt{2\\Le|\\lambda|}\n\\label{estim-H1}\n\\end{align}\nfor any $\\lambda\\in\\C$ with positive real part.\nSince $H_{1,\\lambda}$ and $H_{3,\\lambda}$ can be obtained from $H_{2,\\lambda}$, by taking, $(\\Le,A)=(1,0)$ and $(\\Le,A)=(\\Le,0)$ respectively, we also deduce that\n\\begin{align}\n|H_{1,\\lambda}|\\ge\\Re(H_{1,\\lambda})\\ge\\sqrt{2|\\lambda|},\\qquad\\;\\,|H_{3,\\lambda}|\\ge\\Re(H_{3,\\lambda})\\ge\\sqrt{2\\Le|\\lambda|}\n\\label{estim-H2-H3}\n\\end{align}\nfor the same values of $\\lambda$.\nThanks to \\eqref{estim-H1} and \\eqref{estim-H2-H3}, we can easily estimate the terms\n${\\mathscr I}_j$ $(j=1,\\ldots,5)$. Indeed, since $\\Re(k_1)+1\/2>0$, we obtain\n\\begin{align*}\n|{\\mathscr I}_1|&=\\bigg |\\int_0^{\\infty}f_1(s)e^{-k_1s}ds\\bigg |\\le \\sup_{\\xi>0}e^{\\frac{1}{2}\\xi}|f_1(\\xi)|\\int_0^{\\infty}e^{-\\frac{1}{2}\\Re(H_{1,\\lambda})s}ds \\le C_1|\\lambda|^{-\\frac{1}{2}}\\|\\pmb f\\|_{\\bm{\\mathcal W}}.\n\\end{align*}\nThe other terms ${\\mathscr I}_j$ can be treated likewise and we get\n$\\sum_{j=2}^5|{\\mathscr I}_j|\\le C_2|\\lambda|^{-\\frac{1}{2}}\\|\\pmb f\\|_{\\bm{\\mathcal W}}$ for every $\\pmb f\\in\\bm{\\mathcal W}$ and $\\lambda\\in\\C$ with positive real part.\n\nNext, we turn to the function $D(\\cdot;\\Theta_i,\\Le)$. We observe that\n\\begin{align*}\n|D(\\lambda;\\Theta_i,\\Le)|\\ge [(1-\\Theta_i)\\sqrt{|1+4\\lambda|}-1]|k_{6,\\lambda}-k_{3,\\lambda}||k_{3,\\lambda}-k_{2,\\lambda}|-A\\Le\n\\end{align*}\nfor any $\\lambda\\in\\C$.\nTaking \\eqref{estim-H1} and \\eqref{estim-H2-H3} into account, we can show that\n\\begin{equation}\nC_3\\sqrt{|\\lambda|}\\le |k_{3,\\lambda}-k_{2,\\lambda}|+|k_{3,\\lambda}-k_{6,\\lambda}|\\le C_4\\sqrt{|\\lambda|}\n\\label{estimate-k1-k4}\n\\end{equation}\nfor $\\lambda\\in\\C$ with sufficiently large positive real part. Hence, for such values of $\\lambda$'s we can continue the previous inequality and get\n\\begin{align}\n|D(\\lambda;\\Theta_i,\\Le)|\\ge C_5|\\lambda|^{\\frac{3}{2}}.\n\\label{estimate-D}\n\\end{align}\nSimilarly, $|k_{6,\\lambda}-k_{4,\\lambda}|\\le C_6\\sqrt{|\\lambda|}$\nfor any $\\lambda$ with positive real part and\n\\begin{equation}\n|k_{4,\\lambda}-k_{2,\\lambda}|\\ge \\frac{1}{2}|H_{2,\\lambda}|-\\frac{1}{2}|H_{1,\\lambda}|-\\frac{\\Le-1}{2}\n\\ge \\sqrt{\\frac{\\Le|\\lambda|}{2}}-\\sqrt{\\frac{|\\lambda|}{2}}-\\frac{\\Le-1}{2}\\ge C_7\\sqrt{|\\lambda|},\n\\label{estim-k2-k4}\n\\end{equation}\nif $\\Re\\lambda$ is sufficiently large. From \\eqref{estim-H1}-\\eqref{estim-k2-k4} we infer that\n$|c_1|+|c_3|+|c_6|+|c_8|\\le C_8|\\lambda|^{-1}$ for any $\\lambda\\in\\C$ with $\\Re(\\lambda)\\ge M$ and\na suitable positive constant $M$. Further, observing that\n\\begin{eqnarray*}\n|k_{3,\\lambda}-k_{1,\\lambda}|+|k_{4,\\lambda}-k_{1,\\lambda}|\\ge C_9\\sqrt{|\\lambda|},\\qquad\\;\\,|k_{4,\\lambda}-k_{3,\\lambda}|\\le C_{10}\\sqrt{|\\lambda|},\n\\end{eqnarray*}\nwe are now able to estimate the functions $u$ and $v$ in \\eqref{re-u-negative}-\\eqref{re-v-positive} and show that\n\\eqref{resolvent estimate} holds true. The proof is complete.\n\\end{proof}\n\n\\begin{remark}\n\\rm{It is worth pointing out that, as $\\Le \\to \\infty$, the set ${\\mathcal P}$ degenerates into a vertical line $\\Re \\lambda=-\\Theta_i(1-\\Theta_i)^{-1}-1\/2$. In the limit case, the system is partly parabolic and the semigroup is not analytic, see, e.g., \\cite[Section 1, p. 2435]{GLS10}.}\n\\end{remark}\n\n\\setcounter{tocdepth}{2}\n\\section{The fully nonlinear problem}\n\\label{sect-3}\nOur goal in this section is to get rid of the eigenvalue 0 and then derive a new fully nonlinear problem.\nWe recall that the eigenvalue $0$ is related to the translation invariance of the traveling wave. In a first step, we use here a method similar to that of \\cite{BLS92} or \\cite[p. 358]{Lunardi96}.\n\n\\subsection{Ansatz revisited: elimination of the eigenvalue $0$}\n\\label{subsect-2.3}\nIt is convenient to write System \\eqref{u_1}-\\eqref{u_2} with notation $\\pmb u=(u_1,u_2)$, $\\pmb U=(\\Theta^0,\\Phi^0)$, see Section \\ref{fixed}, in an abstract form:\n\\begin{equation}\n\\label{v}\n\\dot{\\pmb u}=L\\pmb u+\\dot{s}{\\pmb U}^{\\prime}+\\dot{s}{\\pmb u}^{\\prime}.\n\\end{equation}\nNote that, in view of \\eqref{interface-u}, $\\pmb u(\\tau,\\cdot)$ belongs to $D(L)$ for each $\\tau$.\nSince 0 is an isolated simple eigenvalue of $L$, we can introduce the spectral projection $P$ onto the kernel of $L$, defined by\n$P\\pmb f=\\langle\\pmb f,{\\pmb e^{*}}\\rangle\\pmb U^{\\prime}$ for every $\\pmb f\\in\\bm{\\mathcal W}$ and a unique ${\\pmb e^{*}}\\in \\bm{\\mathcal W}^*$, the dual space of $\\bm{\\mathcal W}$, such that $\\langle\\pmb U^{\\prime},{\\pmb e^{*}}\\rangle=1$. For further use, we recall that $P$ commutes with $L$ on $D(L)$.\nWe are going to apply the projections $P$ and $Q=I-P$ to System ($\\ref{v}$) to remove the eigenvalue $0$.\n\n\\medskip\n\\paragraph{\\bf Ansatz 2} We split $\\pmb u$ into $\\pmb u(\\tau,\\cdot)=P\\pmb u(\\tau,\\cdot)+Q\\pmb u(\\tau,\\cdot)=p(\\tau)\\pmb U^{\\prime}+\\pmb w(\\tau,\\cdot)$, i.e.,\n\\begin{align}\nu_1(\\tau, \\xi)=&p(\\tau)\\frac{d\\Theta^0}{d\\xi}(\\xi)+w_1(\\tau, \\xi),\\label{splitting-w1}\\\\\n u_2(\\tau, \\xi)=&p(\\tau)\\frac{d\\Phi^0}{d\\xi}(\\xi)+w_2(\\tau, \\xi),\\notag\n\\end{align}\nwhere $p(\\tau)=\\langle \\pmb u(\\tau),{\\pmb e^{*}}\\rangle $ and $\\pmb w=(w_1,w_2)$. Clearly, $\\pmb w(\\tau,\\cdot)\\in Q(D(L))$ for each $\\tau$.\nIt follows from ($\\ref{v}$) that\n\\begin{equation}\n\\dot{p}=\\dot{s}+\\dot{s}\\langle\\pmb u^{\\prime},{\\pmb e^{*}}\\rangle,\\qquad\\;\\,\n\\dot{\\pmb w}=L\\pmb w+\\dot{s}Q\\pmb u^{\\prime},\n\\label{w}\n\\end{equation}\na Lyapunov-Schmidt-like reduction of the problem. We point out that the above procedure generates a new ansatz slightly different from ansatz 1 (see \\eqref{ansatz1}) that helps us determine the functional framework.\n\nThanks to new ansatz 2, we are going to derive an equation for $\\pmb w$ in the space $\\bm{\\mathcal W}$. Now, the spectrum of the part of $L$ in $Q(\\bm{\\mathcal W})$ does not contain the eigenvalue $0$.\n\n\\subsection{Derivation of the fully nonlinear equation}\nTo get a self-contained equation for $\\pmb w$, we need to eliminate $\\dot{s}$ from the right-hand side of the second equation in \\eqref{w}. For this purpose,\nwe begin by evaluating the first component of \\eqref{w} at $\\xi=0^+$ to get\n\\begin{align}\n\\frac{\\partial w_1}{\\partial\\tau}(\\cdot,0^+)=&(L\\pmb w)_1(\\cdot,0^+)+\\dot{s}(Q\\pmb u')_1(\\cdot,0^+)\\notag\\\\\n=&(L\\pmb w)_1(\\cdot,0^+)+\\dot{s}\\frac{\\partial u_1}{\\partial \\xi}(\\cdot,0^+)+\\dot{s}\\langle\\pmb u',{\\pmb e^{*}}\\rangle\\Theta_i.\n\\label{e}\n\\end{align}\nNext, we observe that the function $w_1$ is continuous (but not differentiable) at $\\xi=0$, since both $\\pmb u$ and $\\pmb U'$ are continuous at $\\xi=0$. Therefore, evaluating \\eqref{splitting-w1} at $\\xi=0$ and recalling that $u_1(\\tau,0)=0$ (see \\eqref{interface-u}), we infer that\n$w_1(\\tau,0)=\\Theta_ip(\\tau)$. Differentiating this formula yields\n\\begin{align}\n\\frac{\\partial w_1}{\\partial\\tau}(\\cdot,0)=\\dot{p}\\Theta_i=\\dot{s}\\Theta_i+\\dot{s}\\langle \\pmb u',{\\pmb e^{*}}\\rangle\\Theta_i,\n\\label{key}\n\\end{align}\nFrom ($\\ref{e}$) and ($\\ref{key}$), it follows that\n\\begin{equation}\n\\label{s}\n\\dot{s}\\Theta_i=(L\\pmb w)_1(\\cdot,0^+)+\\dot{s}\\frac{\\partial u_1}{\\partial \\xi}(\\cdot,0^+).\n\\end{equation}\nTo get rid of the spatial derivatives of $u_1$ from the right-hand side of \\eqref{s}, we use \\eqref{splitting-w1} to write\n\\begin{eqnarray}\n\\label{u_1,w_1}\n\\frac{\\partial u_1}{\\partial \\xi}(\\cdot,0^+)=p\\frac{d^2\\Theta^0}{d\\xi^2}(0^+)+w'_1(\\cdot,0^+)\n=w_1(\\cdot,0)+w'_1(\\cdot,0^+).\n\\end{eqnarray}\nPlugging ($\\ref{u_1,w_1}$) into ($\\ref{s}$), we finally obtain the formula\n\\begin{equation}\\label{shift}\n\\dot s=\\frac{(L\\pmb w)_1(\\cdot,0^+)}{\\Theta_i-w_1(\\cdot,0)-w'_1(\\cdot,0^+)},\n\\end{equation}\nwhich can be regarded as a underlying \\textit{second-order Stefan condition}, see \\cite{BL18}.\nHence, replacing it in ($\\ref{w}$), we get\n\\begin{align*}\n\\frac{\\partial\\pmb w}{\\partial\\tau}\n=&L\\pmb w+\\frac{(L\\pmb w)_1(\\cdot,0^+)}{\\Theta_i-w_1(\\cdot,0)-w'_1(\\cdot,0^+)}Q\\pmb u' \\nonumber\\\\\n=&L\\pmb w+\\frac{(L\\pmb w)_1(\\cdot,0^+)}{\\Theta_i-w_1(\\cdot,0)-w'_1(\\cdot, 0^+)}Q\\bigg (\\frac{w_1(\\cdot,0)}{\\Theta_i}\\pmb{U''}+\\pmb{w'}\\bigg ),\n\\end{align*}\nwhich is a fully nonlinear parabolic equation in the space $\\bm{\\mathcal W}$ written in a more abstract form:\n\\begin{equation}\\label{FNLE}\n\\frac{\\partial\\pmb w}{\\partial\\tau} = L\\pmb w + {F}(\\pmb w), \\quad {\\pmb w}\\in Q(D(L)).\n\\end{equation}\nand is going to be the subject of our attention.\nNote that Equation \\eqref{FNLE} is fully nonlinear since the function $F$ depends on $\\pmb w$ also through the limit at $0^+$ of $L\\pmb w$.\nMoreover, the operator $L$ is sectorial in $Q(\\bm{\\mathcal W})$.\nHence, we can take advantage of the theory of analytic semigroups to solve Equation \\eqref{FNLE}. We refer the reader to \\cite[Chapter 4]{Lunardi96} for further details.\n\n\n\\setcounter{tocdepth}{2}\n\\section{Stability of the traveling wave solution}\\label{stability}\nThis section is devoted to the analysis of the stability of the traveling wave solution $\\pmb U$. Here, stability refers to orbital stability with asymptotic phase $s_{\\infty}$.\t\n From now on, we focus on the asymptotic situation where the Lewis number, $\\Le$, is large and, in this respect, we use the notation $\\varepsilon = 1\/\\Le$ to stand for a small perturbation parameter. Simultaneously, we assume that $\\Theta_i$ is close to the burning temperature normalized at unity, which is physically relevant (see \\cite[Section 3.2, Fig. 5]{BGKS15}). More specifically, we restrict $\\Theta_i$ to the domain $\\frac{2}{3}<\\Theta_i <1$.\n\n\nIn what follows, we introduce \\hbox{$m:= \\Theta_i\/(1-\\Theta_i)$} as the \\textit{bifurcation parameter} which runs in the interval $(2,\\infty)$, due to the choice of $\\Theta_i$.\nWith the above notation, $A =m+\\varep m^2$ and the \\textit{dispersion relation} $D(\\lambda;\\Theta_i,\\Le)$ (see \\eqref{d,r,Le}) in Section 2 reads:\n\\begin{align}\n\\label{dispersion epsilon}\nD_{\\varepsilon}(\\lambda;m) =&-\\frac{1}{4}\\big(\\sqrt{1+4\\varepsilon(m+\\varepsilon m^2+\\lambda)}+\\sqrt{1+4\\varepsilon \\lambda}\\big)\\notag\\\\\n&\\qquad\\times\\bigg(\\frac{1}{\\varepsilon}[\\sqrt{1+4\\varepsilon(m+\\varepsilon m^2+\\lambda)}-1]\\!+\\!1\\!+\\!\\sqrt{1+4\\lambda}\\bigg)\\!\n\\bigg(1\\!-\\!\\frac{\\sqrt{1+4\\lambda}}{1+m}\\bigg )\\!+\\!m\\!+\\!\\varepsilon m^2.\n\\end{align}\n\nThis section is split into two parts. First, we study the stability of the null solution of the fully nonlinear equation \\eqref{FNLE}. Second, we turn our attention to the stability of the traveling wave.\n\n\\subsection{Stability of the null solution of (\\ref{FNLE})}\nTo begin with, we recall that the spectrum of the part of $L$ in $\\bm{\\mathcal W}_Q:=Q(\\bm{\\mathcal W})$ is the set\n\\begin{align*}\n\\left (-\\infty,-{\\textstyle \\frac{1}{4}}\\right ]\\cup\\mathcal{P}\\cup\\{\\lambda\\in\\C\\setminus\\{0\\}:D_{\\varepsilon}(\\lambda;m)=0\\}.\n\\end{align*}\nAs we will show, the roots of the dispersion relation $D_{\\varepsilon}(\\cdot;m)$ are finitely many.\nAs a consequence, there is a gap between the spectrum of this operator and the imaginary axis (at least for $\\varepsilon$ small enough).\nIn view of the principle of linearized stability, the main step in the analysis of the stability of the null solution of Equation \\eqref{FNLE} is a deep insight in the solutions of the dispersion relation. More precisely, we need to determine when they are all contained in the left halfplane and when some of them lie in the right halfplane.\n\nThe limit critical value $m^c=6$ will play an important role in the analysis hereafter.\n\n\\begin{theorem}\n\\label{stability theorem FNLE}\nThe following properties are satisfied.\n\\begin{enumerate}[\\rm (i)]\n\\item\nLet $m\\in (2,m^c)$ be fixed. Then, there exists $\\varepsilon_0=\\varepsilon_0(m)>0$ such that, for $\\varepsilon\\in (0,\\varepsilon_0)$, the null solution of the fully nonlinear problem \\eqref{FNLE} is stable with respect to perturbations belonging to $Q(D(L))$.\n\\item\nLet $m>m^c$ be fixed. Then, there exists $\\varepsilon_1=\\varepsilon_1(m)$ small enough such that, for $\\varepsilon\\in (0,\\varepsilon_1)$, the null solution of \\eqref{FNLE} is unstable with respect to perturbations belonging to $Q(D(L))$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nTo begin with, we observe that the functions $D_{\\varepsilon}(\\cdot,m)$ are holomorphic in $\\C\\setminus (-\\infty,-1\/4]$ and therein they locally converge to the \\textit{limit dispersion relation} $D_0(\\cdot,m)$ defined by\n\\begin{align*}\nD_0(\\lambda;m)=&-\\frac{1}{2}[2(m+\\lambda)+1+\\sqrt{1+4\\lambda}]\\left (1-\\frac{\\sqrt{1+4\\lambda}}{1+m}\\right )+m\\notag\\\\\n=&\\frac{\\sqrt{1+4\\lambda}-1}{4(1+m)}[4\\lambda-(m-2)\\sqrt{1+4\\lambda}+m+2],\n\\end{align*}\t\nas $\\varepsilon\\to 0^+$.\nThe solutions of the equation $D_0(\\lambda;m)=0$ are $\\lambda=0$, for all $m$,\nand the roots of the second-order polynomial $4\\lambda^2+(6m-m^2)\\lambda+2m$, whose real part is not less than $-(m+2)\/4$.\nThis polynomial admits conjugate solutions $\\lambda_{1,2}= a(m) \\pm ib(m)$, where $a(m)=\\frac{1}{8}(m^2-6m)$ and $b(m)= \\frac{1}{8}(m-2)\\sqrt{|8m-m^2|}$, if $m\\in (2,8)$ and real solutions $\\lambda_{1,2}= a(m) \\pm b(m)$ otherwise. The coefficient $a(m)$ is negative whenever $26$. It can be easily checked\nthat ${\\rm Re}(\\lambda_{1,2})\\ge -(m+2)\/4$ for each $m\\in (2,\\infty)$, so that $\\lambda_{1,2}$ solve the equation $D_0(\\lambda; m)=0$. In particular, there are two conjugate purely imaginary roots $\\lambda_{1,2}=\\pm\\sqrt{3}i$ at $m=6$.\n\nWe can now prove properties (i) and (ii).\n\n(i) Fix $\\rho>0$ such that the closure of the disks of center $\\lambda_{1,2}$ and radius $\\rho$ is contained in $\\{\\Re z <0\\}\\backslash (-\\infty,-\\frac{1}{4}]$. Hurwitz Theorem (see, e.g., \\cite[Chapter 7, Section 2]{Conway78}) and the above results show that there exists $\\varepsilon_0 >0$ such that, for $\\varepsilon\\in (0,\\varepsilon_0)$, $D_{\\varepsilon}(\\lambda;m)$ admits exactly two conjugate complex roots $\\lambda_{1,2}(\\varepsilon)$ in the disk $|\\lambda-\\lambda_{i}|<\\rho$ and $\\lambda_{i}(\\varepsilon)$ converges to $\\lambda_i$, as $\\varepsilon \\to 0$, for $i=1,2$. Therefore, all the elements of the spectrum of the part of operator $L$\nin $\\bm{\\mathcal W}_Q$ have negative real parts, which implies that the operator norm of the restriction to $\\bm{\\mathcal W}_Q$ of the analytic semigroup $e^{\\tau L}$ generated by $L$, decays to zero with exponential rate as $t\\to\\infty$. Now, the nonlinear stability follows from applying a standard machinery: the solution of Equation \\eqref{FNLE}, with initial datum $\\pmb w_0$ in a small (enough) ball of $Q(D(L))$ centered at zero, is given by the variation-of-constants-formula\n\\begin{eqnarray*}\n\\pmb w(\\tau,\\cdot)=e^{\\tau L}\\pmb w_0+\\int_0^{\\tau}e^{(\\tau-s)L}F(\\pmb w(s,\\cdot))ds,\\qquad\\;\\,\\tau>0.\n\\end{eqnarray*}\nApplying the Banach fixed point theorem in the space\n\\begin{eqnarray*}\n\\bm{\\mathcal X}^{\\alpha}_{\\omega}\\!=\\!\\bigg\\{\\pmb w\\!\\in\\! C([0,\\infty);\\pmb{\\mathcal W}_Q):\\sup_{\\sigma\\in (0,1)}\\sigma^{\\alpha}\\|\\pmb w\\|_{C^{\\alpha}([\\sigma,1];D(L))}<\\infty: \\tau\\mapsto e^{\\omega\\tau}\\pmb w(\\tau,\\cdot)\\!\\in\\! C^{\\alpha}([1,\\infty);D(L))\\bigg\\},\n\\end{eqnarray*}\nendowed with the natural norm, where $\\alpha$ is fixed in $(0,1)$ and $\\omega$ is any positive number less than the real part of\n$\\lambda_1(\\varepsilon)$, allows us to prove the existence and uniqueness of a solution $\\pmb w$ of \\eqref{FNLE}, defined in $(0,\\infty)$ such that\n$\\|\\pmb w(\\tau,\\cdot)\\|_{\\pmb{\\mathcal W}}+\\|L{\\pmb w}(\\tau,\\cdot)\\|_{\\pmb{\\mathcal W}}\n\\le Ce^{-\\omega\\tau}\\|\\pmb w_0\\|_{D(L)}$ for $\\tau\\in (0,\\infty)$ and some positive constant $C$, which yields the claim. For further details see \\cite[Chapter 9]{Lunardi96}.\n\n(ii) For $m>m^c$, we use again Hurwitz Theorem to show that there exists $\\varepsilon_1=\\varepsilon_1(m)>0$ such that the equation\n$D_{\\varepsilon}(\\lambda,m)=0$ admits a solution with positive real part if $\\varepsilon\\in (0,\\varepsilon_1)$. More precisely, it admits a couple of conjugate complex roots with positive real parts, if $m<8$, a positive root, if $m=8$, and two real solutions if $m>8$. For these values of $\\varepsilon$, the restriction of the semigroup $e^{\\tau L}$ to $\\bm{\\mathcal W}_Q$ exhibits an exponential dichotomy, i.e., there exists a spectral projection $P_+$ which allows to split $\\bm{\\mathcal W}_Q=P_+(\\bm{\\mathcal W}_Q)\\oplus (I-P_+)(\\bm{\\mathcal W}_Q)$. The semigroup $e^{\\tau L}$ decays to zero with exponential rate when restricted to $(I-P)(\\bm{\\mathcal W}_Q)$, whereas the restriction of $e^{\\tau L}$ to $P_+(\\bm{\\mathcal W}_Q)$\nextends to a group which decays to zero with exponential rate as $\\tau\\to-\\infty$. Again with a fixed point technique, we can prove the existence of a nontrivial backward solution\n$\\pmb z$ of the nonlinear equation \\eqref{FNLE}, defined in $(-\\infty,0)$ such that\n$\\|\\pmb z(\\tau,\\cdot)\\|_{\\pmb{\\mathcal W}}+\\|L\\pmb z(\\tau,\\cdot)\\|_{\\pmb{\\mathcal W}}\\le C_{\\omega}e^{\\omega\\tau}$ for $\\tau\\in (-\\infty,0)$ and\nany $\\omega$ positive and smaller than the minimum of the positive real parts of the roots of the dispersion relation.\nThe sequence $(\\pmb z_n)$ defined by $\\pmb z_n=\\pmb z(-n,\\cdot)$ vanishes in $D(L)$ as $n\\to+\\infty$ and the solution $\\pmb w_n$ to \\eqref{FNLE} subject to the initial condition\n$\\pmb w_n(0,\\cdot)=\\pmb z_n$ exists at least in the time domain $[0,n]$, where it coincides with the function $\\pmb z(\\cdot-n,\\cdot)$. Thus,\nthe norm of $\\|\\pmb w_n\\|_{C([0,n];\\pmb{\\mathcal W}_Q)}$ is positive and far way from zero, uniformly with respect to $n\\in\\N$, whence the instability of the trivial solution of\n\\eqref{FNLE} follows. Again, we refer the reader to \\cite[Chapter 9]{Lunardi96} for further results.\n\\end{proof}\n \t\n\\subsection{Stability of the traveling wave}\nWe can now rewrite the results in Theorem \\ref{stability theorem FNLE} in terms of problem\n\\eqref{perturbation T}-\\eqref{interface-Theta-Phi}.\n\\begin{theorem}\n\\label{stability theorem TW}\nThe following properties are satisfied.\n\\begin{enumerate}[\\rm (i)]\n\\item\nFor $m\\in (2,m^c)$ fixed, there exists $\\varepsilon_0=\\varepsilon_0(m)>0$ such that, for $\\varepsilon\\in (0,\\varepsilon_0)$, the traveling wave solution $\\pmb U$ is orbitally stable with asymptotic phase $s_{\\infty}$ $($see \\eqref{s-infty}$)$, with respect to perturbations belonging to the weighted space $D(L)$.\n\\item\nFor $m>m^c$ fixed, there exists $\\varepsilon_1=\\varepsilon_1(m)$ small enough such that, for $\\varepsilon\\in (0,\\varepsilon_1)$, the traveling wave $\\pmb U$ is unstable.\nwith respect to perturbations belonging to the weighted space $D(L)$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n(i) Let us fix $\\pmb w_0\\in Q(D(L))$ with $\\|\\pmb w_0\\|_{D(L)}$ small enough, so that Theorem \\ref{stability theorem FNLE}(i) can be applied.\nDenote by $\\pmb w$ the classical solution to Equation \\eqref{FNLE} which satisfies the initial condition $\\pmb w(0,\\cdot)=\\pmb w_0=(w_{0,1},w_{0,2})$.\nObserve that, since $p=\\Theta_i^{-1}w_1(\\cdot,0)$ (see Subsection \\ref{subsect-2.3}) it follows that the problem \\eqref{v}, subject to the initial condition $\\pmb u(0,\\cdot)=\\Theta_i^{-1}w_{0,1}\\pmb U'+\\pmb w_0$, admits a unique classical solution $(\\pmb u,s)$, where $\\pmb u$ decreases to zero as $\\tau\\to\\infty$, with exponential rate. Moreover, using \\eqref{shift} it is immediate to check\nthat $s(\\tau)$ converges to\n\\begin{equation}\ns_\\infty=\\int_{0}^{\\infty}\\frac{(L\\pmb w)_1(\\tau,0^+)}{\\Theta_i-w_1(\\tau,0)-w'_1(\\tau,0^+)}d\\tau,\n\\label{s-infty}\n\\end{equation}\nas $\\tau\\to\\infty$ (assuming for simplicity that $g$ vanishes at $\\tau=0$).\nWe point out that $s_{\\infty}$ depends on the initial condition.\n\nComing back to problem \\eqref{perturbation T}-\\eqref{interface-Theta-Phi} with initial condition ${\\pmb X}(0)=\\pmb u_0+\\pmb U$ and $g(0)=0$, we easily see that the solution ${\\pmb X}=(\\Theta,\\Phi)$ is defined by\n\\begin{align*}\n&{\\pmb X}=p\\pmb U'+\\pmb w+\\pmb U=\\Theta_i^{-1}w_1(\\cdot,0)\\pmb U'+\\pmb w+\\pmb U,\\\\\n&g(\\tau)=\\tau+\\int_0^{\\tau}\\frac{(L\\pmb w)_1(\\sigma,0^+)}{\\Theta_i-w_1(\\sigma,0)-w'_1(\\sigma,0^+)}d\\sigma,\\qquad\\;\\,\\tau\\ge 0.\n\\end{align*}\nFrom this formula and the above result, the claim follows at once.\n\n(ii) The proof is similar to that of property (i) and, hence, it is left to the reader.\n\\end{proof}\n\n\\setcounter{tocdepth}{2}\n\n\\section{Hopf bifurcation}\n\\label{sect-5}\nThis section is devoted to investigating the dynamics of the perturbation of the traveling wave in a neighborhood, say $(6-\\delta, 6+\\delta)$, of the limit critical value $m^c=6$ (see Section \\ref{stability}). As regards parameter $m$, the situation is more complicated than in Section 4 when it was fixed. Now, the dispersion relation ${D}_{\\varepsilon}(\\lambda;m)$ can be seen as a sequence of analytic functions parameterized by $m$. The main difficulty here is that Hurwitz Theorem does not a priori apply, particularly because of the lack of uniformity of ${D}_{\\varepsilon}(\\lambda;m)$ with respect to $\\varep$ and $m$. We especially find a proper approach to combining $m$ with $\\varep$: we construct in Proposition \\ref{give critical value} a sequence of critical values $m^c(\\varep)$ such that $m^c(0)=m^c$ and apply Hurwitz Theorem to the sequence $D_{\\varep}(\\lambda,m^c(\\varep))$. This proposition will be crucial for proving the existence of a Hopf bifurcation (see Theorem \\ref{Hopf bifurcation theorem}).\n\n\\subsection{Local analysis of the dispersion relation}\\label{p7}\nWe look for the roots of the \\textit{dispersion relation}, see \\eqref{dispersion epsilon},\nin a neighborhood of $m^c=6$ and of $\\lambda = \\pm i\\sqrt{3}$, for $\\varep>0$ small enough. A natural idea is to turn the dispersion relation into a polynomial by squaring, however the price to pay is double: the polynomial will be of high order without algebraic solution, and\nspurious roots therefore appear.\n\nFor convenience, we rewrite the equation $D_{\\varepsilon}(\\lambda;m)=0$ into a much more useful form. Replacing\n$\\sqrt{1+4\\varepsilon(m+\\varepsilon m^2+\\lambda)}+\\sqrt{1+4\\varepsilon \\lambda}$ by\n$4\\varepsilon(m+\\varepsilon m^2)(\\sqrt{1+4\\varepsilon(m+\\varepsilon m^2+\\lambda)}-\\sqrt{1+4\\varepsilon \\lambda})^{-1}$ with some straightforward algebra\nwe obtain the equivalent equation\n\\begin{equation}\n\\sqrt{1+4\\varepsilon \\lambda}-\\frac{1}{1+m}\\sqrt{1+4\\varepsilon(m+\\varepsilon m^2+\\lambda)}\\sqrt{1+4\\lambda}+\\frac{1+\\varepsilon m}{1+m}\\sqrt{1+4\\lambda}\n=\\varepsilon\\frac{1+4\\lambda}{1+m}+1-\\varepsilon.\n\\label{zeta}\n\\end{equation}\n\nIf we denote by $\\zeta$ the right-hand side of \\eqref{zeta} and set\n\\begin{align*}\n\\Sigma_1=&1+4\\varepsilon\\lambda+\\frac{2+6\\varepsilon m+5\\varepsilon^2 m^2+4\\varepsilon\\lambda}{(1+m)^2}(1+4\\lambda),\\\\[1mm]\n\\Sigma_2=&\\frac{1+4\\lambda}{(1+m)^2}\\bigg [(2+6\\varepsilon m+5\\varepsilon^2 m^2+4\\varepsilon\\lambda)(1+4\\varepsilon\\lambda)\n+\\frac{[1+4\\varepsilon(m+\\varepsilon m^2+\\lambda)](1+\\varepsilon m)^2}{(1+m)^2}(1+4\\lambda)\\bigg ],\\\\[1mm]\n\\Sigma_3=&\\frac{[1+4\\varepsilon(m+\\varepsilon m^2+\\lambda)](1+\\varepsilon m)^2}{(1+m)^4}(1+4\\varepsilon\\lambda)(1+4\\lambda)^2.\n\\end{align*}\nSquaring both sides of \\eqref{zeta} and rearranging terms we get the equation\n\\begin{align}\n\\zeta^2-\\Sigma_1=\\frac{2\\sqrt{1+4\\lambda}}{1+m}\\bigg \\{&\\sqrt{1+4\\varepsilon \\lambda}[1+\\varepsilon m-\\sqrt{1+4\\varepsilon(m+\\varepsilon m^2+\\lambda)}]\\notag\\\\\n&-\\frac{1+\\varepsilon m}{1+m}\\sqrt{1+4\\lambda}\\sqrt{1+4\\varepsilon(m+\\varepsilon m^2+\\lambda)}\\bigg \\}.\n\\label{zeta-1}\n\\end{align}\nSquaring both sides of \\eqref{zeta-1} and rearranging terms gives\n\\begin{align}\n(\\zeta^2-\\Sigma_1)^2-4\\Sigma_2=\\frac{8\\sqrt{1+4\\varepsilon \\lambda}(1+4\\lambda)}{(1+m)^2}\\bigg [&\n\\frac{[1+4\\varepsilon(m+\\varepsilon m^2+\\lambda)](1+\\varepsilon m)}{1+m}\\sqrt{1+4\\lambda}\\notag\\\\\n&-\\frac{(1+\\varepsilon m)^2}{1+m}\\sqrt{1+4\\varepsilon(m+\\varepsilon m^2+\\lambda)}\\sqrt{1+4\\lambda}\\notag\\\\\n&-(1+\\varepsilon m)\\sqrt{1+4\\varepsilon\\lambda}\\sqrt{1+4\\varepsilon(m+\\varepsilon m^2+\\lambda)}\\bigg ].\n\\label{zeta-2}\n\\end{align}\nFinally, squaring both sides of \\eqref{zeta-2} and using \\eqref{zeta-1}, we conclude that\n$[(\\zeta^2-\\Sigma_1)^2-4\\Sigma_2]^2-64\\Sigma_3\\zeta^2=0$ or, equivalently, $P_7(\\lambda;m,\\varepsilon)=0$, where\n$P_7(\\cdot;m,\\varepsilon)$ is a seventh-order polynomial (see Appendix \\ref{appendix-C} for the expression of the coefficients of the polynomial).\n\nFinding the eigenvalues of $P_7(\\cdot;m,\\varepsilon)$ is quite challenging. The Routh-Hurwitz criterion (see, e.g., \\cite[Chapter XV]{Gantmakher98}) gives relevant information on the eigenvalues without computing them explicitly, in particular whether the eigenvalues lie in the left halfplane ${\\rm Re} \\lambda <0$, by computing the Hurwitz determinants $\\Delta_j$ ($j=1,\\ldots,6$) associated with $P_7(\\lambda;m,\\varepsilon)$.\nUnfortunately, our double-squaring method produces spurious eigenvalues which render Routh-Hurwitz criterion inefficient.\nHowever, Orlando's formula (see \\cite[Chapter XV, 7]{Gantmakher98}), a generalization of the well-known property for the sum of the roots of a quadratic equation, establishes a relation between the leading Hurwitz determinant $\\Delta_{6}$ and the sums of all different pairs of roots of $P_7(\\lambda;m,\\varepsilon)$. In particular, $\\Delta_{6}=0$ in the case when either $0$ is a double eigenvalue (i.e., $0$ is an eigenvalue with algebraic multiplicity two) or two eigenvalues are purely imaginary and conjugate.\n\nThe following one is the main result of this subsection.\n\n\\begin{proposition}\n\\label{give critical value}\nThere exist $\\varepsilon_0>0$ and $\\delta>0$, and a unique function $m^c: (0,\\varepsilon_0)\\to (6-\\delta, 6+\\delta)$ with $m^c(0)=6$, such that the polynomial $\\widetilde{P}_7(\\lambda;\\varepsilon):=P_7(\\lambda; m^c(\\varepsilon), \\varepsilon)$ has exactly one pair of purely imaginary roots $\\pm i\\omega(\\varepsilon)$, with $\\omega(\\varepsilon)>0$.\nMoreover, $\\omega(\\varepsilon)$ converges to $\\sqrt{3}$ as $\\varepsilon$ tends to $0$.\n\\end{proposition}\n\nWe first need a preliminary technical lemma:\n\\begin{lemma}\\label{technical}\nThere exist $\\upsilon_0>0$ and $\\varepsilon_*>0$ such that, for all $m$ in the interval $[3,7]$ $($to fix ideas$)$, $\\varepsilon\\in (0,\\varepsilon_*)$ and any\npurely imaginary root $i\\upsilon$ of $P_7(\\cdot;m,\\varepsilon)$, with $\\upsilon>0$, it holds that $0<\\upsilon<\\upsilon_0$.\n\\end{lemma}\n\n\\begin{proof}\nWe observe that, if $i\\upsilon$ is a root of $P_7(\\cdot;m,\\varepsilon)$, then, in particular, the imaginary part of $P_7(i\\upsilon;m,\\varepsilon)$, i.e., the\nterm $-a_0\\upsilon^7+a_2\\upsilon^5-a_4\\upsilon^3+a_6\\upsilon$ vanishes.\n\nA straightforward computation (see Appendix \\ref{appendix-C}) reveals that\n\\begin{align*}\n\\Im{P_7(i\\zeta;m,\\varepsilon)}=&-2048(\\varepsilon-1)^4\\varepsilon^2\\zeta^7-8\\varepsilon(m^2+3m+2)\\zeta^5+O(\\varepsilon^2)\\zeta^5\\\\\n&-128(2m^4-7m^2-3m-1)\\zeta^3+O(\\varepsilon)\\zeta^3+a_6\\zeta,\n\\end{align*}\nfor every $\\zeta>0$, where we denote by $O(\\varepsilon^k)$ terms depending only on $\\varepsilon$ such that\nthe ratio $O(\\varepsilon^k)\/\\varepsilon^k$ stays bounded and far away from zero for $\\varepsilon$ in a neighborhood of zero.\nSince $m^2+3m+2$ and $2m^4-7m^2-3m-1$ are both positive for $m\\in [3,\\infty)$, we can estimate\n\\begin{align*}\n|\\Im{P_7(i\\zeta;m,\\varepsilon)}|\n\\ge &[8(m^2+3m+2)-O(\\varepsilon)]\\varepsilon\\zeta^5\\!+\\![128(2m^4-7m^2-3m-1)-O(\\varepsilon)]\\zeta^3\\!-\\!K|\\zeta|,\n\\end{align*}\nwhere $K:=\\max\\{|a_6(m,\\varepsilon)|: m\\in [3,7], \\varepsilon\\in (0,1]\\}$. Hence, we can determine $\\varepsilon_*>0$ such that\n\\begin{align}\n|\\Im{P_7(i\\zeta;m,\\varepsilon)}|\n\\ge & 64(2m^4-7m^2-3m-1)\\zeta^3-K|\\zeta|,\\qquad\\;\\,m\\in [3,7],\\;\\,\\varepsilon\\in (0,\\varepsilon_*).\n\\label{imaginary}\n\\end{align}\nThe right-hand side of \\eqref{imaginary} diverges to $\\infty$ as $\\zeta\\to+\\infty$. From this it follows that there exists $\\upsilon_0>0$ such that\n$|\\Im{P_7(i\\zeta;m,\\varepsilon)}|>0$ for every $\\zeta>\\upsilon_0$ and this clearly implies that $\\upsilon\\le\\upsilon_0$.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Proposition \\ref{give critical value}]\nWe split the proof into two steps.\n\\vskip 1mm\n{\\em Step 1}. First, we prove the existence of a function $m^c$ with the properties listed in the statement of the proposition.\nFor this purpose, we consider the sixth-order Hurwitz determinant ${\\Delta}_6(m,\\varepsilon)$ associated with the polynomial $P_7(\\lambda;m,\\varep)$.\nIt turns out that\n${\\Delta}_6(m,\\varepsilon)=\\varepsilon^2m^2C\\widetilde\\Delta_6(m,\\varepsilon)$ for some positive constant $C$. As $\\varepsilon\\to 0$,\n$\\widetilde\\Delta_6(\\cdot,\\varepsilon)$ converges to the function ${\\Delta}_0$, which is defined by\n\\begin{align*}\n{\\Delta}_0(m)=&-m^{18}+8m^{17}+97m^{16}+42m^{15}-2129m^{14}-9376m^{13}-16811m^{12}\\\\\n&-7866m^{11}+19913m^{10}+31292m^9-4309m^8-55466m^7-66363m^6\\\\\n&-35480m^5-4729m^4+4666m^3+2628m^2+500m+24.\n\\end{align*}\nNoticing that ${\\Delta}_0(6)=0$ and $\\frac{d}{dm}{\\Delta}_0(6)>0$, it then follows from the Implicit Function Theorem that there exist $\\varepsilon_0\\in (0,\\varepsilon_*)$, with $\\varepsilon_*$ given by Lemma \\ref{technical}, $\\delta>0$ and a unique mapping $m^c: (0,\\varepsilon_0)\\to (6-\\delta, 6+\\delta)$ with $m^c(0)=6$, such that $\\widetilde\\Delta_6(m^c(\\varepsilon),\\varepsilon)=0$ and $\\frac{\\partial}{\\partial m}\\widetilde\\Delta_6(m^c(\\varepsilon),\\varepsilon)>0$ for $\\varepsilon\\in (0,\\varepsilon_0)$. Then, upon an application of Orlando formula, it follows that either $0$ is a double root of $\\widetilde P_7(\\lambda;\\varepsilon)$ or there exists at least one pair $\\pm \\omega(\\varepsilon)i$ (with $\\omega(\\varepsilon)>0$) of purely imaginary roots of $\\widetilde P_7(\\lambda;\\varepsilon)$ for every $\\varepsilon\\in (0,\\varepsilon_0)$. The first case is ruled out, since 0 is not a root of $\\widetilde P_7(\\lambda;\\varepsilon)$. Indeed, $a_7(m,\\varepsilon)$ converges to a positive limit as $\\varepsilon$ tends to $0$.\n\n\\vskip 1mm\n{\\em Step 2}.\nNext, we prove that $\\pm\\omega(\\varepsilon)i$ is the unique pair of purely imaginary roots of the polynomial $\\widetilde P_7(\\lambda;\\varepsilon)$ for every $\\varepsilon\\in (0,\\varepsilon_0)$. For this purpose, we begin by observing that $\\widetilde{P}_7(\\cdot;\\varepsilon)$ converges, locally uniformly in $\\mathbb C$ as $\\varepsilon\\to 0$, to the fourth-order polynomial $\\widetilde P_4$, defined by $\\widetilde{P}_4(\\lambda)=-6272(4\\lambda+1)(\\lambda-12)(\\lambda^2+3)$ for every $\\lambda\\in\\mathbb C$.\nBy Hurwitz Theorem, four roots of $\\widetilde{P}_7(\\lambda; \\varep)$, say $\\lambda_1(\\varep)$, $\\lambda_2(\\varep)$, $\\lambda_3(\\varep)$ and $\\lambda_4(\\varep)$ converge respectively to $\\lambda_1(0)=-\\frac{1}{4}, \\lambda_2(0)=12, \\lambda_3(0)= \\sqrt{3}i$ and $\\lambda_4(0)=-\\sqrt{3}i$. More precisely, for $r_1>0$ small enough, $\\lambda_i(\\varepsilon)$ ($i=1,\\ldots,4$) is simple in the ball $B(\\lambda_i(0),r_1)$ for $\\varep\\in (0,\\varepsilon_0)$ (up to replacing $\\varepsilon_0$ with a smaller value if needed).\nAssume by contradiction that there exists a positive infinitesimal sequence $\\{\\varepsilon_n\\}$ such that, for any $n\\in\\N$, ($\\lambda_{5}(\\varepsilon_n),\\lambda_6(\\varepsilon_n)$) is another pair of purely imaginary and conjugate roots of $\\widetilde P_7(\\lambda; \\varepsilon_n)$, different from $\\pm\\omega(\\varepsilon_n)i$. By\nLemma \\ref{technical}, $\\nu(\\varep_n)=|\\lambda_5(\\varepsilon_n)|\\leq \\upsilon_0$ for every $n\\in\\N$. Take a subsequence $\\{\\varep_{n_k}\\}$ such that $\\nu({\\varep}_{n_k})$ converges as $k \\to \\infty$. The local uniform convergence in $\\mathbb C$ of\n$\\widetilde P_7(\\cdot;\\varepsilon_n)$ to $\\widetilde P_4$ implies that $\\nu({\\varep}_{n_k})$ tends to $\\sqrt{3}$ as $k\\to\\infty$. Since the limit is independent of the choice of subsequence $\\{\\varep_{n_k}\\}$, we conclude that $\\nu(\\varep_n)$ converges to $\\sqrt{3}$ as $n\\to\\infty$.\nNext, thanks to Hurwitz Theorem and the fact that $\\lambda_3(\\varep)$, $\\lambda_4(\\varep)$ converge to $\\sqrt{3}i, -\\sqrt{3}i$ respectively, the pair\n($\\lambda_5(\\varepsilon_{n_k}),\\lambda_6(\\varepsilon_{n_k})$) coincides with ($\\lambda_3(\\varepsilon_{n_k}),\\lambda_4(\\varepsilon_{n_k})$) in $B(\\sqrt{3}i,r_1)\\times B(-\\sqrt{3}i,r_1)$. This contradicts the fact that $\\lambda_3(\\varepsilon_{n_k}),\\lambda_4(\\varepsilon_{n_k})$ are both simple. Up to\nreplacing $\\varepsilon_0$ with a smaller value if needed, we have proved that $(\\omega(\\varepsilon)i,-\\omega(\\varepsilon)i)$ is the unique pair of conjugate eigenvalues of\n$\\widetilde P_7(\\cdot;\\varepsilon)$ and $\\lambda_3(\\varepsilon)=\\omega(\\varepsilon)i$ for every $\\varepsilon\\in (0,\\varepsilon_0)$. The proof is now complete.\n\\end{proof}\n\n\n\\setcounter{tocdepth}{2}\n\\subsection{Hopf bifurcation theorem}\n\\label{subsect-5.2}\nFor fixed $0<\\varepsilon<\\varepsilon_0$, $\\varep_0$ and $\\delta$ given by Proposition \\ref{give critical value}, let us consider the fully nonlinear problem \\eqref{FNLE},\nwhere now we find it convenient to write $F(\\pmb w;m)$ instead of $F(\\pmb w)$ to make much more explicit the dependence of the nonlinear term $F$ on the bifurcation parameter $m$.\nAccording to Proposition \\ref{give critical value}, the bifurcation parameter $m$ has a critical value $m^c(\\varep) \\in (6-\\delta,6+\\delta)$. We intend to prove that a Hopf bifurcation occurs at $m=m^c(\\varep)$ if $\\varep$ is small enough. For $m$ close to $m^c(\\varep)$, we are going to locally parameterize $m$ and $\\pmb w$ by a parameter $\\sigma \\in (-\\sigma_0,\\sigma_0)$. To emphasize this dependence, we will write $\\widetilde{m}(\\sigma)$ and $\\widetilde{\\pmb w}(\\cdot,\\cdot;\\sigma)$.\n\n\\begin{theorem}\n\\label{Hopf bifurcation theorem} For any fixed $\\alpha\\in (0,1)$, there exists $\\tilde{\\varep}_0\\in (0,\\varep_0)$, such that whenever $\\varep\\in (0,\\tilde{\\varep}_0)$ is fixed, the following properties are satisfied.\n\\begin{enumerate}[\\rm (i)]\n\\item\nThere exist $\\sigma_0>0$ and smooth functions $\\widetilde{m}$, $\\rho:(-\\sigma_0,\\sigma_0)\\to\\mathbb{R}$, $\\widetilde{{\\pmb w}}:(-\\sigma_0,\\sigma_0)\\to C^{1+\\alpha}(\\R;\\pmb{\\mathcal W})\\cap C^{\\alpha}(\\R;Q(D(L)))$, satisfying the conditions\n$\\widetilde{m}(0)=m^c$, $\\rho(0)=1$ and $\\widetilde{\\pmb w}(\\cdot,\\cdot;0)$ $=0$.\nIn addition, $\\widetilde{\\pmb w}(\\cdot,\\cdot;\\sigma)$ is not a constant if $\\sigma\\neq 0$, and $\\widetilde{\\pmb w}(\\cdot,\\cdot;\\sigma)$ is a\n$T(\\sigma)$-periodic solution of the equation\n\\begin{eqnarray*}\n\\widetilde{\\pmb w}_\\tau(\\cdot,\\cdot;\\sigma) = QL\\widetilde{\\pmb w}(\\cdot,\\cdot;\\sigma) + F(\\widetilde{\\pmb w}(\\cdot,\\cdot;\\sigma);\\widetilde{m}(\\sigma)), \\qquad\\;\\, \\tau \\in \\R,\n\\end{eqnarray*}\nwhere $T(\\sigma)=2\\pi\\rho(\\sigma)\\omega^{-1}$ and $\\omega=\\omega(\\varepsilon)$ is defined in Proposition $\\ref{give critical value}$.\n\\item\nThere exists $\\eta_0$ such that if $\\overline{m} \\in (6-\\delta_0, 6+\\delta_0)$, $\\bar{\\rho}\\in\\R$ and $\\pmb w \\in C^{1+\\alpha}(\\mathbb{R};\\pmb{\\mathcal W})\\cap C^{\\alpha}(\\mathbb{R};Q(D(L)))$ is a $2\\pi\\bar{\\rho}\\omega^{-1}$-periodic solution of the equation\n$\\overline{\\pmb w}_\\tau = QL\\overline{\\pmb w} + F(\\overline{\\pmb w};\\overline{m})$ such that\n\\begin{equation*}\n\\|\\overline{\\pmb w}\\|_{ C^{1+\\alpha}(\\R;\\pmb{\\mathcal W})}+\\|\\overline{\\pmb w}\\|_{C^{\\alpha}(\\R;Q(D(L)))}+|\\bar{m}|+|1-\\bar{\\rho}|\\leq\\eta_0,\n\\end{equation*}\nthen there exist $\\sigma\\in(-\\sigma_0,\\sigma_0)$ and $\\tau_0\\in \\R$ such that\n$\\overline{m}=\\widetilde{m}(\\sigma)$, $\\bar\\rho=\\rho(\\sigma)$ and $\\overline{\\pmb w}=\\widetilde{\\pmb w}(\\cdot+\\tau_0,\\cdot;\\sigma)$. \t\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nWe split the proof into two steps.\n\\vskip 1mm\n\\textsl{Step 1.} Here, we prove that there exists $\\varepsilon_1>0$ such that $\\pm \\omega(\\varepsilon)i$ are simple eigenvalues of $L$ (and, hence,\nof the part of $L$ in $\\pmb {\\mathcal W}_Q=Q(\\pmb {\\mathcal W})$) for every $\\varepsilon\\in (0,\\varepsilon_1]$ and there are no other eigenvalues on the imaginary axis, i.e., we prove that this operator satisfies the so-called resonance condition.\n\nTo begin with, let us prove that $\\pm\\omega(\\varepsilon)i$ are eigenvalues of $L$. In view of Theorem \\ref{thm-2.3}, we need to show that they are roots of the dispersion relation \\eqref{dispersion epsilon}. For this purpose, we observe that the function $\\widetilde{D}_\\varepsilon:= D_\\varepsilon(\\cdot;m^c(\\varepsilon))$ converges to $\\widetilde{D}_0$ locally uniformly in the strip $\\{\\lambda\\in\\mathbb{C}:|\\Re\\lambda|\\le \\ell\\}$ (for $\\ell$ small enough),\nwhere\n\\begin{eqnarray*}\n\\displaystyle\\widetilde D_0(\\lambda)=-\\lambda-\\frac{1+\\sqrt{1+4\\lambda}}{2}+\\frac{1}{14}[(13+2\\lambda)\\sqrt{1+4\\lambda}+1+4\\lambda],\\qquad\\;\\,\\lambda\\in\\mathbb C.\n\\end{eqnarray*}\nThe function $\\widetilde D_0$ has just one pair of purely imaginary conjugate roots $\\pm\\sqrt{3}i$. Hurwitz theorem shows that there exists $r>0$ such that the ball $B(\\sqrt{3}i,r)$ contains exactly one root $\\lambda(\\varepsilon)$ of $\\widetilde D_{\\varep}$ for each $\\varepsilon$ small enough.\nBy the proof of Proposition \\ref{give critical value}, we know that there exists $r_1>0$ such\nthat $\\omega(\\varepsilon)i$ is the unique root of $\\widetilde P_7$ in the ball $B(\\sqrt{3}i,r_1)$. Clearly, $\\lambda(\\varepsilon)$ is a root of the polynomial $\\widetilde P_7$ and, Hurwitz theorem also shows that $\\lambda(\\varepsilon)$ converges to $\\sqrt{3}i$ as $\\varepsilon\\to 0^+$. Therefore, for $\\varepsilon$ small enough, both\n$\\lambda(\\varepsilon)$ and $\\omega(\\varepsilon)i$ belong to $B(\\sqrt{3}i,r_1)$ and, hence, they do coincide.\nThe same argument shows that $-\\omega(\\varepsilon)i$ is also a root of $\\widetilde D_\\varepsilon$. We have proved that there exists $\\varepsilon_1\\le\\varepsilon_0$ such that $\\omega(\\varepsilon)i$ and $-\\omega(\\varepsilon)i$ are both eigenvalues of $L$ of every $\\varepsilon\\in (0,\\varepsilon_1]$. In particular, $\\pm\\omega(\\varepsilon)i$ are simple roots of\nthe function $\\widetilde D_{\\varepsilon}$ and there are no other eigenvalues of $L$ on the imaginary axis.\n\nTo conclude that $\\pm\\omega(\\varepsilon)i$ are simple eigenvalues of $L$ for each $\\varepsilon\\in (0,\\varepsilon_1]$,\nwe just need to check that their geometric multiplicity is one. For this purpose, we observe that the proof of Theorem \\ref{thm-2.3} shows that the eigenfunctions associated with the eigenvalues $\\pm\\omega(\\varepsilon)i$ are given by\n\\begin{eqnarray*}\n\\begin{array}{lll}\n\\displaystyle u(\\xi)=c_1e^{k_1\\xi}+\\frac{A}{H_{1,\\lambda}}\\bigg (\\frac{e^{k_3\\xi}}{k_3-k_2}-\\frac{e^{k_3\\xi}-e^{k_1\\xi}}{k_3-k_1}\\bigg )c_3,\\quad &v(\\xi)=c_3e^{k_3\\xi}, &\\xi<0,\\\\[3mm]\nu(\\xi)=c_6e^{k_2\\xi}, &v(\\xi)=c_8e^{k_6\\xi}, &\\xi\\ge 0\n\\end{array}\n\\end{eqnarray*}\nwith $k_j=k_{j,\\pm\\omega(\\varepsilon)i}$ and the constants $c_1$, $c_3$, $c_6$ and $c_8$ are determined through the equation \\eqref{matrix} (with $\\lambda=\\pm\\omega(\\varepsilon)i$) where $F_1=\\ldots=F_4=0$.\nSince the rank of the matrix in \\eqref{matrix} is three at $\\lambda=\\pm\\omega(\\varepsilon)i$, it follows at once that the geometric multiplicity of\n$\\pm\\omega(\\varepsilon)i$ is one.\n\n\\vskip 1mm\n\\textsl{Step 2:} Now, we check the nontransversality condition. We begin by observing that, for every $\\varepsilon\\in (0,\\varepsilon_1]$, the function $D_{\\varepsilon}$\nis analytic with respect to $\\lambda$ and continuously differentiable with respect to $m$ in $B(\\sqrt{3}i,r)\\times (6-\\delta,6+\\delta)$, where $r$ is such that\nthe ball $B(\\sqrt{3}i,r)$ does not intersect the half line $(-\\infty,-1\/4]$.\nWe intend to apply the Implicit Function Theorem at $(\\omega(\\varep)i,m^c(\\varep))$ for $\\varep$ small enough.\nIn this respect, we need to show that the $\\lambda$-partial derivative of $D_{\\varepsilon}$ does not vanish at $(\\lambda_3(\\varep),m^c(\\varep))$.\nTo this aim, we observe that\n\\begin{eqnarray*}\n\\lim_{\\varepsilon\\to 0^+}\\frac{\\partial D_{\\varepsilon}}{\\partial\\lambda}(\\omega(\\varepsilon)i,m^c(\\varepsilon))=\\frac{\\partial D_0}{\\partial\\lambda}(\\sqrt{3}i,6)=\\frac{5\\sqrt{3}i-3}{49}.\n\\end{eqnarray*}\nTherefore, there exists $\\varepsilon_2\\leq\\varepsilon_1$ such that, if $\\varepsilon\\in (0,\\varepsilon_2]$, the $\\lambda$-partial derivative of $D_{\\varepsilon}$ at $(\\omega(\\varepsilon)i,m^c(\\varepsilon))$ does not vanish. Then, it follows from the Implicit Function Theorem that\nfor each $\\varepsilon\\in (0,\\varepsilon_2]$, there exist $\\delta_{\\varepsilon}>0$, $r_{\\varepsilon}0$\nand in the negative side if\n\n$1+\\alpha_{1}p_{1}+\\alpha_{2}p_{2}+...,+\\alpha_{n}p_{n}<0$\n\nEach component of the Orientation Vector of a point P in X space,is on the positive side or negative side of each plane.(The`Orientation Vector', is defined in the next section, below.)\n\n\\subsection{Definitions and Terminology: }\n\n\\textbf{Orientation Vector}: Suppose we have a point P in n-dimension\nspace \\textbf{(also called X-space)} and suppose we are given 3-planes.\nwe define a Orientation Vector as a Hamming vector whose components\nprecisely specify on which side the point P lies with respect to the\n3 planes. Example: if the point P lies on the positive side of plane\n1, negative side of plane 2 and positive side of plane 3 , then we\ndefine the Orientation Vector associated with P as $Ov(P)$ and define\n$Ov(P)\\equiv(1,-1,+1)$. Similarly a point Q which lies on the negative\nside of plane 1, negative side of plane 2 and positive side of plane\n3 will have an Orientation Vector $Ov(Q)\\equiv(-1,-1,1)$ . Points,\nwhose Orientation Vector differ from another by at least one component\ncan be said to be separated by at least one plane.\n\n\\textbf{Q Space or Hamming space: }We will also call the space spanned\nby the Orientation Vector as Hamming Space or Q space. Since each\npoint P in X-space has an Orientation (Hamming) Vector associated\nwith it, we can imagine that all the points in X space are mapped\nto a point in Hamming space. Of course this mapping is many to one,\nbut the fact to notice is the following: Let P be some point in X-Space,\nthen all points, R, in X space which are not separated from P by planes\nwill all have the same Orientation Vector as P ie. $Ov(P)=Ov(R)$\nand we describe this by saying: {}``Points P and R belong to the\nsame `quadrant''' this statement is certainly true for the {}``images''\nof P and R in Q-Space, but we will loosely use this terminology for\nthe points in X-space and say P and R are in the same {}``quadrant'',\nwhat we really mean is that P and R are points in X-Space and that\nthey are not separated by planes; Sometimes we will use the term \\textbf{neighbors}\nand say P and R are neighbors. The point to remember is that P and\nR will be neighbors if and only if $Ov(P)=Ov(R)$. Therefore if one\nwishes to find out if two points A and B are separated in X-Space\nwhich contains planes, all one needs to do is to compare their Orientation\nVectors: $Ov(A)$ with $Ov(B)$.\n\n\\textbf{A Saturated Plane:} We consider a plane in $n$ dimension\nspace to be {}``saturated'' if it has already been constrained to\npass through $n$ points and hence cannot be adjusted to pass through\na new point, the coefficients of such a plane are completely determinable.\nEg. A plane in 3 dimension gets saturated if it is made to pass through\nthree points.\n\nWe will suppose we are given all the data containing $N_{f}$ points, and that all \nall their coordinates in $n$ dimension space are known to us. Our task is then to find the planes numbering $q_f$, that can separate all these points from one another and also to find all the coefficients of the equations which define each one of these planes.\n\n\\subsection{Input Output Requirements}\n\n\\textbf{Input to the Algorithm} We are given a set G containing a\ntotal of $N_{f}$ points in $n$ dimensional space. That is we know\nthe coordinates of all these points. These points may also belong\nto different classes, so they may be given a label or `color'. However,\nsince we are isolating all points from one another regardless of their\nclass, the labels do not matter here, but will be useful in certain\napplications.\n\n\n\\textbf{Without, too much of a loss of generality we will assume that the coordinates of all the N points in G are rational numbers, i.e they are either integers or fractions.} We will see that this assumptions makes the coefficients of all the $q$ planes that separate the $N$ in G, also rational numbers. \n\n\\textbf{Output Items of the Algorithm:} The equations of the planes\nwhich divide each point in such a way that every point is separated\nfrom another. And an integer $q_{f}$ which is equal to the\ntotal number of planes required. The equations of the $q^{th}$ plane will be of the form:\n\\begin{equation}\n\\tau +\\alpha_{q1}x_{1}+\\alpha_{q2}x_{2}+....+\\alpha_{qn}x_{n}=0\\quad \\quad (q= 1,2,..,q_f)\n\\end{equation}\n where all the $\\alpha_{qj}$ are coefficients of the plane all these coefficients are determined by the algorithm and $\\tau$ is a known number nearly equal to unity. \n\n\\textbf{ Storage Requirements for running the algorithm:} A set Set\nS which will contain a list of $N$ points and $q$ planes. S will contain the Orientation Vectors of all the points in S and the identification numbers (labels) for the planes, along with the coefficients which define each plane. S contains an array V for storing Orientation Vectors; and `Counter': An integer number. \n\nIn all stages of the algorithm\nS, will contain only those points, (N in number) which are completely\nseparated from one another by the q planes contained in S. This condition\nof separability of the N points in S is never violated even though\nN and and q change as the algorithm progresses. When the Algorithm\nends S will contain the necessary output.\n\nAnother Set T which contains points and their Orientation Vectors\nand an array M :{}``List of Midpoints'' which are the coordinates\nof the mid points of certain line segments. At each stage of the algorithm\nthis set T, will contain pairs of points one not in S and the other\nin S. We are sure that each such point has only one such neighbor (in S),\nbecause the points in S are always separate from one another and hence\ntwo points in the same quadrant cannot both belong to S. Each such\npoint stored in T will also store the coordinate of the point which\nis the midpoint of the line joining it to its neighbor and the line\nsegment with its neighbor. The points in T are the candidates waiting\nto be put in S but they first have to be separated from all the points\nin S. We shall see that as soon as T has a collection of n pairs of\nsuch points, then a new plane is drawn which separates all these points\nso that they become eligible to be incorporated in S. (Later on, we will permit a given point in S to have two or even three neighbors in T, these too will be candidates to be put in S, but for a first reading of this algorithm, it is simpler to assume that a point in S can have at most one neighbor in T.) \n\nWe also need another set D, this is the `Dust-Bin' set, it removes\nall `accumulation' points when detected from set G, (as explained\nlater this is done so that the algorithm does not go into an infinite\nloop; which may happen if G contains a sequence of points converging to an `accumulation' point or `limit' point - we assume that such points do not exist in G).\n\n\\medskip{}\n\n\n\n\\subsection{Steps of the Algorithm}\n\nStep 1: Initially collect a small number of initial points numbering\n$N_{0}$ and choose a set of $q_{0}$ planes that separate each one\nof these $N_{0}$ points and put these planes in S. The coefficients\ndefining the equations of these $q_{0}$ planes should be determined\n(or randomly chosen but ensuring that they separate the $N_{0}$),\ncall $N=N_{0}$ and $q=q_{0}$. Store all Orientation Vectors of points\nin S in array V. In addition we need a set T which will contain points\nwhich cannot become immediate members of S but are prospective members\nand will become members of S eventually. Set T will contain points\nwhich are neighbours of a point which already is a member of S. To\nmake things simple we will assume that at the start, all the $q_{0}$\nplanes are saturated. (We do not want to adjust any of these planes)%\n\\footnote{To avoid `starting troubles', choose $q_{0}$ and $N_{0}$ s.t. $n<2^{q_{0}}$\nand $N_{0}>n$%\n}. Put Counter = 0.\n\nStep 2: If no more points in G go to step 7, else: Randomly choose\na new point from G, which could be a candidate point to be put in\nS, from the remaining points not in S, go to Step 3.\n\nStep 3: Check if this new point is in a new `quadrant', this involves\nfinding its Orientation Vector and comparing with those in S. If the\npoint is in a new quadrant, put this point in S and store its Orientation\nVector in V, put $N=N+1$ go to Step 2, if not, it means it has a\nneighbor in its quadrant. Go to step 4.\n\nStep 4: (You will come here only if the current point has a neighbor\nin S. Notation and procedure for this Step: We keep count of the number\nof members of S which have first neighbors. Such points are called $a_{i}$,\nits first neighbor will be called $b_{i}$, if $a_{i}$ has a second\nneighbour this will be called $c_{i}$ and the third will be called\n$d_{i}$ .) First: Find the `distance' of this new point from its neighbor, call this$\\delta$ \\textbf{If} $\\delta < \\delta_{th}$\nthen remove the new point from G itself and put it in set D, and Go To Step\n2; \\textbf{else} Put this new point in Set T. Check if this new point is in a quadrant which has already a pair in T, if not, put $Counter=Counter+1$ define $i=Counter$, call the point with which this new point is a neighbor as $a_{i}$. However, if this new point already is a neighbor of some point $a(j)$ already in S and if $b(j) \\& c(j)$ exist call this new point $d(j)$, if only $b(j)$ exists call the new point $c(j)$ go to Step 2. However, if $b_{j}$ , $c_{j}$ as well as $d_{j}$ exists this means the new point will become a 4th neighbor of $a(j)$, a situation which we do not permit, so put back the new point in G and go to step 2. If $c_{i}$ and if $d_{i}$ do not exist, it means this point $a_{i}$ has only the present point as neighbor. \n\n Calculate the mid point of the \\textquotedbl{}segment\\textquotedbl{}$(a_{i},b_{i})$,\nif not already calculated, and call it $m_{i}$, add the coordinates\n$m_{i}$, in a {}``List of Midpoints''.( Note $Counter$ only keeps track of the first neighbors of the the $a's$.)\n\nIf $Counter=n$ go to Step 5, if less than n, go to Step 2.\n\nStep 5: Since $Counter=n$, this means you have collected n points\nin T, each of which have to be separated from their neighbors. However\nwe, first re-check if all the $n$ points collected in T are in different\nquadrants (This step is only cautionary, and not necessary if step 4 has been done properly),if say $b_{i}$ and $b_{k}$ in T\nare in the same quadrant, then mark one of them say $b_{k}$\nas $c_{i}$, if the latter doesn't exist, else as $d_{i}$ (If both\n$c_{i}$and $d_{i}$both exist there is no place for $b_{k}$ put it\nback in G , restore the count of points in G as well as restore in\neither case$Counter=Counter-1$, check similarly for other $b's$\nand then go to Step 2.)%\n\\footnote{We have permitted three points in T to belong to the same quadrant\nbut not 4 or more. This is just to simplify the algorithm, anyway,\nsuch events are extremely rare. Since, for large $n$, if there are\n$q$ planes there are $2^{q}$ quadrants, hence the chances of two\npoints randomly chosen to be in the same quadrant is $O(1\/2^{q})$.\nWe permit max. no of neighbors,3, just to ensure that even in the\nfreakiest conditions the algorithm comes to a halt successfully. Of\ncourse the the other possibility is the presence of points of `accumulation',\nhowever this does not happen when points represent integers.%\n}\n\nNow in this step we will separate the n pairs collected in T by introducing\na new plane which passes through all the $n$ mid points, whose coordinates\nare available in {}``List of Midpoints'' in the array M: {}``List\nof Midpoints'' and determine the coefficients of the new plane, by\nsolving the$n$ constraint eqs. to pass it through the $n$ midpoints;%\n\\footnote{The coefficients of the plane containing the n midpoints, can be found\nby using Gauss elimination, Gram-Schmidt evaluation or by using the\nQR algorithm, the last is more useful just in case the n mid points\nfall in a plane of n-1 dimension or less, then the rank of the coefficient\nmatrix will become less than n. This will happen rarely, and even\nif it does, it only means we can accomodate another point (or points)\nin T with a neighbor in S, thus making the rank n. In such a case\nall the points in T (even though they are now more than n) can be\nseparated by a single plane - the algorithm can proceed.%\n} call this plane as plane number $q+1$ and put $q=q+1$, and include\nthis new plane in S along with all the $n$ points labeled $b_{j},j=1,2..,n)$\nand their coordinates, put $N=N+n$,\n\nStep 6: Now check the neighbors of all the $a_{i}$ ,$i=1,2,..,Counter$, in T,\nif only $c_{i}$ exists then it gets promoted to first neighbour.\nBut there is a slight complication after $a_{i}$ and $b_{i}are$\nseparated $c_{i}$ can be a neighbor either of $a_{i}$ or $b_{i}$\nso if it is the former call $c_i$ as $b_i$ else call $b_i$ as\n$a_i$ and $c_i $ as $b_i$ the same kind of investigation needs\nto be done for $d_i $ because it can now belong to the old $ a_i $ \nquadrant or belong to the new $b_i $ quadrant, in either case the\n$d_i$ gets promoted as second neighbour and it will be called $c_{i}$.\nCalculate the new mid point $m_{i}$ for the just created segment\n$(a_{i},b_{i})$ and store.\n\nAfter finishing this task, the number, $r$ of second neighbours that\nwere promoted as first neighbours will be known, after drawing the\nplane $q+1$, then call $Counter = r$.\n\nUpdate V, the Orientation Vectors of points now stored in S wrt to\nthis new plane (their dimensions become $q+1$); Clear the data in\nM: {}``List of Midpoints'' of all points T you have transfered to\nS and remove data, of these $k$ points that have been transfered\nto S, from set T (most of the time $ k=n$) and go to Step 2.\n\nStep 7: (You will come here only if no more points are left in G $N_{f}=0$\nand Counter is not yet $=n$ ) If Counter = 0, Stop else introduce\na new plane passing through the midpoints of segments collected so\nfar, (T will contain as many points as the value of Counter), since\nCounter $0$ and $Usf(z)=0$ if $z\\le0$ and define $U_{1}=Usf(z_{1})$,\nthen $U_{1}$ is 1 if the point P is on the +ve side of plane 1 and\nis zero if it is on the -ve side of plane 1. the same goes for the\nother planes $E_{2},E_{3},..E_{q}$. So the output array $(U_{1},U_{2},U_{3},...U_{q})$\nis a binarized version of the orientation vector $Ov(P)$. Therefore\nour {}``Storage Plan'' for each point L in is to use a binary representation\nof the Orientation Vector $Ov(L)$ of a point $L$ as its label as\nshown in the receptacle in the figure and then store information about\nL in the space next to this code.\n\n{}``Retrieval Plan'': Present an approximate image of L say P to\nthe network. It is then possible to immediately retieve the information\nabout L when a nearby point P has the same OV as L ie it is in the\nsame quadrant as L in X-space. Thus the coordinates $(x_{1},x_{2}...x_{n})$\nof P when presented to the neural engine retrieves L. The retrieval\njust takes $q.n$ multiplication $q.n$ additions. \n\n\n\\subsection{Application to Medical Data}\n\nA similar application can be thought of in medicine. In this case\na person P may be represented as a point in n-dimensional feature\nspace and the $n$ numbers in the array $(x_{1},x_{2}...x_{n})$ may\nrepresent, values of platelet count, WBC count, RBC count, MCV, MCH,\netc. a total of $n$ variables. So a doctor may be looking at patient\nP, and wondering if there is any other person in the data base whose\nmedical condition is similar to that of the present patient P. All\nthe doctor has to do is to input $(x_{1},x_{2}...x_{n})$ to the Retrieval\nsystem shown in Fig 1, and a L will be extracted s.t $Ov(L)=Ov(P)$,\nhence it is very likely that P and L share similar ailments since\nthey share the same quadrant. The doctor can then examine the entire\ncase sheet of person L which is retrieved by the the repository shown\nto the extreme right.\n\nThanks to the advancement of computer technology, algorithms such\nas this one could be very attractive. In order to be aware that such\npossibilities could soon become realities, let us obtain an approximate\nestimate of the numbers involved: Taking the case of medical records;\nassuming $n=50,N=10^{10}$ For such a problem since $n$ is large\nit is reasonable to expect that approx. $q=40$, we see that the number\nof calculations for finding these planes and separating all the N\npoints is approx $2\\times10^{13}$ multiplications, Since computers\nhave achieved around 1 TFLOPS this translates to around 20 seconds.\nTo retrieve the data: for example the history of patient L, which\nis similar to patient P, from a data base of 10 billion, would take\n$q.n$ multiplications i.e. 2000 multiplications, which is practically\ninstantaneous.\n\n\n\\section{Properties and Interesting Aspects of the Algorithm}\n\nIn these two subsections we speak about one of the properties that\nthe algorithm has and some interesting aspects of the algorithm.\n\n\n\\subsection{The Algorithm follows Shannon's Principle}\n\nClaude Shannon, held the view that a bit is an information and one\nmust use just the right amount of bits necessary to solve a problem.\n(A mathematical Theory of Communication Bell Sys. Tech. J. pp. 379-423,\nand 623-656, 1948).\n\nWe show below that the algorithm gathers and uses the information\noptimally: We will define a completed `Stage' of the algorithmic process\nas that state when a new plane say the $q^{th}$ has been inducted\ninto S along with the other points which it has just separated and\nwhich are now in S along with all the Orientation Vectors of the points\nnow in S. To get to the next `Stage' the algorithm transfers enough\nnumber of points and garners just sufficient information for it to\ndraw the $(q+1)^{st}$ plane. We will now prove that the new information\ncollected in bits is exactly equal to that which will be stored after\nthe next stage is completed that is after the $(q+1)^{st}$ plane\nis drawn. No extra bits are collected, so the algorithm always makes\noptimum use of information it collects `Stage' by `Stage' right up\nto the end when the problem is completely solved.\n\nLet us imagine: we are at a `Stage' when we had just added the $q^{th}$\nplane which then separated all $N$ points in S, since we have calculated\nall the $N$, Orientation Vectors w.rt. each of the planes we have\nused $Nq$ bits of information to arrive at this `Stage'. Now let\nus say as we proceed with the algorithmic process: We added $k$ new\npoints which happened to fall in empty quadrants and $n$ new points\nwhich are paired with the old. When the $(q+1)^{st}$ plane is drawn\nall these $(k+n)$ new points are added to $N$, to make the number\nof points in S equal to $N'=N+k+n$. But since we need to determine\nthe Orientation Vectors of these new points wrt the $(q+1)$ planes\nnow in S, we would be acquiring $(k+n).(q+1)$ bits of information,\nin addition,since we have to update the Orientation Vectors of the\nold $N$ points wrt this new $(q+1)$plane, we have to add one bit\nfor each such old vector (because the old orientation vectors were\nof dimension $q$ and now it has to be increased by one to accommodate\nthe new plane) so we have acquired $N$ bits. So the new bits of information\nacquired is: $N+(k+n).(q+1)$. So if we add this to the old $Nq$\nbits already acquired we have a total of $Nq+N+(k+n).(q+1)=(N+k+n).(q+1)$\nbits which is exactly equal to $N'.(q+1)$ bits now stored as Orientation\nVectors by the $N'$ points now in S. So we see, that in the process\nof acquisition and usage and then storage of bits of information there\nhas been no loss nor any redundancy. And even though the points in\nG are picked at random, Shanon's principle has been followed, strictly,\nin an optimal manner in going from $q$ planes to $q+1$ planes and\nthen onwards till the final $q_{f}$ plane is determined and drawn.%\n\\footnote{In the author's opinion, this one of the truly beautiful features\nof this algorithm.%\n}\n\nThe algorithm uses the geometrical properties of $n-$dimension space,\nto judiciously choose a plane passing through $n$ mid-points simultaneously,\nat an appropriate time, to make all this possible.\n\n\n\\subsection{Some Interesting Aspects of the Algorithm}\n\nWe will now speak about a few interesting aspects of the algorithm:\n\n1. Once you have solved the problem, there is no need to start from\nthe very beginning, if at some later date you need to separate a new\nset of $N_{k}$ points. All you have to do is include this new set\ninto set G (which is presently empty) and start running the algorithm\nfrom Step 2. Then these new points will be inducted into S one by\none and new planes added when ever necessary, till all the points\nare separated and G is once again empty.Also since for large dimension\n$n$ we have the relationship $q=O(log_{2}(N)$, when you have solved\nthe problem of $N$ points with $q$ planes then if at some later\ndate the number of points become $2N$ then $q\\rightarrow q+1$ only\nand when $N\\rightarrow N+4N$ then $q\\rightarrow q+2$. This is a\nhuge advantage.\n\n2. Suppose at some future time you need to increase the dimension\nof data say from $n$ to $n+1$ ? Once again there is no need to worry,\nassume that all the existing points have the same coordinates for\nthe first $n$ components but for the $(n+1)$ st coordinate we define\nits value as zero. Do the same thing for the planes, the $(n+1)$\ncoefficient of each plane is defined to be zero. That is for any existing\nplanes we just define the coefficient $\\alpha_{n+1}=0$ eg. see Eq(1).\nNow all the data has become $(n+1)$ dimensional and there is no need\nto change anything else and as and when new data points which are\nnow dimension $n+1$ comes in, we put them into G and the algorithm\ncan proceed from Step 2, just as before.\n\n3. Both the points 1. and 2, above raises interesting possibilities.\nSuppose each data point of $n$ dimension, represented an image of\ndimension $n$, you can incorporate new images at any point of time.\nIn fact you can incorporate images of larger size.\n\n4. Let us extend 3. a bit further: Now if you want to incorporate\na different kind of data which are $r$ dimensional, let us say they\nare different because they are data pertaining to spoken words, again\nthere is no problem! Just increase the dimension of data to dimension\n$n+r$, so each point is an entity in $n+r$ dimension space. The\nold data will live in the first $n$ dimensions with their components\nfrom $n+1$ to $n+r$ defined as zero; whereas the new data pertaining\nto words will again be defined in this $n+r$ dimension space but\nwill have their first $n$ components as zero and have their components\nfrom $n+1$ to $n+r$ as mostly non zero.\n\n5. So we see new data can always be added not only of the same type\nbut of different types and the old data is never discarded only the\ndimension of space increases and this can go on from generation to\ngeneration. The set S will be the repository of all knowledge in perpetuity!\nAs they say Knowledge never dies!\n\n\n\\section{Conclusion}\n\nIn this paper we have reported the discovery of a new algorithm for\nseparating a given set of $N_{f}$ points by planes in n-dimensional\nspace. The algorithm is noniterative and will always halt successfully.\nIt has the property of restart, if new points are needed to be separated\nthe algorithm can continue from where it left off. And at some later\nstage if the dimension of the data is increased ($n$ to $n+r$) is\nincreased the algorithm can still continue from where it left off,\n(after some adjustments) and tackle the new data points which are\nof a higher dimension. The algorithm is insensitive to the type of\ndata; the points may represent, images, words or any other type. It\ncan handle even the types are mixed as discussed in Section 4.\n\nA rigorous proof has been provided in the body of the paper and a worked example is given in Appendix A.\n\n The computational Complexity is of $O(n.Nlog(N))+O(n^{3}log(N))$, where $N$ is the given number of points and $n$ is the dimension of space.\n\n\\textbf{An Informal Essay on the New Algorithm and its Future Applications is contained in Appendix B}\n\n\\section{Acknowledgements}\n\nThe author thanks the management of Sreenidhi Institute of Science\nand Technology for their sustained support. He proclaims his grateful\nthanks to his wife Suhasini, for her willingness to be a sounding\nboard on innumerable occasions and listen to his monologues without\nwhich he does not think this work would have ever been done.\n\n\n\\section{References}\n\n1. William B. Johnson and Joram Lindenstrauss: Extensions of Lipschitz\nmappings on to a Hilbert Space, Contemporary Mathematics, 26, pp 189-206\n(1984)\n\n2. Ralph P. Boland and Jorge Urrutia: Separating Collection of points\nin Euclidean Spaces, Information Processing Letters, vol 53, no.4,\npp, 177-183 (1995)\n\n3. K.Eswaran:A system and method of classification etc. Patents filed\nIPO No.(a) 1256\/CHE July 2006 and (b) 2669\/CHE June 2015\n\n4. R.A. Fischer: {}``The statistical utilization of multiple\nmeasurements'', Annals of Eugenics, 8, 376-386 (1938); also Annals\nEugenics, 7, 179-188 (1936)\n\n5. P.C. Mahalanobis: {}``On the generalized distance in statistics'',\nProc. Nat. Inst. of Sc. India, 12, 49-55 (1936)\n\n6. McCulloch, W. and Pitts, W. A: {}``Logical calculus of the\nideas immanent in nervous activity'', Bulletin of Mathematical Biophysics,\n5:115\\textendash{}133. (1943)\n\n7. A. N. Kolmogorov: On the representation of continuous functions\nof many variables by superpositions of continuous functions of one\nvariable and addition. Doklay Akademii Nauk USSR, 14(5):953 - 956,\n(1957). Translated in: Amer. Math Soc. Transl. 28, 55-59 (1963).\n\n8. Paul Werbos: {}``Beyond Regression: New Tools for Prediction\nand Analysis in the Behavioral Sciences'', PhD thesis, Harvard University,\n1974\n\n9. Rumelhart, D. E., Hinton, G. E., and R. J. Williams: Learning\nrepresentations by back-propagating errors. Nature, 323, 533\\textendash{}536\n(1986).\n\n10. J. Schmidhuber: Deep Learning in Neural Networks: An Overview.\n75 pages, www.arxiv.org\/abs\/1404.7828 (2014).\n\n11. Yoshua Bengio: Learning Deep Architectures for AI. Foundations\nand Trends in Machine Learning: Vol. 2: No. 1, pp 1-127 (2009).\n\n12. K. Eswaran: {}``On the storage and retrieval of primes using\nn-dimensional geometry'', sent for publication.\n\n13. J.C. Hawkins with S. Blakeslee: {}``On Intelligence'',\nPubl. Henry Holt and Co. NY. (2004)\n\n14. D. George and J.C. Hawkins: Trainable hierarchical memory\nsystem and method, January 24 2012. URL https:\/ www.google.com patents\nUS8103603. US Patent 8,103,603\n\n\\section{APPENDIX:A }\n\n\n\\section*{WORKED EXAMPLE}\n\n\\begin{figure}[htp]\n \\begin{center}\n \\includegraphics[scale=0.50]{set-of-pointstobe-separated-rev.png}\n\n\\caption{Fig Shows the Example Problem which was used to demonstrate the working of our algorithm}\n\n\\label{fig:fig-b}\n\\end{center}\n\\end{figure} \n\n\\textbf{We will now show the working of our algorithm in complete\ndetail. The algorithm was used to separate the points shown in Fig } \\ref{fig:fig-b}\n\nIt was discovered that the above points which are 29 in number in\n2d space, can be separated by 8 planes, by using the algorithm. We\nwill now show how this was done. \n\nHowever, instead of using unlabeled points we will use Fig \\ref{fig:fig-c} which\nhas a label for each point but in the same position. (Fig \\ref{fig:fig-b} and Fig\n\\ref{fig:fig-c} are identical except for labels).\n\nThe 29 points belong to three classes, (class a,b,c). \n\nWe have numbered each point by a number, these numbers is the sequence\nwith which each point was chosen to be in set S (or T) when we solved\nthe problem. We now retain the labels while we describe the process\nthat was followed so that the method can be better understood. \n\nWe describe the process in Stages.\n\n\\medskip{}\n\n\\begin{figure}[htp]\n \\begin{center}\n \n\\includegraphics[scale=0.50]{beginning1-rev.png}\n\\caption{Fig shows initial set of points to be separated.}\n\n\\label{fig:fig-c}\n\\end{center}\n\\end{figure} \n\n\n\\medskip{}\n\n\n\\textbf{STAGE1:}\n\nThe first stage is when we started with an initial set of points which\nare separated by planes.\n\nIn our commentary below we use the present tense in describing what\nwas actually already done, this change of tense from past to present\nhas been adopted so that the commentary reads better. \n\n\\medskip{}\n\\begin{figure}[htp]\n \\begin{center}\n \n\\includegraphics[scale=0.50]{Start-Stage-rev.png}\n\\caption{Fig Start of the algorithm: Initial contents of S}\n\n\\label{fig:fig-i}\n\\end{center}\n\\end{figure} \n\nThis initial set of planes are two in number and numbered 1 and 2\nas shown in Fig \\ref{fig:fig-i}\nbelow. And the initial set of points, three in number are\n$a_{1},b_{2},c_{3}$. We store the coefficients of the two equations\nfor the planes 1 and 2 in S. The equations for the two planes 1 and\n2 will be of the form :\n\n\\begin{equation}\n1+\\alpha_{1}x+\\alpha_{2}y=0\n\\end{equation}\n\\begin{equation}\n1+\\beta_{1}x+\\beta_{2}y=0\n\\end{equation}\n\n\nWe compute the Orientation Vectors of these three points which are\nshown in Fig \\ref{fig:fig-i}. and store in S and put $q=2$ and $N=3$.\nThe first component of the Orientation vector of $a_1$ is $+1$ because it lies on the positive side of the normal of plane 1, indicated by black arrow, similarly the second component is also $+1$ because $a_1$ is also on the positive side of plane 2, therefore we write $Ov(a_1)=(+1,+1) $, the other Orientation vectors of $b_2$ and $c_3$ are depicted in Fig \\ref{fig:fig-i}. The first component of $Ov(c_3)$ is $-1$, because it lies on the negative side of plane 2 but its second component is $+1$ because it lies on the positive side of plane 2, hence $Ov(c_3)=(-1,+1)$, the OV of point $b_2$ is also shown in the Fig \\ref{fig:fig-i}.\nNotice all the three point have different Orientation Vectors and therefore they all are in separate quadrants, of course this is as it should be, otherwise they would not have qualified to be members of S. \n \nNow applying\nStep 2 of our algorithm we choose point $a_{4}$, and then go to Step\n3, and check whether this point is in the same quadrant as the other\nthree points, in order to do this we have to find the Orientation\nVector of this point: This is done by substituting the $(x,y)$ of\nthe point $a_{4}$ in Eq. (2), for plane 1, the lhs will evaluate\nto some -ve value which will signify that $a_{4}$ is on the -ve side\nof plane 1 , then we substitute $(x,y)$ of the point $a_{4}$ in\nEq. (3) which is the equation for plane 2, the lhs will again evaluate\nto some -ve value which will signify that $a_{4}$ is on the -ve side\nof plane 2. In this manner we obtain the Orientation Vector of point $a_{4}$\nas $Ov(a_4) =(-1,-1)$. \n\nIn n dimension space this is the only way we can determine\nOrientation Vectors. And determining an orientation vector of a point\nin n-space where q planes are present would involve evaluating q linear\nequations of type Eq.(1), and therefore would need a total of $q.n$\nmultiplications and $q.n$ additions.\n\n However when n = 2 (or 3) and\nwe have a diagram such as the one below present, we can write down\nthe Orientation Vectors by sight, we can see\nthat $a_{4}$ is on the -ve side of the plane 2 because it is on the\nopposite side of the normal direction of plane 1, which is indicated\nby the black arrow on the top of the figure, similarly $a_{4}$ is\non the -ve side of the plane 2 because it is on the opposite side\nof the normal direction of plane 2, which is indicated by the black\narrow in the middle right corner of the figure, hence we can conclude, correctly, that $Ov(a_4) =(-1,-1)$. \\textbf{From now on\nwe will write down the Orientation Vectors by sight. }\n\nWe now complete the process given in Step 3 with regard to this new\npoint $a_{4}$, we see that by comparing the Orientation Vectors of\nthis point $a_{4}$ with the Orientation Vectors of the points which\nare currently in S, namely $a_{1},b_{2},c_{3}$ , and finding that\nit is different we are sure that $a_{4}$ is in its own quadrant space\nand thus point $a_{4}$ is added to S, and its Orientation Vector is\nstored in V and we put $N=N+1=4$ and go to Step 2. We now see that\nthere are no more free regions where a point randomly chosen from G that is points from Fig \\ref{fig:fig-c} can be put in S.\nAnd hence when we go to Step 2 and choose a new point, we will be directed to go to Step 3, as we shall soon see in Stage 2.\n \n\n\\medskip{}\n\\medskip{}\n\n\n\\textbf{STAGE 2}\n\n We are now in Step 2, we randomly select a point say $a_{5}$ notice\nit has a neighbor $a_{1}$ . This is discovered by sight, but in actuality \nif we are in $n$ dimension space, we calculate the Orientation Vector\nof this new point, $Ov(a_{5})=(1,1)$ and compare with all other Orientation\nVectors of points already in S, namely $a_{1},b_{2},c_{3},a_{4}$\nand then discover that $Ov(a_{1})=Ov(a_{5})=(1,1)$, this implies\nthat in Q-space $a_{1}$and $a_{5}$ are mapped to the same point,\nhence in X-space they are not separated by any planes thus we discover\nthat point $a_{1}$ is a neighbor of the randomly chosen point $a_{5}$.\n\nThe above is the Standard procedure of discovering whether a new randomly\nchosen point has a neighbor in S, this procedure is done in Step\n3 for all randomly chosen points. \\textit{This Standard procedure\nof discovering neighbors in S, is assumed to be always adopted,} \\textbf{\nthough for the purpose of this example we will from hence forth, do\nthe {}``neigbour detection'' in S, by sight. }Now that $a_{5}$ has\na neighbor we go to Step 4, calculate the mid point coordinate $m_{1}$ and\nstore it in the List of Midpoints and put $a_{5}$ in T. Put $Counter=Counter+1=1$\nand then go to Step 2. We now choose a new point at random say, $c_{6}$\nit has a neighbor point $c_{3}$ which is already in S, we find the\nmidpoint $m_{2}$ put $c_{6}$ in T and increase $Counter=Counter+1=2$, \nbut this time we cannot go back to Step 2. Since we are in two dimension\nspace (n=2), we need to immediately separate the points in T from\ntheir respective neighbors in S. As can be seen that the algorithm initiates procedures to start including a new plane, in S, as soon as $Counter=n$ (where $n$ is the dimension of Space).\n\n\\footnote{In n dimension space we could have gone back to Step 2 and collected\nsome more points in T, till Counter becomes n, indicating that n points\nhave been collected in T, each having one neighbor in S, and then\nit will be time to separate the n points by introducing a new plane\nwhich passes through the n mid points $m_{1},m_{2},...,m_{n}$ which\nare stored in the {}``List of Midpoints''. The calculation of the\ncoefficients of this new plane is done in Step 5 and the order of computations\nto find the newplane (say) by using Gaussian elimination is $O(n^{3})$,\nso if we have to intoduce $q_{f}$ planes in the problem the multiplications\nare in the order of $O(n^{3}).q_{f}$.%\n}\nTo proceed with our algorithm we need to go to Step 5. Step 5 says\nthat we must now find the equation to a new plane which passes through\nthe mid points $m_{1}$ and $m_{2}$, This is easliy done by assuming\nthat the equation to this new plane is :$1+\\beta_{1}x+\\beta_{2}y=0$;\nwith unknown coefficients $(\\beta_{1},\\beta_{2})$ then these two\nunknown coefficients can be found by using the condition that the\nline must pass through $m_{1}$and $m_{2}$, whose coordinates are\n$(m_{1x},m_{1y})$ and $(m_{2x},m_{2y})$ to obtain the eqs:\n\n\\begin{equation}\n1+\\beta_{1}m_{1x}+\\beta_{2}m_{1y}=0\n\\end{equation}\n\n\n\\begin{equation}\n1+\\beta_{1}m_{2x}+\\beta_{2}m_{2y}=0\n\\end{equation}\n\n\nSolving the above we determine $(\\beta_{1},\\beta_{2})$ which completely\ndefines the equation for plane 3, which is now include S and we drawn this line as shown in Fig \\ref{fig:fig-d}. Since the coefficients $(\\beta_{1},\\beta_{2})$ are known the direction of the normal is also known, the normal is also drawn (black arrow) in the figure.\n\nNow $q=q+1=3$. Going to Step 6, we check to see if all the points\nare in their own quadrants (ie are separated) by actually updating\nthe finding the Orientation Vectors of the old points wrt to the 3\nplanes. The Orientation Vectors are now of dimension 3 and given by:\n\n$ov(a_{1})=(1,1,-1); ov(b_{2})=(1,-1,-1);$ $ov(c_{3})=(-1,1,1);$\n$ ov(a_{5})=(1,1,1); ov(c_{6})=(-1,1,-1)$.\n\nWe see that all the Orientation Vectors \n\nof $a_{1},b_{2},c_{3},a_{4},a_{5},c_{6}$ are all different so that\nthey are all in their own quadrants in Q-space and hence separated\nfrom one another in X-space. This ofcourse, is plainly visible by\nsight in Fig \\ref{fig:fig-d}. \n\nNow we complete the tasks indicated in Step 6. we clear the ``List\nof Mid Points'', include the new points $a_{5},c_{6}$ in S, include\nplane 3 in S put q=3, Clear the set T because its contents $a_{5},c_{6}$\nhave been transferd to S. Then we go to Step 2. The situation arising\nis shown in Fig \\ref{fig:fig-d} below: \n\n\\begin{figure}[htp]\n \\begin{center}\n \n\\includegraphics[scale=0.50]{Stage-2-rev.png}\n\\caption{This is the situation after completing Stage 2}\n\n\\label{fig:fig-d}\n\\end{center}\n\\end{figure} \n\n\n\\medskip{}\n\n\n\\medskip{}\n\n\n\\textbf{STAGE 3: }\n\nWe are now in Step 2. Since no more points can be added,the situation\nis exactly the same as that in the beginning of Stage 2, so we repeat\nwhat was done in stage 2 except that we randomly choose two new points\n$b_{7}$ and $c_{8}$ which have neighbors $b_{2}$ and $c_{6}$ which\nare already in S. So we follow the steps similar to that described\nin STAGE 2 and end up with a new plane, viz. plane number 4, put q=q+1=4; and S\nnow contains 8 points all their Orientation Vectors in the new 4 dimensional\nq space are all different and therefore all the 8 points are separated\nby these 4 planes. See fig \\ref{fig:fig-e}.\n\n\\begin{figure}[htp]\n \\begin{center}\n \n\\includegraphics[scale=0.50]{Stage-3-rev.png}\n\\caption{This is the situation after completing Stage 3}\n\n\\label{fig:fig-e}\n\\end{center}\n\\end{figure} \n\n\n\n\n\n\\medskip{}\n\\medskip{}\n\n\n\\textbf{STAGE 4;}\n\nFrom now on we will hurry through the steps.\n\nWe are in Step 2. Two new points: $a_{9}$ which is neighbor of $a_{4}$ and\n$a_{10}$ which is a neighbor of $a_{1}$ are introduced and a new\nplane plane No. 5 is introduced and transfered to S.\n\nAfter this we see that the next point $c11$ occurs in an empty quadrant\nhence it is immediately added to S, The next two points $a_{12},b_{13}$\nhave neighbors $a_{10}$ and $b_7$ resp. and therefore need a new plane, viz plane 6, and are all transfered to\nS along with plane 6. The next point $c_{14}$ happens to be in its\nown quadrant so is included in S. The next two points $a_{15},c_{16}$\nhave neighbors $b_{13}$ and $a_4$ in S and determine a new plane 7 and are transferred\nto S along with 7. See Fig \\ref{fig:fig-f}\n\n\n\\begin{figure}[htp]\n \\begin{center}\n \n\\includegraphics[scale=0.50]{simple-example-5-rev.png}\n\\caption{Situation after planes 5,6,and 7 are introduced in Stage 4}\n\n\\label{fig:fig-f}\n\\end{center}\n\\end{figure} \n\n\n\n\\medskip{}\n\\medskip{}\n\n\n\\textbf{STAGE 5:}\n\nWe are in Step 2: 7 planes have been just added and S contains 16\npoints. All the Orientation Vectors are now dimension 7. They are\nall different for the 16 points, as may be checked tediously or by\nsight.\n\nThe new randomly chosen points $a_{17,}b_{18},c_{19},c_{20},b_{21},a_{22}$\nall are in their own quadrants and can be directly included in S with\nout troubling ourselves to add a new plane see Fig \\ref{fig:fig-g}. (This sort of situation\nwhen we can include a whole sequence of points before we introduce\na new plane occurs when the number of planes q increase, because\nthe number of quadrants in q space increase. For q planes we have\n$2^{q}$ quadrants in Q space, therefore there is space for more points\nin q space.)\\footnote{ The situation in large $n$ dimension space is much easier, this is because whenever you introduce a plane say from $q$ to $q+1$ the number of quadrants in X-space double from $2^q$ to $2^{q+1}$, this doubling in X-space stops after $q=n$ after which you will get some confined regions. Now an interesting question arises: If have just added a plane (say) the $q^{th}$ and you already have N points in S, how many points can you add to S before you need the next plane? The answer for large $n$ is that you can on the average add $O(N)$ points.}\n After this we arrive at $b_{23}$ and $a_{24}$ which\nhave neighbors already in S namely $c_{3}$ and $c_{6}$ . These determine\nthe last plane plane 8. See Fig \\ref{fig:fig-g}\n\n\\medskip{}\n\n\\begin{figure}[htp]\n \\begin{center}\n\\includegraphics[scale=0.50]{simple-example-7-rev2.png}\n\n\\caption{Eight points and one plane are introduced in Stage 5}\n\n\\label{fig:fig-g}\n\\end{center}\n\\end{figure} \n\n\n\\medskip{}\n\n\n\\textbf{STAGE 6:}\n\nWe are in Step 2 and have just introduced plane 8. There are 24 points\nin S and 8 planes.\n\nWe see that we now randomly choose the sequence of points $c_{25},a_{26},b_{27},c_{28}$\nand $a_{29}$ all of them are in their own quadrants and are already\nseparated from other points in S and from each other and hence can be\nincluded in S without adding any more planes. And now there are no\nmore points in G and so we have $N=29$and $q=8$. The Final solution\nis given in Fig \\ref{fig:fig-h}\n\n\\medskip{}\n\\begin{figure}[htp]\n \\begin{center}\n\\includegraphics[scale=0.50]{simple-example-final-8-rev.png}\n\n\\caption{After Stage 6 the final configuration all 29 points separated by 8 planes.}\n\n\\label{fig:fig-h}\n\\end{center}\n\\end{figure} \n\n\\newpage\n\\subsection{Tackling New data}\n\nIn this subsection we will, for the sake of completeness, tackle some of the situations that we had\ndiscussed before, in section 4,but with reference\n to this \\textbf{Worked Example.}\n\n\nHow shall we tackle new points? Suppose we have once solved a problem that is, we have separated $N$ points given in an original set G, in n dimension space and have found that $q$ planes have done the job. After this we make a retrieval and storage engine as depicted in Fig \\ref{fig:fig-a} and discussed in Section 3. We begin to use the retrieval engine for some time, and afterwards we encounter new data. \n\n\\textbf{Case 1: New Data has the same dimension as the old data} \n\nWe have already spoken about this situation: We had said that there is no need to start from the very beginning, The algorithm can start from where it left off.We just add these new points to Set G, which until now was empty, and start the algorithm from Step 2 keeping the Set S containing the latest number of points and planes which we had (i.e. $N=N_f$ and $q=q_f$, with all the $Ov's$), just before we encounter this new data. We already depicted the situation, in our example, Figure \\ref{fig:fig-h} , we have separated 29 points with 8 planes. Now we encounter 2 more points (say) $c_{30}$ and $a_{31}$. We just continue as before (assuming just for convenience that both these have neighbors): treat these two as temporary candidates in T with neighbors as shown in \\ref{fig:fig-j} and draw a new plane number 9. This separates all the 31 points and we have an additional plane, and these two new points all of which now can be included in S. Notice new quadrants are created, these new quadrants could capture some new points, which are are still in G. The algorithm continues from Step 2 till G is empty and then Stops. If G becomes empty at $N=31$ then the situation in S is as shown in Fig \\ref{fig:fig-j}.\n\n\n\n\\medskip{}\n\\medskip{}\n\\begin{figure}[htp]\n \\begin{center}\n\\includegraphics[scale=0.50]{simple-example-final-8-RESTART.png}\n\n\\caption{Two New Points encounered: Resuming from last step.}\n\n\\label{fig:fig-j}\n\\end{center}\n\\end{figure} \n\n\\medskip{}\n\n\\textbf{Case 2: New Data does not have the same dimension as the old data, but has one more dimension $n=n+1$} \n\nSuppose we are as before in a situation depicted by Fig. \\ref{fig:fig-h} i.e. S contains 29 points and 8 planes. Now we encounter three new points of dimension $n=n+1$, since all our original data is 2d we can assume that all the points are in the X-Y plane. We assume that these three new points belong to the new data set and are of dimension $n+1$. \n\nWe tackle the problem by first converting all the 2-d data in S to 3 d data, this is simply done by embedding the 2d data in 3d space. With reference to Fig \\ref{fig:fig-c}\n\nStep C1: Convert all the 2-d coordinates by defining for each point whose original coordinates are $(x_j,y_j), (j=1,2..N)$ a new coordinate $(x_j,y_j,z_j), (j=1,2,...N)$ such that $ (x_j=x_j; y_j=y_j, z_j=0) , (j=1,2...N).$\n\n\n\nStep C2: Convert all the 2d equations of the $q$ planes to corresponding equations valid as 3d equations eg if the equation of the $2$nd plane is (say):\n\n\\begin{equation}\n1+\\beta_{1}x+\\beta_{2}y=0\n\\end{equation}\nwe should convert it to : \n\\begin{equation}\n1+\\beta_{1}x+\\beta_{2}y + \\beta_{3} z = 0\n\\end{equation}\nand define \n\\begin{equation}\n \\beta_3 = 0\n\\end{equation}\n this process should be done for all the $q$ planes.\n\nBy this process a 2d plane will become a 3d plane, however its normal will be in the X-Y plane. If we assume that X-Y plane to be `horizontal' then all the planes will become vertical walls containing the original 2d lines. These planes are now shown as blue lines. All the original quadrants at this stage will become 3-d regions defined by vertical walls shown as blue lines. \n\n\\medskip{}\n\\begin{figure}[htp]\n \\begin{center}\n\\includegraphics[scale=0.50]{simple-example-final-8-Dimension-increase.png}\n\n\\caption{The dimension of Problem has increased by one to n=3}\n\n\\label{fig:fig-k}\n\\end{center}\n\\end{figure} \n\nWe start the algorithmic process: We go to Step 2 of the algorithm after putting $Counter=0$ and $n=3$ and putting A,B,C in G. And S will contain $q$ 3d planes and N points along with their `Orientation Vectors'. For simplicity we will assume that none of the three points A,B,C fall in a new `quadrant'\\footnote{If they do just transfer them to S} so they will all end up in T and each with their respective neighbors for example the neighbor of A and B are shown to be : $b_{13}$ and $a{4}$ resp. We will have three mid points which we call $m_A, m_B, m_C$. Now since $Counter = 3 = n$, we go to Step 5 and find the equation to the plane which passes through these three midpoints. The new `genuine' 3d plane\\footnote{What we mean by `genuine' is that this $(q+1)$st plane is the first plane among all the planes in S, whose normal is not lying in the X-Y plane. Of course as we proceed with the algorithm and be adding points in 3d space there will be many such planes in S.} along with the old 8 planes will now separate all the N=29 points from A,B,C and from each other. \\footnote{Here we assume for simplicity that no two points of A,B and C have the same neighbor.} So we can now add A,B,C to S and include this new plane so $N=32$ and $q=9$ and calculate all the Orientation Vectors of the 32 points (i.e. update V) and then go to Step 2 of the Algorithm after clearing T and the `List of Midpoints'.\n\n So we see now S has 32 points all separated by 9, 3d planes if there are more points in G, the algorithm proceeds or else stops. Fig \\ref{fig:fig-k}\n\nWe have thus seen how the algorithm can restart from the last step made earlier, even when the dimension of the new points are increased by one.\n\n\n\n\\textbf{END OF WORKED EXAMPLE}\n\n\n\\section{APPENDIX B : An Informal Essay on the New Algorithm and its Future Applications}\n\n\n\\textbf{\\large What the Algorithm Does}: Imagine we are given a set\n$G$ of N points in n-dimensional space. Basically the algorithm finds\nplanes, in n-dimension space such that they can separate all the N\npoints, in such a manner that every point is separated from every\nother point by at least one plane. The output of the algorithm is\na Set S containing the points and also the equations of the planes\nthat separate them. (In general for large dimension space the number\nof planes $q,$required is approx.$log2(N)$).\n\n\\medskip{}\n\n\n\\textbf{\\large How it\\textquoteright{}s Done:} Let us imagine the\nSet $G$ as an n-dimensional $X-$Space, containing $N$ points. We\ncreate another n-dimensional $X$-Space called $S$. We then transfer\npoints randomly from $G$ to $S$ one by one, so that they occupy\nthe same coordinate position in $S$ as they had occupied in $G$\n(their coordinates do not change) and also planes are drawn in S.\nThe algorithm makes sure that after a new plane is drawn, all the\npoints in S at this stage are separated. The algorithm proceeds Stage\nby Stage transferring, new points from $G$ and drawing new planes\nin S till eventually S contains all the N points as well as the q\nplanes needed to separate all the N from one another. \n\n\\medskip{}\n\n\n\\textbf{\\large Beauty: }There is a certain beauty in the algorithm,\nthere are no redundancies and duplications. (Ex. if it required say\n97 bits of information to draw a plane, then it will acquire exactly\nall these 97 bits of information and draw the plane, then store the\n97 bits, before it seeks more information to draw another plane),\nSec 4.1, p. 15, contains the proof. It adopts Shannon\\textquoteright{}s\nprinciple that each bit is an information, so use it as far as possible,\nbefore you seek another bit of information, (but these are technicalities\nthat I will pass on for some other time).\n\n\\medskip{}\n\n\nJust to put the paper in its proper perspective, we list its contents: \n\\begin{enumerate}\n\\item It has a very strict mathematical proof, of how $N$ points in n-dimensional\nspace can be separated by q planes.\n\\item The Complexity is of $Nlog2N$. This itself is somewhat unique because\nvery few algorithms have this efficiency to name a few:(i) the FFT,\ncomplexity: $N.log(N)$, (ii) Euclidean algorithm of GCD complexity:\n$log(N)$, $N$ being the larger of the two numbers, (iii) Quick Sort\nalgorithm complexity on the average $N.log(N)$ worst case $O(N^{2})$,\n(iv) Gauss elimination for solving $N$ linear equations $O(N^{3})$,\n(vi) Primality testing algorithm of Agarwal et al, Complexity $O((logN)^{6})$,\n(vi) RSA algorithm $O(N^{3})$, here $N$ is the number of bits.\n\\item It contains the conditions under which the algorithm can be made to\nwork along with a proof. \n\\end{enumerate}\nWe have Simple Worked example in the Appendix A and outlined applications\nin Sec 3 and its interesting properties in Sec 4.\n\n\\medskip{}\n\n\n\n\\section*{A brief essay on the origins and relevance of the work}\n\n\\medskip{}\n\n\n\n\\subsection*{About separation of clusters vs. the separation of points }\n\nMany researchers, from statisticians and neural network scientists\nhave long been trying to separate clusters by planes or discriminant\nfunctions. This has been ever since the time of Fisher {[}4{]} and\nMahalanobis {[}5{]} and McCulloch and Pitts{[}6{]}, Kolmogorov {[}7{]},\nWerbos{[}8{]}, and Rumelhart, Hinton and Williams {[}9{]} and others\n{[}10{]}, also the Deep Learning people{[}11{]}, famous names in the\nfields. But all have had great difficulties in large n-dimensional\nspace and they found that it is not easy. So the phrase like \\textquotedbl{}The\nCurse of Dimensionality\\textquotedbl{} and \\textquotedbl{}NP Hard\nComplexity\\textquotedbl{}, has become a part of folk lore. However,\nfrom the very beginning it was never very easy to separate clusters\nby planes. This is mostly because, a cluster is NOT well defined,\nevery cluster has its own shape and in $n$-dimensions you could have\nlong thin filaments and all kinds of snake like dragon like shapes\nwhich constitute a cluster. Though statisticians try to approximate\nthe shape of each cluster as ellipsoids or even simple spheres, clusters\nin general would require more parameters to define their shapes than\nthat required to define the planes which are supposed to separate\nthem! So all along it was, perhaps, very naive of all of us to have\ntried to separate clusters when such entities are not mathematically\nwell defined. I felt that it is far better to separate the individual\npoints and to use the enormous space and degrees of freedom that is\navailable in $n$-dimension space to separate each point, rather than\ntry to separate clusters which will never be well defined. This was\nthe genesis of the idea that gave an impetus to do the kind of research\nwork reported in this paper. As an illustration, in the last section\nwe have considered a cluster of 29 points in 2-d, see Fig 3, \nand used the algorithm to separate all the 29 points using 8 planes\nsee Fig 9, (it can be proved that the theoretical minimum for\n29 points in 2-d, is 7 planes).\n\nAnother fact, that only adds to the prospect of success in this new\ndirection is that: for large n dimension space where $2^{n}>N$ ($N$\nbeing the number of points), you will find the number of planes $q$,\nneeded for separating $N$ points is far less than the number of points\nitself, in fact $q=O(log2(N)).$\n\nImagine: Even a highly reduced small passport size image of $30X30$\npixels is a point in a $900$ dimension space. And in such a space\nthere are $2^{900}$ quadrants i.e. approximately $10^{270}$ quadrants.\nAnd even if you put one image in one quadrant (thus automatically\nseparating them from others) you will never be able to fill up this\nspace: There are only about $10^{85}$ atoms in the universe so how\nwill you make so many photographs?\n\nSo you see this paper makes the {}``Curse of dimensionality'' into\na very Great Boon and Blessing! It is this aspect which has induced us to\n write this Short Note.\n\nYou may ask: Why did not the author speak about all this, in the main\nbody of the paper?\n\nThe answer is: We wanted all the readers to concentrate on the algorithm\nand its proof and not rile them with matters not germane to the task\non hand; for after all a mathematical paper makes its greatest impact\nby cold logic and rigid proofs rather than tall talk and philosophy.\n\\ldots{}.. \n\nIn Sec 3.2 where we describe a possible application to medical records\nof 10 billion people- the number of planes, $q$, you require would\nbe $30$ or $40$ planes. Most of the time you will require very few\nplanes to separate all$N$ points. This is counter intuitive but will\nalways happen, for large $n$ the number of planes $q$ will be $O(log2N)$.\nThe following `explains the phenomena\\textquoteright{} : Consider\nan empty $n$-dimensional space, void of planes, then if you put one\nplane, it divides the space to 2 `quadrants\\textquoteright{}, the\n2nd plane will divide the space to 4 `quadrants\\textquoteright{},\nthe 3rd to 8 and the qth plane to$2^{q}$ `quadrants\\textquoteright{}.\n(The doubling stops only when $q=n$ , afterwards some closed regions\nare formed), so you see you have sufficient number of planes $q$\nto handle $N$ points even if you put one point in a single quadrant\nall you need is $log2(N)$planes.\n\n\\medskip{}\n\n\n\\textbf{\\large OTHER APPLICATIONS }{\\large \\par}\n\nIn all these applications one must some how employ mappings to reduce\nmulti-dimensional data, temporal data or any other kind of data to\nimage points in n-dimensional space, it is only then that the methods\nof the algorithm described in this paper can be usefully employed\nfor classification or decision making. An example illustrating this method is given in Ref.[12], by which any $n-$digit prime number can be depicted as a point in $n-$ dimension space; and therefore a prime number repository for storage and easy retrieval of primes can be created. The figure 12, \n below shows how all 2-digit prime numbers can be considered as points in 2-d space and separated by just 10 planes.\n\n\\medskip{}\n\\begin{figure}[htp]\n \\begin{center}\n\\includegraphics[scale=0.5]{primes-below-100-10-best-so-far.png}\n\\caption{Separation of 2-digit primes. Each prime is given a unique (x,y) \/\/ coordinate, eg, 53 is put in position (3,5)}\n\n\\label{fig:fig-m}\n\\end{center}\n\\end{figure} \n\\medskip{}\n\n\n\\textbf{\\large 1. Chess Games:}We can treat each chess game as a point\nin $2n$-dimension. Its coordinates being $(w1,b1,w2,b2,...,wn,bn)$\nthe $n$ moves made by white and black (it is not hard to convert\nthe moves $w1,b1,$ etc. into numbers so the above array which represents\na single chess game, is represented as a single point in 2n-dimension\nspace). It is possible to store numerous chess games, say $N$, for\neasy retrieval using the same technique as described for medical data.\nThe number of planes involved would be only $q=O(log2N)$. \n\n\\medskip{}\n\n\n\\textbf{\\large 2. Storing of sequences:}\\textbf{ (for easy retrieval\nand for making predictions using historical data)}\n\nIn the above cases the data is static and not dynamic. In many life\nsituations it is necessary to tackle dynamic data viz. sequences,\nlike a sequence of events, tracking of time signal, or even such matters\nas health monitoring of machines, (eg. see Jeff Hawking {[}13{]}-{[}14{]},\nwho has underlined the importance of storing and recalling a sequence\nof events in order to further research in artificial intelligence).\n\n\\medskip{}\n\n\nWe then need a method using which we can compare two sequences, say,\none a shorter sequence, $c(1),c(2),c(3),...,c(s)$,\nof length, $s$, with a longer sequence $h(1),h(2),h(3),..,h(r)$\nof length $r$, which is stored in the memory. The $c(j)$ may be\nthe `condition\\textquoteright{} of a machine in the year, $j$, of\nits working life. So that the possible `future` values of the shorter\nsequence$c(j)$ can be predicted by comparing it with a longer sequence$h(r)$,\nwhich is the record of some machine which has already lived its life\nand whose records are now stored in a memory along with the historical\nrecords of many such machines. We must imagine that the longer second\nsequence has been extracted from the data base, by the retrieval engine\n(see Sec 3.2), because it happens to have its first $s$ values somewhat\nclose to the $s$ values of the shorter sequence. \n\n\\medskip{}\n\n\nWe describe a method of mapping a sequence to points in a $n$-dimensional\nspace. This mapping is necessary to tackle dynamic situations and\ntracking\/memorizing a sequence of events. The data contained in the\nsequence are then `mere\\textquoteright{} points in $n$-dimensional\nspace which are then separated by using the algorithm using $q$ planes,\nthe Orientation Vectors are used to store the data in a repository\nSec 3.2). But in order to perform all these tasks we need a scheme\nto convert a sequence to points in $n$-dimensional space; we now\ndemonstrate how this can be done. \n\nWe will suppose$n$ is the max number of years that each record has.\nIt is possible to store the sequence as points in an $n$ dimensional\nspace using the following scheme. We define the coordinates in $n$-dimension\nspace for each member of the sequence as follows:\n\n$(c(1),0,0,0,...,0)$ : \\qquad{}\\qquad{}$n-1$ zeros;\n\n$(c(1),c(2),0,...,0)$:\\qquad{}\\qquad{}$n-2$ zeros\n\n$(c(1),c(2),c(3),0,0,...,0)$: \\qquad{}\\qquad{} $n-3$ zeros \n\n$.........$\n\n$(c(1),c(2),c(3),...,c(s),0,...,0):$ \\qquad{}\\qquad{}$(n-s)$ zeros \n\n\\medskip{}\n\n\nNote: Each of the above is a point in $n$-dimension space. Hence\nthe sequence of points given above is like a world-line in $n$-space.\n(Even though, we considered the $c(j)$ as a single number; there\nis no difficulty if this is (say) $m$ dimensional \\textendash{} then\nthe actual value of the bigger space will $n.m$ dimensional i.e.\nbe $n\\rightarrow n.m$ . And a similar scheme for the sequence$h$:\n\n$(h(1),0,0,0,...,0)$:\\qquad{}\\qquad{} $n-1$zeros;\n\n$(h(1),h(2),0,...,0):$\\qquad{}\\qquad{} $n-2$ zeros\n\n$(h(1),h(2),h(3),0,0,...,0):$\\qquad{}\\qquad{} $n-3$ zeros\n\n$......$\n\n$(h(1),h(2),c(3),...,h(n):$\\qquad{}\\qquad{} We assume $r=n$.\n\n\\medskip{}\n\n\nSimilarly, we can think of the sequence of points above as a world-line\nin n-space. The problem of comparing two sequences $c$ and $h$ has\nbeen reduced to the comparison of two world-lines. Since we have separated\nevery point such as the $h`s$ by using\nplanes, we can easily retrieve any sequence such as $h$ by using\nthe retrieval engine. Now, suppose we are presented a point $P$ (as\ndescribed in Sec 3.2), whose coordinates are $(c(1),c(2),c(3),...,c(s),0,...,0)$\n; the retrieval engine will retrieve a point $L$ whose coordinates\n$(h(1),h(2),h(3),...,h(s),0,...,0),$\nare closest to point $P$. This implies you have detected another\nworld-line $h$, which had had similar experiences in its first$s$\nepisodes of its `life' as that of world-line $c$; and hence it is\nquite possible that the fate of $c$ would be similar to what befell\n$h.$ Thus making it possible to retrieve the entire record $(h(1),h(2),c(3),...,h(n)$,\nthus enabling us to predict the possible values of $c(j),$ when $j>s$\nby looking at the other values $h(s+1),h(s+2),h(s+3),...,h(n)$\nwhich are now been made available. \n\n\\medskip{}\n\n\nIn the above example, you could also think of world line $c$ as the\nmedical record of some person who is presently living and is $s$\nyears old. And the world line $h$ as the medical record of some person\nwho had probably lived and died, but whose first $s$ years of life she\/he\nhad a similar medical history as that of $c$. This is how we can\npredict the `future life' of a presently occurring sequence of events\nby using the repository containing historical data of sequences that\nhave occurred in the past.\n\n\\medskip{}\n\n\n\\textbf{\\large 3. Other Possibilities:} We could similarly convert\ndecision trees and\/or logical trees to sequences which can then be\nmapped as points in a large $n$ dimension space, so we see the algorithm\ncan help in decision making and imitative learning etc. of large complex\ndata.\n\n\\medskip{}\n\n\nEND OF BRIEF ESSAY \n\n\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe Ryu-Takayanagi proposal \\cite{Ryu:2006bv,Ryu:2006ef} for holographic entropy and the covariant generalization \\cite{Hubeny:2007xt} by Hubeny, Rangamani, and Takayangi (HRT) relate the area of certain codimension-2 bulk extremal surfaces $\\Sigma$ to corresponding von Neumann entropies $S(\\rho_D)$ for the dual CFT. Each entropy involves a reduced density matrix $\\rho_D$ defined by restricting the CFT to a globally hyperbolic domain $D$. The main requirement is that, interpreting $D$ as a region of the conformal boundary of the asymptotically-AdS$_{d+1}$ bulk, the intersection $\\Sigma \\cap D$ must coincide with the boundary $\\partial C$ of a Cauchy surface $C$ in $D$. In addition, $\\Sigma$ must be homologous to $C$ and there should be no other such surface $\\Sigma'$ of smaller area. In such contexts, these proposals state\n\\be\n\\label{RT}\nS_{\\mathrm{ren}}(\\rho_D) = \\frac{\\mbox{Area}_{\\mathrm{ren}}(\\Sigma)}{4G_N}.\n\\ee\nOn both sides, the subscript ``ren'' indicates that divergent quantities have been renormalized in corresponding ways.\n\nWhile there is now an impressive amount of data supporting these conjectures (see e.g. \\cite{Ryu:2006bv,Ryu:2006ef,Headrick:2007km,Wall:2012uf,Faulkner:2013yia,Hartman:2013qma,Hartman:2013mia,Lewkowycz:2013nqa} and further references in \\cite{Nishioka:2009un}), much of the evidence remains rather qualitative.\nThis is especially true in the time-dependent context. As a result, it leaves open the question of what conditions might be required for \\eqref{RT} to hold quantitatively. We focus below on the possibility that analyticity of the bulk spacetime may be important, and on related questions involving complex extremal surfaces. Understanding such issues may be important for properly interpreting recent work using Ryu-Takayanagi and HRT to study the relationship between bulk and boundary notions of localization \\cite{Bousso:2012sj,Czech:2012bh,Hubeny:2012wa} and to derive bulk dynamics from that of the CFT \\cite{Nozaki:2013vta,Lashkari:2013koa,Faulkner:2013ica,Swingle:2014uza}.\n\nOur study is motivated by two observations.\nThe first is that all attempts \\cite{Fursaev:2006ih,Faulkner:2013yia,Hartman:2013mia,Lewkowycz:2013nqa,Fursaev:2014tpa} to provide general derivations of \\eqref{RT} make use of both Euclidean path integrals and the bulk saddle-point approximation. This structure inherently relies on some measure of analytic continuation, and suggests that one may find cases where intrinsically complex saddles dominate the path integral. While the arguments in these works (and in particular \\cite{Lewkowycz:2013nqa}) are phrased in the static context of the original Ryu-Takayangi proposal \\cite{Ryu:2006bv,Ryu:2006ef}, the only crucial ingredient appears to be the existence of a well-defined -- not necessarily real -- asymptotically-Euclidean section. As noted in e.g. \\cite{Balasubramanian:2014hda}, for any spacetime with this property analytic continuation to the real Lorenzian section will imply the HRT conjecture so long as the real Lorentzian extremal surface provides the most relevant saddle point. This suggests that \\eqref{RT} might apply only to analytic spacetimes and, furthermore, that even in this case it may generally require the use of complex extremal surfaces.\n\nThe second observation is an explicit example of the concerns raised by the first. Recall (see e.g. \\cite{Balasubramanian:1999zv,Kraus:2002iv}) that two-point functions of heavy quantum fields may be approximated by $e^{-mL}$, where $L$ is the proper length of a geodesic connecting the points and $m$ is the relevant mass. Since geodesics are extremal surfaces of codimension $d$ in a $(d+1)$-dimensional spacetime, this geodesic approximation shares formal similarities with the holographic entanglement proposal. Furthermore, it can be derived from the stationary phase approximation to the Euclidean path integral, and the fact \\cite{Holzhey:1994we} that CFT von Neumann entropies may be computed from twist operator correlation functions may provide a tight connection to holographic entanglement for $d=2$ (with corresponding generalizations from geodesics to other minimal surfaces when $d > 2$). But for the geodesic approximation one can show that analyticity is indeed generally required \\cite{Louko:2000tp} and that complex geodesics play critical roles in certain contexts \\cite{Fidkowski:2003nf}.\n\nThough this concern has been understood for some time, there is a surprising lack of discussion in the literature. This may be due in part to the lack of known examples. Indeed, to our knowledge no complex codimension-2 surfaces have been previously identified that satisfy appropriate boundary conditions in any spacetime. We overcome this obstacle below by exhibiting families of complex codimension-2 surfaces in standard $(d+1)$-dimensional planar black holes corresponding as in \\cite{Maldacena:2001kr} to thermofield double states in dual CFTs on $\\mathbb{R}^d$. We investigate the Ba\\~nados-Teitelboim-Zanelli (BTZ) solution, Schwarzschild-AdS$_{d+1}$ black holes for $3 \\le d \\le 7$, and Schwarzschild-Lifshitz black holes \\cite{Taylor:2008tg}. We work in the maximally analytically extended spacetimes, where the real Lorentzian section has two asymptotic regions. The dual CFT thus lives on two copies of $\\mathbb{R}^d$. The surfaces we consider are anchored on both boundaries at some spatial location $x^\\perp$ and some time $t_b$, much as in \\cite{Hartman:2013qma}. They would thus be appropriate for computing the entropy of the CFT on a pair of half $(d-1)$-planes ending at $x_\\perp$ at the time $t_b$, with one half-plane in each copy of $\\mathbb{R}^d$. For this case, the globally hyperbolic domain $D$ mentioned in the introduction is just the corresponding pair of Rindler-like wedges with each origin of Rindler coordinates located at $t_b, x^\\perp$. In all cases we identify complex extremal surfaces satisfying boundary conditions relevant to the holographic entanglement conjectures. For Schwarzschild-AdS and Schwarzshild-Lifshitz we find families where the real part of the area is smaller than for corresponding real extremal surfaces.\n\nWe begin by discussing the status of \\eqref{RT} for complex surfaces in section \\ref{sec:interp}.\nThe area of a complex surface is generally complex, while entropies must be real. We must therefore modify \\eqref{RT} if complex surfaces turn out to be relevant. This issue remains confusing, but for the present work we choose to study a straw-man model that replaces $A_{\\mathrm{ren}}$ in \\eqref{RT} by its real part.\n\nSection \\ref{sec:setup} then explains our general approach to finding the desired complex surfaces and studying their properties. This is largely a transcription of the method used for complex geodesics in \\cite{Andrade:2013rra}, which in turn builds on many other works. However, we take the opportunity to make certain improvements and corrections. The technique applies to surfaces of any codimension $n$, and we study complex geodesics in Schwarzschild-AdS$_{d+1}$ as an illustration of the general method. The results for $d \\neq 4$ appear to be new, and for $d > 4$ indicate that real geodesics in the Lorentz-signature spacetime can fail to dominate even on surfaces invariant under time-reflection symmetry (where analytic continuation between Euclidean and Lorentzian signatures is in some sense trivial). This emphasizes that complex surfaces could be important even in the original Ryu-Takayanagi context of static bulk spacetimes and not just in the more general time-dependent HRT context.\n\nComplex codimension-2 surfaces for planar BTZ, Schwarzschild AdS$_{d+1}$ (with $3 \\le d \\le 7$), and Schwarzschild-Lifshitz are studied in section \\ref{sec:examples}. The BTZ case yields a complete analytic solution showing that all complex extremal surfaces are in some sense higher copies of the real HRT surfaces. It follows that the same is true for global AdS${}_3$, of which BTZ is just a subset, and also for Poincar\\'e AdS${}_3$. Schwarzschild-AdS$_{d+1}$ is more interesting, and exhibits several qualitatively-different families of complex extremal surfaces. We identify two families where the qualitative behavior of $\\mathrm{Re} \\ A_{\\mathrm{ren}}$ matches expectations for the dual CFT entropy on our half-planes. For the family called contour $C$ below, $\\mathrm{Re} \\ A_{\\mathrm{ren}}$ is notably less than for the corresponding real extremal surfaces. It is thus plausible that the dual CFT entropy is indeed controlled by these complex surfaces. Our brief study of Schwarzschild-Lifshitz indicates results analogous to those for Schwarzschild-AdS.\n\nWe close with a summary and some final discussion in section \\ref{sec:discussion}. In particular, we note that all complex extremal surfaces in our spacetimes lie on what are naturally called secondary sheets of an associated Riemann surfaces. This feature may make it difficult for the associated saddles to contribute to the stationary phase approximation of the relevant path integrals.\n\n\n\n\\section{Entropy from complex areas?}\n\\label{sec:interp}\n\nAs noted above, if complex surfaces are indeed relevant to the Ryu-Takayanagi or HRT conjectures, the formula \\eqref{RT} will require modification. The issue is that the imaginary part of $A_{\\mathrm{ren}}$ is generally non-zero while the von Neumann entropy is real by definition. Now, since complex numbers enter only by analytic continuation from a real spacetime, complex extremal surfaces must appear in what one might call complex-conjugate pairs satisfying identical boundary conditions with complex-conjugate renormalized areas~$A_\\mathrm{ren}$ and~$A_\\mathrm{ren}^*$. The two members of each pair are obtained by analytically continuing along corresponding paths but in opposite directions. One might thus hope to combine~$A_\\mathrm{ren}$ and~$A_\\mathrm{ren}^*$ in some way to give a real entropy $S$.\n\nThe question is just how this should be done. In parallel with the geodesic approximation to two-point functions, it is natural to interpret $A_\\mathrm{ren}\/4G_N$ as a saddle-point approximation to the logarithm of a partition function. One might then expect a pair of relevant saddles $s_1,s_2$ to give\n\n\\be\n\\label{eq:combinesaddles}\nS_\\mathrm{ren} = -\\ln \\left( C(s_1) e^{-A_\\mathrm{ren}(s_1)\/4G_N} + C(s_2) e^{-A_\\mathrm{ren}(s_2)\/4G_N} \\right),\n\\ee\nwhere the factors $C(s_1), C(s_2)$ represent finite $G_N$ corrections that in particular include fluctuation determinants from quantum fields propagating on the classical spacetimes $s_1,s_2.$ \n\nFor $A_{\\mathrm{ren}}(s_1) = A_{\\mathrm{ren}}(s_2)^*$ (and presumably $C(s_1) = C(s_2)^*$) the entropy becomes\n\\be\nS_\\mathrm{ren} = \\frac{\\mathrm{Re} \\, A_\\mathrm{ren}}{4G_N} - \\ln 2|C(s_1)| - \\ln \\cos \\left(-\\frac{\\mathrm{Im} \\, A_{\\mathrm{ren}}}{4G_N} + \\phi \\right),\n\\ee\nwhere the phase~$\\phi$ is defined by $C(s_1) = |C(s_1)|e^{i\\phi}$. But for small~$G_N$, where the formula~\\eqref{RT} holds, the cosine oscillates rapidly. This will often give $S_\\mathrm{ren}$ an unphysical imaginary part. It is not a priori clear whether one should think of this imaginary part as being of order $1\/G_N$ or instead being bounded but rapidly changing as $G_N \\rightarrow 0$. In the latter case it would be problematic only at the level of subleading corrections, and we might content ourselves with using\n\\begin{equation}\n\\label{eq:real1}\nS_\\mathrm{ren} \\approx \\frac{\\mathrm{Re} \\, A_\\mathrm{ren}}{4G_N}\n\\end{equation}\nat leading order in $1\/G_N$.\n\nInterestingly, the actual form of the Lewkowycz-Maldacena argument \\cite{Lewkowycz:2013nqa} for \\eqref{RT} -- or indeed any replica argument with a saddle-point approximation -- appears to lead to result somewhat different from \\eqref{eq:combinesaddles}\\footnote{This point was brought to our attention through a conference presentation by Matt Headrick \\cite{HeadrickTalk}, who in turn learned it from private discussion with Rob Myers \\cite{MyersPrivate}.}. This occurs because it is the Renyi entropies $S_n = -\\frac{1}{n-1} \\ln \\mathrm{Tr} \\, \\rho^n$ (for integer $n$) that are directly given by partition functions, and for which the saddle-point approximation is then used. The von Neumann entropy is finally computed by analytically continuing to all real $n$ and using\n\\be\n\\label{eq:RenyiTovN}\nS_{\\mathrm{ren}} = \\lim_{n \\rightarrow 1} S_n = - \\lim_{n \\rightarrow 1} \\frac{1}{n-1} \\ln \\mathrm{Tr} \\rho^n ,\n\\ee\nrenormalizing each expression as needed. In the saddle point approximation we have $\\mathrm{Tr} \\, \\rho^n \\approx e^{-I_n\/4G_N}$ for some $I_n$. If the von Neumann entropy is to be finite, $I_n$ must vanish at $n=1$. So, for fixed $G_N$, as $n \\rightarrow 1$ we may write\n\\be\ne^{-I_n\/4G_N} = 1 - (n-1) \\frac{1}{4G_N}\\left.\\frac{dI_n(s)}{dn}\\right|_{n=1} + \\cdots,\n\\ee\nwhere $s$ now denotes a family of saddles with one for each $n$. If two such families are relevant, we have\n\\begin{subequations}\n\\bea\nS_n =& -\\frac{1}{n-1} \\ln \\left( C_n(s_1) e^{-I_{n}(s_1)\/4G_N} + C_n(s_2) e^{-I_{n}(s_2)\/4G_N} \\right) \\\\\n =& -\\frac{1}{n-1} \\ln \\left( C_n(s_1) \\left[ 1 - (n-1) \\frac{1}{4G_N}\\left.\\frac{dI_n(s_1)}{dn}\\right|_{n=1} + \\cdots \\right] \\right. \\\\\n &+ \\left. C_n(s_2) \\left[ 1 - (n-1) \\frac{1}{4G_N} \\left. \\frac{dI_n(s_2)}{dn}\\right|_{n=1} + \\cdots \\right] \\right).\n\\eea\n\\end{subequations}\nA finite von Neumann entropy requires the normalization $C(s_1) + C(s_2) =1$. Taking $n \\rightarrow 1$ thus yields\n\\be\n\\label{eq:MLlimit}\nS_{\\mathrm{ren}} = \\frac{1}{4G_N} \\left.\\left( C_1(s_1) \\frac{dI_n(s_1)}{dn} + C_1(s_2) \\frac{dI_n(s_2)}{dn} \\right)\\right|_{n=1},\n\\ee\nwhere we have neglected a term involving $dC_n\/dn$ which is subleading at small $G_N$.\n\nFurthermore, in any such argument, the saddle at $n=1$ is taken to be known and fixed; indeed, it should give the bulk dual of the original mixed state $\\rho$. Thus $s_1$ and $s_2$ both approach this fixed saddle as $n \\rightarrow 1$. As a result, if the saddle-point approximation continues to hold as $n \\rightarrow 1$, the fluctuation contributions $C_1(s_1)$, $C_1(s_2)$ must agree at $n=1$. The constraint $C_1(s_1) + C_1(s_2) =1$ then requires both to be $1\/2$. Since obtaining \\eqref{RT} in the case of a single extremal surface requires $A_{\\mathrm{ren}} = dI_n(s_1)\/dn|_{n=1}$, with two extremal surfaces the argument gives\n\\begin{equation}\n\\label{eq:straw}\nS_{\\mathrm{ren}} = \\frac{A_{\\mathrm{ren}}(s_1) + A_{\\mathrm{ren}}(s_2) }{8G_N}\n\\end{equation}\nso long as each surface leads to a corresponding family of saddles for $\\mathrm{Tr} \\, \\rho^n$ for all $n$. Thus the area in \\eqref{RT} has been replaced with the average of the two areas. For $A_{\\mathrm{ren}}(s_1) = A_{\\mathrm{ren}}(s_2)^*$ this is equivalent to taking the real part; i.e., the final conclusion is essentially identical to \\eqref{eq:real1}.\n\nThe result \\eqref{eq:straw} appears to be physically incorrect. As a concrete example, consider the black hole quotients of AdS${}_3$ described in \\cite{Brill:1995jv,brill:1998pr,Aminneborg:1997pz,Aminneborg:1998si} that have a single asymptotically-AdS region (which asymptotes to global AdS${}_3$). Such spacetimes were called AdS geons in \\cite{Louko:1998hc}, which suggested that they are dual to pure CFT states. This was later argued in detail by \\cite{Maldacena:2001kr,Skenderis:2009ju}. This is consistent with the fact that any Cauchy surface for the conformal boundary is homologous in the bulk to the empty set. So minimizing over real extremal surfaces leads to $S =0$ as desired. But the bifurcation surface of the black hole horizon is another extremal surface, this time of positive area. Averaging the two as in \\eqref{eq:straw} would give $S > 0$ and contradict the description as a pure state.\n\nIt remains possible that \\eqref{eq:straw} might nevertheless be salvaged by including in the average further extremal surfaces not yet identified. Complex extremal surfaces could contribute negatively and cancel the positive contribution from the extremal surface at the horizon. But this seems unlikely and, even if true, would make the entanglement conjectures extremely difficult to use in practice. One instead expects that the saddle-point phase approximation simply fails near $n=1$, as this is typically the case when one varies parameters so as to make two saddles coincide.\n\nThe above discussion mostly serves to illustrate our ignorance of how \\eqref{RT} should be modified to accommodate complex extremal surfaces. While we have discussed the problem at the level of the von Neumann entropy, the replica discussion above makes it clear that the issue is already present at the level of the Renyi entropies. The point is that $\\mathrm{Tr} \\rho^n$ must be positive definite for any quantum system. But writing\n\\begin{equation}\n\\mathrm{Tr} \\rho^n = e^{-I_n\/4G_N} + e^{-I^*_n\/4G_N}\n\\end{equation}\nfor a complex conjugate pair of saddles one finds that the sign of the right-hand side oscillates quickly as $G_N \\rightarrow 0$ when the action $I_n$ is not real. One could choose to take this as an indication that only saddles with real action can contribute to Renyi entropies in the semiclassical limit, and thus that only extremal surfaces with real areas could contribute to von Neumann entropies. But other possibilities may exist. For example, we recall that in some contexts \\cite{Marolf:1996gb} carefully studying contours of integration can show that the correct semi-classical approximation is $e^{-|S|}$. It would be very interesting if a similar conclusion might somehow apply here.\n\nSince we found two arguments above leading us to replace $A_{\\mathrm{ren}}$ in \\eqref{RT} with its real part,\nwe adopt this hypothesis for discussion purposes below. To emphasize the uncertainty in this conclusion, we refer to this suggestion as the straw-man proposal\\footnote{\\label{foot:subadd} It would be very interesting to understand whether our straw man proposal -- or indeed any other proposal involving complex extremal surfaces -- satisfies well known properties of entropies like strong subadditivity. This property has been shown to hold in \\cite{Headrick:2007km} and \\cite{Wall:2012uf} for the original Ryu-Takayanagi and HRT proposals based solely on real extremal surfaces, but it is far from clear that they continue to hold for complex generalizations.}.\nWe will consider each complex conjugate pair separately and not attempt to further combine the results from various pairs. We also comment on the relative size of $\\mathrm{Re} \\, A_{\\mathrm{ren}}$ for various such complex pairs, though we refrain from stating whether this means that any such pair necessarily dominates the result. Indeed, given a set of saddles it is typically difficult to determine whether the contour of integration can be deformed to pass through them in such a way that they can actually contribute to the desired saddle-point approximation. We defer further discussion of this issue to section \\ref{sec:discussion}.\n\n\n\n\\section{Method and Analytic Structures}\n\\label{sec:setup}\n\nWe now outline our general procedure for finding complex extremal surfaces. After a brief introduction to the spacetimes of interest, the basic techniques are presented in section \\ref{subsec:surf} generalizing methods used to study complex geodesics in \\cite{Andrade:2013rra} (based on e.g. \\cite{Festuccia:2005pi,Festuccia:2008zx,Hartman:2013qma}). Relevant analytic structures are discussed in section \\ref{subsec:complexsurf}. We consider extremal surfaces $\\Sigma$ of general codimension $n$, and we illustrate the method in section \\ref{subsec:geoSAdS} by studying complex geodesics in Schwarzschild AdS$_{d+1}$.\n\nAs noted above, for simplicity we study $(d+1)$-dimensional spacetimes describing planar black holes with AdS-like asymptotics in each of two asymptotic regions. We therefore restrict to spacetimes of the form\n\\be\n\\label{eq:general}\nds^2 = -f(r) dt^2 + \\frac{dr^2}{g(r)} + r^2 dx_{d-1}^2,\n\\ee\nwhere~$f(r)$ and~$g(r)$ each have a simple zero at some~$r = r_h > 0$ corresponding to a horizon with inverse temperature\n\\be\n\\label{eq:beta}\n\\beta = \\frac{4\\pi}{\\sqrt{f'(r_h)g'(r_h)}}.\n\\ee\nWe assume our spacetimes to have timelike conformal boundaries at $r = \\infty$, though we make no further assumption about the large~$r$ behavior of~$f$ and~$g$. In particular, we allow both asymptotically AdS$_{d+1}$ and asymptotically Lifshitz spacetimes \\cite{Kachru:2008yh} restricted to $z \\ge 1$ (so that the null energy condition is satisfied~\\cite{Hoyos:2010at}). We assume that $f$, $g$, and $f\/g$ are analytic functions of $r$ everywhere on the complex plane except perhaps at $r=0$ and $\\infty$. In the Lifshitz case, $r=0, \\infty$ will be branch points so that it is better to say that $f$, $g$, and $f\/g$ are analytic on appropriate Riemann surfaces.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[page=1]{Complex_surfaces_v10-pics.pdf}\n\\caption{A conformal diagram of our spacetimes. The asymptotic regions are located in the left and right regions. The imaginary part of the time coordinate $t$ is constant in each wedge, and $t$ has period~$t \\sim t + i\\beta$. We consider extremal surfaces anchored at the points indicated on each boundary.}\n\\label{fig:wedges}\n\\end{figure}\n\n\n\\subsection{Extremal Surfaces}\n\\label{subsec:surf}\n\nTo study surfaces $\\Sigma$ of codimension $n$, it is useful to divide the $(d-1)$ coordinates $x$ into two families\n\\begin{subequations}\n\\bea\n\\left\\{x^\\perp\\right\\} &= \\left\\{x^1,\\ldots,x^{n-1}\\right\\}, \\\\\n\\left\\{x^\\parallel\\right\\} &= \\left\\{x^n,\\ldots,x^{d-1}\\right\\}.\n\\eea\n\\end{subequations}\nWe will require $x^\\perp$ to be constant on the boundary $\\partial \\Sigma$ of $\\Sigma$, and by translation invariance we may take $(x^\\perp)|_{\\partial \\Sigma} =0$. This fixes $n-1$ boundary conditions, so it remains only to specify a time coordinate on $\\partial \\Sigma$.\n\nThe horizon at~$r = r_h$ divides the spacetime into four wedges, and we can use the Schwarzschild-like coordinates $t,r$ of~\\eqref{eq:general} in all four wedges by analytic continuation. This prescription causes the imaginary part of $t$ to shift by ~$i\\beta\/4$ every time a horizon is crossed, as shown in figure~\\ref{fig:wedges}, and imposes a periodicity~$t \\sim t+i\\beta$. We thus require $\\Sigma$ to stretch between the two boundaries, with $t|_{\\partial \\Sigma} = t_b$ on the right and~$t|_{\\partial \\Sigma} = -t_b + i\\beta\/2$ on the left. We take take~$t_b$ to be a real parameter specifying the desired boundary conditions and more generally use $\\Delta t$ to denote the time difference between the two ends of any extremal surface with $(x^\\perp)|_{\\partial \\Sigma} = 0$. It will sometimes be useful to break~$\\Delta t$ into its real and imaginary parts by writing~$\\Delta t = -2t_R + i t_I$ so that surfaces satisfying our boundary conditions have~$t_R = t_b$ and~$t_I = \\beta\/2$.\n\nSince our boundary conditions are invariant under translations in $x^\\parallel$ we assume $\\Sigma$ to share this symmetry. Thus the problem reduces to finding $(t, r, x^\\perp)$ as functions of a single parameter $\\lambda$ which we specify below. In fact, since momentum conservation requires $x^\\perp$ to be monotonic in $\\lambda$, the fact that $x^\\perp$ vanishes on both boundaries implies $x^\\perp =0$ on all of $\\Sigma$ so that we need only solve for the two embedding functions ~$(t,r) = (T(\\lambda), R(\\lambda))$. The area functional then becomes\n\\be\n\\label{eq:area}\nA = V_{d-n} \\int d\\lambda \\, R^{d-n} \\sqrt{-f(R) \\dot{T}^2 + \\frac{\\dot{R}^2}{g(R)}} \\equiv V_{d-n} \\int d\\lambda \\, \\mathcal{L},\n\\ee\nwhere~$V_{d-n}$ is the volume of the~$x^\\parallel$ space and dots denote derivatives with respect to~$\\lambda$.\n\nSince~$T$ is cyclic in~\\eqref{eq:area}, its conjugate momentum (hereafter referred to as energy) is a constant of motion:\n\\be\n\\label{eq:E}\nE = -\\frac{\\partial \\mathcal{L}}{\\partial \\dot{T}} = \\frac{R^{2(d-n)} f(R)}{\\mathcal{L}} \\, \\dot{T}.\n\\ee\nNote that~$E$ may be complex for complex surfaces $\\Sigma$. Finally, we invoke the reparametrization freedom of \\eqref{eq:area} to choose $\\lambda$ to satisfy~$\\mathcal{L} = R^{d-n}$. This constraint serves as the remaining equation of motion, which using~\\eqref{eq:E} can be written as the Newtonian particle-in-a-potential problem\n\\be\n\\label{eq:Rdot}\n\\dot{R}^2 + V_\\mathrm{eff}(R) = 0, \\mbox{ where } V_\\mathrm{eff}(R) = -g(R) - \\frac{E^2 g(R)}{R^{2(d-n)} f(R)}.\n\\ee\n\nWe have thus reduced the system to quadratures. In particular, since we allow complex $R$ and $T$, given any contour~$\\gamma$ in the complex $R$ plane we can solve \\eqref{eq:Rdot} and \\eqref{eq:E} for $dT\/dR$ and integrate to find a $T(R)$ that solves the equations of motion\\footnote{\\label{error} This point was not correctly discussed in \\cite{Andrade:2013rra}, which instead claimed that each complex geodesic had a preferred turning point. This is not generally true, but does not affect the final results of \\cite{Andrade:2013rra}.}. The only question is whether the associated complex extremal surface satisfies our boundary condition. I.e., we must require both ends of the contour $\\gamma$ to approach $R = \\infty$ along the real axis and then compare the total elapsed time\n\\be\n\\label{eq:deltat}\n\\Delta t \\equiv -2t_R + i t_I = \\int_\\gamma \\frac{E}{R^{d-n} f(R) \\sqrt{-V_\\mathrm{eff}(R)}} \\, dR\n\\ee\nwith $-2t_b + i \\beta\/2$.\n\nA similar calculation gives the renormalized area of the surface as\n\\be\n\\label{eq:areacontour}\nA_\\mathrm{ren} = \\lim_{\\epsilon \\to 0} \\left(V_{d-n} \\int_{\\gamma_\\epsilon} \\frac{R^{d-n}}{\\sqrt{-V_\\mathrm{eff}(R)}} \\, dR + A_\\mathrm{ct}(\\epsilon)\\right),\n\\ee\nwhere~$\\epsilon$ is a UV regulator, $A_\\mathrm{ct}(\\epsilon)$ is a counterterm that cancels the~$\\epsilon$-divergent terms in~$A$, and~$\\gamma_\\epsilon$ is a regulated contour that runs to~$R = r_h\/\\epsilon$ rather than~$R \\to \\infty$. Since the renormalized area is an on-shell action, \\eqref{eq:deltat} and \\eqref{eq:areacontour} satisfy the Hamilton-Jacobi relation\n\\be\n\\label{eq:HJ}\ndA_\\mathrm{ren} = - V_{d-n} E \\ d (\\Delta t),\n\\ee\nwhich can also be checked directly. This structure precisely parallels that of complex geodesics; see e.g. \\cite{Festuccia:2005pi} and the recent review in \\cite{Andrade:2013rra}.\n\nSince $V_\\mathrm{eff}(R)$ generally vanishes at several values of $R$, the function $\\sqrt{-V_\\mathrm{eff}(R)}$ defines a non-trivial Riemann surface over the complex $R$ plane. There may also be additional branch points at $R=0$ and at $R = \\infty$ (for the Lifshitz case). The branch points of $\\sqrt{-V_\\mathrm{eff}(R)}$ will be denoted $R_{\\mathrm{branch}}(E)$. So long as $f$ and $g$ have no branch points themselves (i.e., except for the Lifshitz case), the Riemann surface for $\\sqrt{-V_\\mathrm{eff}(R)}$ has precisely two sheets.\n\nBecause the sign of $\\sqrt{-V_\\mathrm{eff}(R)}$ in \\eqref{eq:deltat} determines the sign of $\\dot{R}$, our boundary conditions require it to take opposite values at the two ends of $\\gamma$. In particular, in the non-Lifshitz case allowed contours $\\gamma$ thus run between endpoints $R = \\infty$ on opposite sheets of the Riemann surface for $\\sqrt{-V_\\mathrm{eff}(R)}$, and without loss of generality we may take them to run from the negative branch to the positive branch. Examples of such contours are shown in figure \\ref{fig:integrationcontour}. In the limit where the contour is deformed to tightly circle some branch point, it is natural to think of the branch point as a turning point of the trajectory. This is the case for contours along the real $R$-axis -- such as the one shown in figure \\ref{fig:integrationcontour}(b)-- that describe real extremal surfaces in either Euclidean or Lorentzian signature.\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[]{\n\\includegraphics[page=2]{Complex_surfaces_v10-pics.pdf}\n}\n\\hspace{0.5cm}\n\\subfigure[]{\n\\includegraphics[page=3]{Complex_surfaces_v10-pics.pdf}\n}\n\\caption{The branching structure of the integrands of~\\eqref{eq:deltat} and~\\eqref{eq:areacontour} in the complex~$R$-plane, and sample contours of integration~$\\gamma$. The number of branch points depends on the precise form of~$V_\\mathrm{eff}$; here we draw four, as for geodesics in~$d = 3$ AdS-Schwarzschild. The branch points correspond to zeros of~$V_\\mathrm{eff}$ and often an additional branch point at $R=0$. We introduce branch cuts in order to draw figures; the solid and dashed portions of~$\\gamma$ indicate segments that run on different sheets of the associated Riemann surface. For convenience we choose the branch cuts to run radially inward, connecting all other branch points $R_{\\mathrm{branch}}$ directly to $R=0$. We adopt this convention even when $R=0$ is not a branch point -- in effect momentarily introducing an artificial branch point whose effects must disappear from the final results. Figure~(a) shows the generic (complex~$E$) case in which all the branch points lie at complex~$R$. Figure~(b) shows the special case in which~$E$ is real, in which case at least one of the branch points lies on the positive~$R$-axis. The extremal surface corresponding to the indicated contour~$\\gamma$ is then equivalent to a real extremal surface which may be described as having a turning point at the encircled branch point. The integrand for~$\\Delta t$ may also have poles at other values of~$R$, but these are not shown.}\n\\label{fig:integrationcontour}\n\\end{figure}\n\nOf course, smooth deformations of the contour $\\gamma$ that preserve the endpoints will not change \\eqref{eq:deltat} or \\eqref{eq:areacontour}. Two contours related in this way will be said to describe equivalent extremal surfaces, with inequivalent surfaces at given $E$ corresponding to homotopically distinct contours on the Riemann surface for $\\sqrt{-V_\\mathrm{eff}(R)}$.\n\n\n\n\\subsection{Analytic Structure of $\\Delta t(E)$ and $A_\\mathrm{ren}(E)$}\n\\label{subsec:complexsurf}\n\nOne would like to use \\eqref{eq:deltat} and \\eqref{eq:areacontour} to define $A_\\mathrm{ren}$ as a function of $t_b$. But in general there will be multiple inequivalent extremal surfaces for a given $t_b$. As a result, $A_\\mathrm{ren}(t_b)$ is in fact properly defined on a multi-sheeted Riemann surface. A useful way to deal with this complication is to work directly with $\\Delta t(E)$ and $A_\\mathrm{ren}(E)$ as described by \\eqref{eq:deltat} and \\eqref{eq:areacontour}. While $\\Delta t(E)$ and $A_\\mathrm{ren}(E)$ are again defined on non-trivial Riemann surfaces, their structure is closely related to that of the branch points $R_{\\mathrm{branch}}(E)$ for $\\sqrt{-V_\\mathrm{eff}(R)}$. This structure is again like that of the geodesic case presented in \\cite{Andrade:2013rra}, though our discussion below corrects some minor errors in \\cite{Andrade:2013rra} related to footnote \\ref{error}.\n\nIndeed, the functions \\eqref{eq:deltat} and \\eqref{eq:areacontour} are analytic in $E$ so long as the contour~$\\gamma$ can be deformed to avoid branch points~$R_\\mathrm{branch}(E)$ or poles. But at certain critical energies two branch points will merge. Contours $\\gamma$ that run between these branch points will be said to be pinched as $E$ becomes critical, and can no longer be deformed to avoid them. Mergers of three or more branch points do not occur for the examples considered below.\n\nWhen the integration contour is pinched we divide the critical energies into two classes, which we denote $E_c$ and $E'_c$. The former ($E_c$) are energies where the merging branch points are both simple roots of~$V_\\mathrm{eff}$ (with no other coincident singularities\\footnote{Section \\ref{subsec:SAdS} will describe a case where two simple roots of $V_\\mathrm{eff}$ merge with a non-branching singularity (a pole) at $R=0$.}), so that~$V_\\mathrm{eff}$ develops a double root at $E_c$. Thus as $E \\rightarrow E_c$, each integrand becomes structurally similar to $|R - R_\\mathrm{branch}|^{-1}$ so that the integrals $\\Delta t(E)$ and $A_\\mathrm{ren}(E)$ diverge. Careful examination shows that when the contour $\\gamma$ is pinched at such $E_c$, the functions $\\Delta t(E)$ and $A_\\mathrm{ren}(E)$ both behave like $C \\ln (E-E_c)$ near $E_c$ for some complex coefficient $C$. So both have logarithmic branch points at $E_c$. In contrast, the $E'_c$ are energies where roots of~$V_\\mathrm{eff}$ moves to $R=0$ or (for Lifshitz) to $R = \\infty$. In general, $\\Delta t(E)$ and $A_\\mathrm{ren}(E)$ do not diverge at such $E'_c$, though they do have branch points there.\n\nWhen the integration contour is not pinched, $\\Delta t(E)$ and $A_\\mathrm{ren}(E)$ remain analytic even when roots merge; such situations are neither $E_c$'s nor $E'_c${}'s and will not be called critical. Since we will see below that different sheets of our Riemann surface are associated with different contours $\\gamma$, this means that the identification of a given energy $E$ as being critical (or not) will vary as one moves from one sheet to another.\n\nSince $\\Delta t(E)$ diverges at the $E_c$, we expect the large time behavior of at least some families of extremal surfaces to be determined by the $E_c$. As for the geodesic case \\cite{Festuccia:2005pi}, for a family of extremal surfaces with $\\Delta t \\rightarrow \\infty$ as $E \\rightarrow E_c$, the Hamilton-Jacobi relation \\eqref{eq:HJ} immediately yields a linear relationship between $\\Delta t(E)$ and $A_\\mathrm{ren}(E)$. This can also be seen from the fact that both behave like $\\ln (E-E_c)$. In particular, for codimension-2 extremal surfaces (i.e.~$n = 2$), one has\n\\be\n\\label{eq:entanglementvelocity}\n\\frac{A_\\mathrm{ren}}{4G_N} = S_\\mathrm{ren} = -\\frac{V_{d-2} E_c}{4G_N} \\, \\Delta t + \\cdots \\equiv -\\frac{1}{2} s v V_{d-2} \\Delta t + \\cdots,\n\\ee\nwhere~$s = r_h^{d-1}\/4 G_N$ is the thermal entropy density,~$v$ is a constant, and~$\\cdots$ denote subleading terms in~$\\Delta t$. For surfaces of this type that dominate the HRT prescription, the constant~$v$ is a speed characterizing the rate of growth of the entanglement entropy; see e.g. \\cite{Hartman:2013qma,Liu:2013iza,Liu:2013qca}. It is interesting that the relation \\eqref{eq:entanglementvelocity} is linear for asymptotically Lifshitz spacetimes (and, indeed, for more general asymptotics) as well as for the asymptotically AdS case. This speed was recently computed in \\cite{Alishahiha:2014cwa} along with other properties of Schwarzschild-Lifshitz black holes.\n\n\n\\begin{figure}[t]\n\\centering\n\\subfigure{\n\\includegraphics[page=4]{Complex_surfaces_v10-pics.pdf}\n}\n\\hspace{0.5cm}\n\\subfigure{\n\\includegraphics[page=5]{Complex_surfaces_v10-pics.pdf}\n}\n\\caption{Sample integration contours $\\gamma_1', \\gamma_2'$ for~\\eqref{eq:deltat} and~\\eqref{eq:areacontour} that define secondary Riemann sheets of~$\\Delta t(E)$. Both contours are obtained from $\\gamma$ in figure~\\ref{fig:integrationcontour} by exchanging the branch points in quadrants $1$ and $3$. For $\\gamma_1'$ the originally-encircled branch point passes below the other during the exchange, while for $\\gamma_2'$ it passes above. At each step, the contour must be deformed to keep it smooth on the associated Riemann surface; it must avoid both branch points and poles, though for simplicity we show only the former.}\n\\label{fig:deformedcontours}\n\\end{figure}\n\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[page=6]{Complex_surfaces_v10-pics.pdf}\n\\caption{A sample choice of branch cut structure used to define a single sheet of~$\\Delta t$ and~$A_\\mathrm{ren}$ in the complex~$E$-plane; the particular structure shown here is that of e.g.~geodesics in Reissner-Nordstr\\\"om AdS$_5$ or codimension-2 extremal surfaces in Schwarzschild-AdS$_7$. The branch points shown here correspond to the critical energies~$E_c$ at which the contour of integration $\\gamma$ for ~\\eqref{eq:deltat} and~\\eqref{eq:areacontour} becomes pinched between two roots of~$V_\\mathrm{eff}$ that coincide, and are therefore energies at which~$|\\Delta t|$ and~$|A|$ diverge.}\n\\label{fig:Ecuts}\n\\end{figure}\n\nTracing a closed contour in the complex~$E$-plane around one of the branch points of~$\\Delta t(E)$ results in movement\nfrom one sheet of~$\\Delta t(E)$ to another. Traveling around such a contour corresponds to swapping two of the roots of~$V_\\mathrm{eff}$, so one can think of constructing a secondary sheet of~$\\Delta t(E)$ by simply changing the contour of integration in~\\eqref{eq:deltat} to a new contour~$\\gamma'$, where the new contour is obtained from the original contour~$\\gamma$ by exchanging two of the branch points in figure~\\ref{fig:integrationcontour} without allowing the contour to cross any branch points or poles. Examples of resulting contours are shown in figure~\\ref{fig:deformedcontours}.\n\nIn order to draw diagrams, we find it useful to cut the resulting Riemann surfaces into sheets. It is convenient to do so by introducing branch cuts that run radially outward from branch points at any $E_c, E'_c$ to $E = \\infty$; see figure~\\ref{fig:Ecuts}. It is also convenient to introduce a notion of principal vs. secondary sheets. We take the principal sheet to be the one containing those extremal surfaces that lie entirely within either the real Lorentzian or real Euclidean sections of the complexified spacetime. For all examples below, it is consistent to take both such families of surfaces to lie on a single sheet. It is natural to ask whether the principal sheet is preferred in any physical sense over the secondary Riemann sheets, but we defer discussion of this question to section \\ref{sec:discussion}.\n\nThe above structure makes the identification of extremal surfaces straightforward. The boundary conditions require that~$\\Delta t = -2t_b + i\\beta\/2$, so extremal surfaces satisfying the boundary conditions correspond to the contours~$t_I = \\beta\/2$ (mod~$\\beta$) in the complex~$E$-plane. Since~$\\Delta t(E)$ is analytic (except at branch points and poles), so long as the derivative does not vanish the inverse function $E(\\Delta t)$ is also analytic and defines a good conformal map. Thus~$t_R$ must change monotonically along these contours when the derivative is non-zero; vanishing derivative is generally signalled by the intersection of multiple contours. The contours~$t_I = \\beta\/2$ may be found by numerically integrating \\eqref{eq:deltat}, for example by using \\texttt{Mathematica}'s built-in \\texttt{NIntegrate} command which is capable of performing contour integrals in the complex plane. Below, we use the structure of such contours to probe the associated complex extremal surfaces.\n\n\n\\subsection{A Cautionary Tale: Geodesics in Schwarzschild-AdS}\n\\label{subsec:geoSAdS}\n\nTo illustrate the above techniques, we pause to discuss complex geodesics (the case $n=d$) in Schwarzschild-AdS$_{d+1}$. We have studied only cases with $d \\le 7$, though we expect the results for $d \\ge 8$ to resemble those found for $d=5,6,7.$ We find interesting distinctions between the cases $d=3$, $d =4$, and $d \\ge 5$. The case $d=4$ was discussed in \\cite{Fidkowski:2003nf}, though to our knowledge the results for $d\\neq 4$ are new. In particular, one might have hoped that since the $t=0$ surface is common to both the Euclidean and Lorentzian sections, geodesics in this surface would always provide good saddle points for path integral with $t_b=0$. But we will see that Schwarzschild-AdS$_{d+1}$ for $d \\ge 5$ provides a counter-example\\footnote{This might be expected from the analysis of \\cite{Fidkowski:2003nf}, which argued that perturbing the $d=4$ case would produce this result. Changing $d=4$ to $d=5$ is such a perturbation, though so is changing $d=4$ to $d=3$ (which yields very different results as shown in figure \\ref{fig:geodesiccontours}).}.\n\nFor definiteness, we first consider $d=4$ as in \\cite{Fidkowski:2003nf} so that we have\n\\be\nf(r) = g(r) = \\frac{r^2}{\\ell^2}\\left(1-\\frac{r_h^4}{r^4}\\right).\n\\ee\nThe function~$V_\\mathrm{eff}$ is as in~\\eqref{subeq:Veff}, and one finds~\\cite{Fidkowski:2003nf}\n\\be\n\\Delta t(E) = \\frac{\\beta}{2 \\pi}\\left[\\ln\\left(\\frac{\\mathcal{E}^2\/2 - \\mathcal{E} + 1}{\\sqrt{1+\\mathcal{E}^4\/4}}\\right) - i\\, \\ln\\left(\\frac{-\\mathcal{E}^2\/2 + i\\mathcal{E} + 1}{\\sqrt{1+\\mathcal{E}^4\/4}}\\right)\\right],\n\\ee\nwhere ~$\\mathcal{E} \\equiv E\\ell\/r_h$ and~$\\beta = \\pi \\ell^2\/r_h$. Note that~$\\Delta t$ has branch points at~$\\mathcal{E}^4 = -4$. Sketching the contours of~$t_I = \\beta\/2$ in the center panel of figure~\\ref{fig:geodesiccontours}, one finds a contour along the real~$E$-axis corresponding to real geodesics, and two complex contours that start and end on the branch points\\footnote{In fact, these contours spiral infinitely many times around the branch points, so they actually move off of the principal sheet of~$\\Delta t(E)$.}. Taking again~\\eqref{eq:areact} for the area regulator, one finds that the regulated length diverges as the contours approach the branch points.\n\nThe presence of complex contours is generic and independent of dimension. In figure~\\ref{fig:geodesiccontours} we sketch the contours on the principal sheet for the three cases~$d = 3$,~$d = 4$, and~$d \\geq 5$. Note that there are always two sets of contours: a contour along the real~$E$-axis corresponding to real geodesics, and a set of complex contours that end at the branch points.\n\nFor $d\\ge 5$ the real geodesics have properties very similar to those found in \\cite{Fidkowski:2003nf} for $d=4$. In particular, the renormalized action diverges to $-\\infty$ at finite $t_b$. If these were the relevant saddle points for the path integral, this would imply a boundary to boundary two-point function $e^{-m {\\cal L}_{\\mathrm{ren}}}$ that diverges at finite $t_b$. This cannot happen in a good field theory, and even the small $t_b$ behavior is suspicious. The fact that the arrow on the real contour runs to the left in the right panel of figure \\ref{fig:geodesiccontours} means that $t_b$ increases in that direction and thus by the Hamilton-Jacobi relation \\eqref{eq:HJ} that $t_b = 0$ would be a local \\emph{minimum} of the resulting two-point function. But on physical grounds it should be a local maximum; see e.g. \\cite{Festuccia:2005pi,Festuccia:2008zx,Hartman:2013qma,Marolf:2013dba}. We conclude that there must be some obstacle to deforming the path integral contour of integration to make use of the real Lorentz-signature geodesics. Instead, it is the complex geodesics shown in the right-most panel of figure \\ref{fig:geodesiccontours} that give physically reasonable behavior, and which in particular end at branch points for which ${\\cal L}_{\\mathrm{ren}}$ diverges to positive infinity as $t_b \\rightarrow \\pm \\infty.$ The story is similar to that in \\cite{Fidkowski:2003nf} for $d=4$ except that the complex $t_I = \\beta\/2$ contours do not pass through $E=0$, and\nthe correct complex geodesics now differ in action from the real Lorentzian geodesic even at $t_b =0$. Indeed, we find that the complex geodesics with $t_b =0$ have smaller action\\footnote{\\label{foot:sub} We stress, however, that the real $t_b=0$ geodesic appears not to provide even a subdominant contribution. If the path integral contour could be deformed to use this geodesic in the saddle point approximation, then by continuity the same should be true of real geodesics with $t_b \\neq 0$. But the action of the real geodesics clearly has smaller real part in the limit where it approaches $-\\infty$, so in that limit the real geodesics would become the dominant saddles.}.\n\n\\begin{figure}[t]\n\\centering\n\\subfigure{\n\\includegraphics[page=7]{Complex_surfaces_v10-pics.pdf}\n}\n\\hspace{0.5cm}\n\\subfigure{\n\\includegraphics[page=8]{Complex_surfaces_v10-pics.pdf}\n}\n\\hspace{0.5cm}\n\\subfigure{\n\\includegraphics[page=9]{Complex_surfaces_v10-pics.pdf}\n}\n\\caption{The structure of the~$t_I = \\beta\/2$ contours for geodesics in Schwarzschild-AdS$_{d+1}$; arrows denote the direction of increasing~$t_b$. From left to right, the figures show~$d = 3$,~$d = 4$, and~$d \\geq 5$. Note that there is always a contour along the real~$E$-axis, which for~$d \\geq 5$ is disconnected from the two complex ones. The complex contours spiral into the branch points.}\n\\label{fig:geodesiccontours}\n\\end{figure}\n\n\n\n\\section{HRT Surfaces in Planar Black Holes}\n\\label{sec:examples}\n\nWe now turn to codimension-2 extremal surfaces ($n=2$), which are our primary interest. In particular, we apply the above methods to identify and study such surfaces in the maximally-extended planar BTZ, Schwarzschild-AdS$_{d+1}$, and Schwarzschild-Lifshitz spacetimes, each of which is dual to a thermofield double state on $\\mathbb{R}^d$ in parallel with the discussion in \\cite{Maldacena:2001kr}. In all cases, we consider the class of surfaces described in section~\\ref{sec:setup} which satisfy boundary conditions appropriate to computing the entropy of a pair of half $(d-1)$-planes in opposite components of the thermofield double state. These are bulk surfaces that stretch from one of the two conformal boundaries to the other as shown in figure \\ref{fig:wedges}. We are mostly interested in the Schwarzschild-AdS$_{d+1}$ case (section \\ref{subsec:SAdS}), but study BTZ as an analytically-solvable warm-up in section \\ref{subsec:BTZ}. We also use Schwarzschild-Lifshitz to probe possible dependence on boundary conditions in section \\ref{subsec:Lifshitz}. Of course, since $d=2$ for BTZ, geodesics and codimension-2 surfaces coincide in that context.\n\n\n\\subsection{HRT in BTZ}\n\\label{subsec:BTZ}\n\nA planar version of the BTZ spacetime~\\cite{Banados:1992wn} may be defined by taking $d=n=2$ and\n\\be\nf(r) = g(r) = \\frac{r^2}{\\ell^2}\\left(1-\\frac{r_h^2}{r^2}\\right).\n\\ee\nThe metric \\eqref{eq:general} then describes a region of global AdS${}_3$ and contains no singularities. One might thus argue that a better name for this region is AdS${}_3$-Rindler, but we use the term planar BTZ to emphasize that it is the unique 3-dimensional analogue of planar Schwarzschild-AdS${}_{d+1}$ for $d \\ge 3$.\n\nFor this case one finds\n\\begin{subequations}\n\\bea\n\\label{subeq:Veff}\nV_\\mathrm{eff}(R) &= -f(R) - E^2, \\\\\n\\label{subeq:DeltatBTZ}\n\\Delta t &= \\beta \\left[-\\frac{1}{\\pi} \\, \\mathrm{arctanh} \\, \\mathcal{E} + \\frac{i}{2}\\right],\n\\eea\n\\end{subequations}\nwhere again~$\\mathcal{E} \\equiv E\\ell\/r_h$ and now~$\\beta = 2\\pi \\ell^2\/r_h$. Taking the area regulator to be\n\\be\n\\label{eq:areact}\nA_\\mathrm{ct} = - 2\\ell \\ln \\left(\\frac{1}{\\epsilon}\\right),\n\\ee\nwe obtain\n\\be\n\\label{eq:ABTZ}\nA_\\mathrm{ren} = \\ell \\ln \\left(\\frac{4}{1-\\mathcal{E}^2}\\right).\n\\ee\n\nThe simple form of the expressions~\\eqref{subeq:DeltatBTZ} and~\\eqref{eq:ABTZ} allows one to write~$A_\\mathrm{ren}$ as an explicit function of~$\\Delta t$. But in order to illustrate the general procedure, we continue to treat~$A_\\mathrm{ren}$ and~$\\Delta t$ as separate functions parametrized by~$\\mathcal{E}$. In order to find geodesics connecting the two boundaries of the BTZ black hole, we require~$t_I = \\beta\/2 \\ (\\rm{mod} \\ \\beta)$. This condition will clearly be satisfied for real~$\\mathcal{E} \\in (-1,1)$. These energies correspond to the usual real geodesics, so we will call this the principal ~$t_I = \\beta\/2$ contour. At the endpoints~$\\mathcal{E} \\to \\pm 1$ we find~$t_b \\to \\pm \\infty$. Moreover,~$A_\\mathrm{ren}$ is real and diverges to $+\\infty$ at the endpoints. Indeed, on the principal~$t_I = \\beta\/2$ contour~$A_\\mathrm{ren}$ has a global minimum at $t_b =0$. It then increases monotonically as one moves away from this value. This agrees with the expected behavior of the entanglement entropy at large times. One can also check that certain results are quantitatively correct \\cite{Hartman:2013qma}. Since these surfaces are geodesics it is also natural to compare $e^{-m {\\cal L}_{\\mathrm{ren}}}$ with two-point functions, and one finds corresponding agreement \\cite{Festuccia:2005pi}.\n\nHowever, we may also consider the full Riemann surfaces defined by~$\\Delta t$ and~$A_\\mathrm{ren}$. These are obtained by a simple analytic continuation of the~$\\mathrm{arctanh}$ and logarithm, so that each of the resulting sheets can be labeled by an integer~$m$:\n\\begin{subequations}\n\\bea\n\\Delta t_m &= \\beta \\left[-\\frac{1}{\\pi} \\, \\mathrm{arctanh} \\, \\mathcal{E} + \\frac{(2m+1)i}{2}\\right], \\\\\nA_{\\mathrm{ren},m'} &= \\ell \\ln \\left(\\frac{4}{1-\\mathcal{E}^2}\\right) + 2m'\\pi i \\ell.\n\\label{eq:BTZReA}\n\\eea\n\\end{subequations}\nThe union of all such sheets yields the full Riemann surface. There are now many contours for which $t_I = \\beta\/2 \\ \\ (\\mathrm{mod} \\ \\beta)$. These contours are labeled by $m$ and all project to the interval $\\mathcal{E} \\in (-1,1)$ along the real line in the complex $E$ plane. We see that $t_b(E)$ is independent of $m$, while $A_{\\mathrm{ren}}(E)$ (and thus $A_{\\mathrm{ren}}(t_b)$) differs from its values on the principal ($m'=0$) contour only by a $t_b$-independent purely-imaginary constant. So all choices of $m'$ would lead to the same entropies under the straw-man proposal of section \\ref{sec:interp}.\n\nAs noted above, the spacetime we called planar BTZ is really just a subset of global AdS${}_3$ (described in Rindler-like coordinates). Thus our surfaces immediately define complex extremal surfaces in AdS${}_3$. If $(t,r,\\theta)$ are the usual global coordinates, these surfaces intersect the boundary at $(t, \\theta=0)$ and $(t, \\theta = \\pi)$. For given $m$ above, they are all related by global time translations; the nontrivial time-dependence of the area in \\eqref{eq:BTZReA} is entirely due to the transformation between the global AdS${}_3$ and BTZ conformal frames. One may also describe these surfaces in the Poincar\\'e patch.\n\n\n\\subsection{HRT in Schwarzschild-AdS}\n\\label{subsec:SAdS}\n\nWe now turn to the more interesting case of Schwarzschild-AdS$_{d+1}$. We again set~$n = 2$ and take\n\\be\nf(r) = g(r) = \\frac{r^2}{\\ell^2}\\left(1-\\frac{r_h^d}{r^d}\\right).\n\\ee\nWe identify the critical~$E_c$ and the corresponding coincident branch points~$R_{\\mathrm{branch}}$ by requiring~$V_\\mathrm{eff}(R_{\\mathrm{branch}}) = 0 = V'_\\mathrm{eff}(R_{\\mathrm{branch}})$, which gives\n\\be\n\\label{eq:Ecrit}\nE_c = \\pm e^{2\\pi i m\/d} \\sqrt{\\frac{d}{d-2}}\\left(\\frac{d-2}{2(d-1)}\\right)^{(d-1)\/d} \\, \\frac{r_h^{d-1}}{\\ell}\n\\ee\nfor~$m = 1, \\ldots, d$. By numerically integrating~\\eqref{eq:deltat}, we find for all~$3 \\leq d \\leq 7$ that the only~$E_c$ on the principal sheet of~$\\Delta t(E)$ are the two real ones, which form a pair of points on the real axis with opposite signs. We also find that the only~$t_I = \\beta\/2$ contour on this sheet connects these $E_c$ by running along the real axis, as shown in figure~\\ref{fig:sheets}(a) for $d=4$. This contour corresponds to the real surfaces studied in~\\cite{Hartman:2013qma}. As in that work, taking\n\\be\nA_\\mathrm{ct} = -\\frac{2\\ell r_h^{d-2}V_{d-2}}{d-2} \\frac{1}{\\epsilon^{d-2}}\n\\ee\nshows that $A_\\mathrm{ren}$ increases as one moves along this contour away from $t_b=0$ and diverges to positive infinity as the branch points are approached (where $t_b \\rightarrow \\pm \\infty$). Though we have studied only $d \\le 7$, we expect similar behavior for larger values of $d$.\n\n\nThe secondary sheets turn out to contain much more structure. For simplicity, we will focus in detail on the case~$d=4$, though we will briefly comment on the cases~$d = 3$ and~$d = 6$ as well\\footnote{For even $d$ the analysis is simplified by working in terms of a new variable~$w = (r_h\/r)^2$; thus is $d = 6$ more tractable than~$d = 5,7$.}. In Appendix~\\ref{app:expansion}, we express the integrals~\\eqref{eq:deltat} and~\\eqref{eq:areacontour} for~$d = 4$ in terms of standard elliptic integrals, which we will use to obtain various approximations.\n\nFor~$d = 4$, we see from~\\eqref{eq:Ecrit} that there are only four critical energies $E_c$. These $E_c$ lie on the real and imaginary axes, and are related to one another by multiples of the phase~$e^{i\\pi\/2}$. In addition, there is another critical energy $E_c' = 0$ at which two roots of~$V_\\mathrm{eff}(R)$ coincide at $R=0$. Though $R=0$ is not a branch point of the integrands of~\\eqref{eq:deltat} and~\\eqref{eq:areacontour} for $d=4$, it remains a singularity; in this case a pole for $E \\neq 0$. Thus the functions~$\\Delta t(E)$ and~$A_\\mathrm{ren}(E)$ will generally have branch points at $E=0$ though they will not diverge there.\n\nLet us now analytically continue off the principal sheet through one of the branch cuts shown in figure~\\ref{fig:sheets}(a) onto what we now call sheet \\#2. As shown in figure~\\ref{subfig:sheets1}, we find a sheet with branch points at all four of the~$E_c$ as well as at~$E_c'=0$. The choice of direction is arbitrary for the branch cut ending at~$E_c'=0$; we find the choice shown in the figure convenient.\n\nThe new purely imaginary $E_c$ lead to interesting behavior. This is perhaps best studied by using\nexpression~\\eqref{eq:deltatelliptic} to show that near~$E_c = -i\\sqrt{2}\/3^{3\/4} \\, r_h^3\/\\ell$,\n\\be\n\\label{eq:logsing}\n\\Delta t = -\\frac{i \\beta}{2^{3\/2} \\cdot 3^{1\/4} \\, \\pi} \\ln\\left(\\mathcal{E} - \\mathcal{E}_c\\right) + C + \\mathcal{O}\\left(\\mathcal{E} - \\mathcal{E}_c\\right),\n\\ee\nwhere~$\\beta \\equiv \\pi \\ell^2\/r_h$,~$\\mathcal{E} \\equiv \\ell E\/r_h^3$, and~$C$ is a (complex) constant. In particular, we see that taking~$|E-E_c|$ arbitrarily small makes~$t_I$ arbitrarily large and that and $t_R$ increases uniformly as one travels around this~$E_c$. Thus there are an infinite number of contours satisfying~$t_I = \\beta\/2$ (mod~$\\beta$) circling near these $E_c$, crossing to higher and higher sheets with each cycle; these contours thus form an infinite family of ``helical contours''. Some examples are shown in figure \\ref{fig:sheets}.\n\n\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[]{\n\\includegraphics[page=10]{Complex_surfaces_v10-pics.pdf}\n}\n\\hspace{0.5cm}\n\\subfigure[]{\n\\includegraphics[page=11]{Complex_surfaces_v10-pics.pdf}\n\\label{subfig:sheets1}\n}\n\\hspace{0.5cm}\n\\subfigure[]{\n\\includegraphics[page=12]{Complex_surfaces_v10-pics.pdf}\n}\n\\caption{Schematic drawings where solid lines with arrows (red in color version) show contours with~$t_I = \\beta\/2$~$(\\mathrm{mod} \\ \\beta)$ for codimension-2 extremal surfaces on various sheets of~$\\Delta t(E)$ for Schwarzschild-AdS${}_5$. Arrows on the contours show directions of increasing~$t_b$ and dashed lines indicate loci where $t_b=0$. Panel (a) shows the principal sheet. Here the\nonly contour lies along the real~$E$-axis, so on this sheet only the familiar real extremal surfaces satisfy our boundary conditions. Analytically continuing through the right-hand branch cut in the direction indicated by the vertical arrow takes us to sheet \\#2, shown in~(b). Note the infinite family of helical contours that circle the branch points on the imaginary axis, as well as new contours and branch points. Analytically continuing through the right-hand branch cut takes us to sheet \\#3, shown in~(c). The contour labeled~$A$ on sheet~\\#2 continues through this cut onto sheet \\#3. Aside from the real contour on the principal sheet, only the two contours marked $B$ and $C$ on sheet \\#3 are physically acceptable near $t_b=0$ under the straw-man proposal of section \\ref{sec:interp}. All other segments of complex contours shown above cross $t_b=0$ when $\\mathrm{Re} \\ E \\neq 0$. In addition, on\nhelical contours $\\Re \\ A_{\\mathrm{ren}}$ remains unphysically bounded at large $t_b$.}\n\\label{fig:sheets}\n\\end{figure}\n\n\nReturning to sheet \\#2, we also find the additional contours shown in figure~\\ref{fig:sheets}(b). Two contours start at the branch point on the negative real axis and leave through branch cuts, while the contour in the first quadrant enters and exits through branch cuts. Tracking this contour through a branch cut onto a third sheet (\\#3), we find that it continues and crosses yet another branch cut. On this third sheet, we also find a variety of new contours. We will focus on the contour labeled~$B$ in figure~\\ref{fig:sheets}, which starts at the branch point~$E_c'$ and ends at the branch point on the positive real axis. This contour resembles a deformed version of the original real contour, and we expect additonal such deformed contours to appear as one probes more of the Riemann surface.\n\nFor the~$d = 3$ case, the only contour on the principal sheet is again the real one. In this case there are no contours on sheet \\#2 with $t_I = \\beta\/2 \\ ({\\rm mod} \\ \\beta)$, and in particular no analogue of the helical contours in figure \\ref{fig:sheets}(b). However, we expect that new contours could be found on higher sheets. For~$d = 6$, we once more find that the only contour on the principal sheet is the real one. On sheet \\#2 there are analogues of the helical contours for~$d = 4$ that now spiral into the the complex~$E_c$ of \\eqref{eq:Ecrit}. We also find an analogue of the contour in the upper left quadrant of figure \\ref{fig:sheets}(b), again terminating at an $E_c$ on the negative real axis. We have not examined higher sheets.\n\nIt is clearly of interest to investigate the areas of the extremal surfaces along our contours. For simplicity we limit this discussion to $d=4$. Following the straw-man hypothesis of section \\ref{sec:interp}, we focus on the real part $\\mathrm{Re} \\ A_{\\mathrm{ren}}(E)$. Were this real part to describe the CFT entropy on our pair of half-planes, the time-reflection symmetry of the dual CFT thermofield-double state would require a corresponding symmetry of the relevant $\\Re \\ A_{\\mathrm{ren}}$'s. In particular, if a single smooth contour is to provide the relevant surfaces near $t_b=0$, then the derivative with respect to $t_b$ must vanish there. The Hamilton-Jacobi relation \\eqref{eq:HJ} then requires that $\\Re \\ E$ vanish as well; i.e., $t_b$ could vanish only on the imaginary $E$ axis. Of the complex contours shown in figure \\ref{fig:sheets}, only the two marked $B$ and $C$ have vanishing $\\Re \\ E$ at $t_b=0$.\n\nOf course, the symmetry of the spacetime under time-reversal implies that any contour must have a time-reversed image somewhere on the Riemann surface -- though this will generically lie on some yet-unexplored Riemann sheet. One can clearly combine the $t_b > 0$ part of one contour with the $t_b < 0$ part of its image to define time-symmetric $\\mathrm{Re} \\ A_{\\mathrm{ren}}$. But with non-vanishing $\\Re \\ E$ at $t_b =0$, the time-derivative is discontinuous at $t_b=0$; one would then need to rely on surprising sub-leading corrections in $1\/G_N$ to match the physically expected vanishing of $dS_\\mathrm{ren}\/dt_b$ in the CFT. Furthermore, choosing to keep the surfaces with smallest $\\mathrm{Re} \\ A_{\\mathrm{ren}}$ would necessarily force $\\mathrm{Re} \\ A_{\\mathrm{ren}}$ to have a local maximum at $t_b=0$; see figure \\ref{fig:localmax}.\n\n\n\n\\begin{figure}[t]\n\\centering\n\n\\includegraphics[page=13]{Complex_surfaces_v10-pics.pdf}\n\\caption{The small $t_b$ part of a generic smooth real function (solid) with non-vanishing slope at $t_b =0$ and its time-reversed image (dashed). Taking the minimum of the two (red parts in color version) defines a function with a local maximum at $t_b=0$ where the derivative is discontinuous.}\n\\label{fig:localmax}\n\\end{figure}\n\n\nIn contrast, as discussed in e.g. \\cite{Hartman:2013qma} the thermofield-double nature of the CFT state strongly suggests that the entropy should be a mininum at $t_b=0$ followed by monotonic increase with $|t_b|$ to diverge as $t_b \\rightarrow \\pm \\infty$. From the Hamilton-Jacobi relation \\eqref{eq:HJ} and the arrows in figure \\ref{fig:sheets}, we see that this correctly describes the behavior of $\\mathrm{Re} \\ A_{\\mathrm{ren}}$ along contours $B$ and $C$. But it fails at various points along other contours. In particular, for helical contours \\eqref{eq:logsing} and \\eqref{eq:HJ} imply that $\\mathrm{Re} \\ A_{\\mathrm{ren}}(E)$ oscillates with each cycle and remains bounded as $t_b \\rightarrow \\pm \\infty$. The large $t_b$ regimes of these contours are particularly problematic, as there $\\mathrm{Re} \\ A_{\\mathrm{ren}}(E)$ is clearly smaller than for any physically acceptable contour. Under suitable extensions of the straw man proposal, the comments in footnote \\ref{foot:sub} about the implications of such behavior for the geodesic approximation would thus apply here as well and indicate that even finite $t_b$ pieces of these contours cannot be relevant to the dual CFT entropy.\n\nFor the above reasons we discuss only contours $B$ and $C$ in detail. These contours are defined only for $t_b>0$ and $t_b < 0$ respectively, and since at $t_b=0$ they reach the $E=0$ branch point there is no simple notion of an extension through~$t_b = 0$. But each must have a time-reversed copy as discussed above, and this copy will also reach the $E=0$ branch point at $t_b=0$. So it is natural to glue $B$ and $C$ at $t_b=0$ to their respective time-reversed copies. Since $E(t_b)=0$, extending $B$ and $C$ in this way defines contours where $A_{\\mathrm{ren}}$ is at least $C^1$, which continue to meet the above physical expectations.\n\nWe begin with $B$. As shown in figure~\\ref{fig:contour2} (left), to good accuracy the function~$\\Re \\ A_\\mathrm{ren}(t_b)$ along $B$ agrees with that along the real contour on the principal sheet. It would be interesting to understand whether the tiny discrepancy near $t_b\/\\beta \\sim 0.1$ is a numerical artifact. While this is beyond the scope of the present work, it is straightforward to study the small- and late-time regimes perturbatively at leading order. In particular, the Hamilton-Jacobi relation (or alternatively,~\\eqref{eq:entanglementvelocity}) guarantees that the late-time growth of~$A_\\mathrm{ren}(t_b)$ will be identical along the two contours since both approach the same~$E_c$. At small $E$ we can expand the elliptic integrals~\\eqref{eq:deltatelliptic} and~\\eqref{eq:Aelliptic} to find\n\\bea\n\\label{eq:tbexp}\nt_b &= \\frac{\\beta}{2\\pi} \\, \\mathcal{E} + \\mathcal{O}(\\mathcal{E})^3, \\\\\n\\Re \\, A_\\mathrm{ren} &= \\frac{\\ell r_h^2 V_2}{2} \\, \\mathcal{E}^2 + \\mathcal{O}(\\mathcal{E})^4,\n\\eea\nso that\n\\label{eq:Arenexp}\n\\be\n\\label{eq:Aexp}\n\\Re \\, A_\\mathrm{ren} = \\frac{2r_h^4 V_2}{\\ell^3} \\, t_b^2 + \\mathcal{O}(t_b)^4\n\\ee\nalong both contours. Thus $B$ agrees with the real contour to this order.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.475\\textwidth]{Contour_RealA_post.png}\n\\includegraphics[width=0.475\\textwidth]{Contour_RealA_negt.png}\n\\caption{The plots show~$\\Re \\ A_\\mathrm{ren}(t_b)$ for contour $B$ (lower curve in left panel, blue in color version) and contour $C$ (lower curve in right panel, blue in color version) in comparison with that on the real contour (upper curve in both panels, red in color version). Contour $C$ clearly has smallest $\\Re \\ A_\\mathrm{ren}$. Near $t_b\/\\beta = 0.1$ contour $B$ also appears to have $\\Re \\ A_\\mathrm{ren}$ slightly smaller than for the real contour, though a more careful analysis would be required to show that this is not an artifact of our numerics.}\n\\label{fig:contour2}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.475\\textwidth]{Contour_ImagA_post.png}\n\\includegraphics[width=0.475\\textwidth]{Contour_ImagA_negt.png}\n\\caption{The imaginary parts~$\\Im \\, A_\\mathrm{ren}(t_b)$ along contours~$B$ (left) and~$C$ (right). The noise at larger values of~$t_b$ is a numerical artifact, likely due the failure of~$\\Im \\, A_\\mathrm{ren}(E)$ to be continuous at~$E_c$. The function $\\Im \\, A_\\mathrm{ren}(E)$ does admit direction-dependent limits at $E_c$ that make $\\Im \\, A_\\mathrm{ren}(t_b)$ continuous there for real $t_b$, but a small error in the location of our contour near $E_c$ can translate into a large error in $\\Im \\, A_\\mathrm{ren}$. We have also excised portions near~$t_b = 0$ which exhibit numerical noise.}\n\\label{fig:imagA}\n\\end{figure}\n\nContour $C$ is even more interesting. The Hamilton-Jacobi relation again guarantees the late-time growth to be identical to those above, and\n~\\eqref{eq:deltatelliptic} and~\\eqref{eq:Aelliptic} again yield \\eqref{eq:tbexp}, \\eqref{eq:Arenexp}, and \\eqref{eq:Aexp}. But for $t_b\\neq 0$ figure~\\ref{fig:contour2} clearly shows the associated $\\Re \\ A_\\mathrm{ren}(t_b)$ to be smaller than for real extremal surfaces. It is thus plausible that the associated entropy of the dual CFT is controlled by the complex surfaces contour $C$, and not by the original real extremal surfaces.\n\nFor completeness we also include plots of the imaginary part of $A_{\\mathrm{ren}}$ along $B$ and $C$ in figure \\ref{fig:imagA}. Expansions analogous to those above show that $\\Im \\, A_\\mathrm{ren} = 2 + {\\cal O}(t_b^4)$ near $t_b=0$ for both contours, and since they end at real $E_c$ the imaginary parts are again much smaller than $\\Re \\, A_\\mathrm{ren}$ at large $|t_b|$. As a result, for large $t_b$ we have $|A_\\mathrm{ren}| \\sim \\Re \\, A_\\mathrm{ren}$ and using $|A_\\mathrm{ren}| \\sim \\Re \\, A_\\mathrm{ren}$ gives the same result as taking the absolute value.\n\n\n\\subsection{Lifshitz}\n\\label{subsec:Lifshitz}\n\nIn order to investigate possible dependence on boundary conditions, we now briefly consider the Schwarzschild-Lifhshitz black holes of \\cite{Taylor:2008tg}. The spacetimes are characterized by the spacetime dimension $d+1$, a choice of dynamical scaling exponent $z$, and a horizon radius $r_h$. Since $z=1$ is just the Schwarzschild-AdS case already studied in section \\ref{subsec:SAdS}, we assume $z \\neq 1$ below. In order to respect the null energy condition we consider only $z > 1$~\\cite{Hoyos:2010at}. We also restrict to rational $z$.\n\nWe will find that these spacetimes follow the same pattern seen above. The only $t_I = \\beta\/2$ contour on the principal sheet describes real extremal surfaces, but complex contours appear on secondary sheets. We refer to the contour on the principal sheet as the real contour below. For an infinite class of special cases, an analytic argument allows us to identify contours on certain secondary sheets that are simply related to the real contour: the associated extremal surfaces satisfy the same boundary conditions (i.e., they have same $\\Delta t$) while $A_{\\mathrm{ren}}$ differs from that on the real contour by a phase. For appropriate choices, such families satisfy our qualitative physical expectations (minimum at $t_b=0$ and monotonic increase to infinity with $|t_b|$) for use as an HRT surface. However, in such cases $\\mathrm{Re} \\, A_{\\mathrm{ren}} (t_b)$ is always smaller than for the corresponding real extremal surface.\n\nWe now begin the calculations.\nFrom~\\cite{Taylor:2008tg} one sees that the desired spacetimes satisfy\n\\be\n\\label{eq:Lifshitz}\nf(r) = \\left(\\frac{r}{\\ell}\\right)^{2z}\\left(1-\\left(\\frac{r_h}{r}\\right)^{d+z-1}\\right), \\quad g(r) = \\left(\\frac{r}{\\ell}\\right)^2\\left(1-\\left(\\frac{r_h}{r}\\right)^{d+z-1}\\right).\n\\ee\nWe therefore find\n\\begin{subequations}\n\\bea\n\\label{subeq:deltatLifshitz}\n\\Delta t &= \\frac{\\alpha\\beta}{4\\pi} \\int_\\gamma \\frac{\\mathcal{E}}{\\rho^{z-1} \\left(\\rho^\\alpha-1\\right)\\sqrt{-\\widetilde{V}_\\mathrm{eff}(\\rho)}} \\, d\\rho, \\\\\nA &= V_{d-2} \\ell \\, r_h^{d-2} \\int_\\gamma \\frac{\\rho^{d-2}}{\\sqrt{-\\widetilde{V}_\\mathrm{eff}(\\rho)}} \\, d\\rho,\n\\eea\n\\end{subequations}\nwhere~$\\alpha \\equiv d+z-1$,~$\\beta = 4\\pi\\ell^{z+1}\/\\alpha r_h^z$,~$\\rho \\equiv R\/r_h$,~$\\mathcal{E} \\equiv \\ell^z E\/r_h^{\\alpha-1}$, and\n\\be\n\\widetilde{V}_\\mathrm{eff}(\\rho) = -\\frac{1}{\\rho^{2(\\alpha-2)}} \\left(\\rho^{2(\\alpha-1)} - \\rho^{\\alpha-2} + \\mathcal{E}^2\\right).\n\\ee\nWe regulate the area with\n\\be\nA_\\mathrm{ct} = -\\frac{2V_{d-2}\\ell r_h^{d-2}}{(d-2)\\epsilon^{d-2}}.\n\\ee\n\nThe critical energies are\n\\be\n\\mathcal{E}_c = \\pm (1)^{1\/\\alpha}_n \\sqrt{\\frac{\\alpha}{\\alpha-2}} \\left(\\frac{\\alpha-2}{2(\\alpha-1)}\\right)^{(\\alpha-1)\/\\alpha},\n\\ee\nwhere~$(1)^{1\/\\alpha}_n$ is the~$n^\\mathrm{th}$ root of~$x^\\alpha = 1$. If~$\\alpha$ is irrational, there are an infinite number of such roots and the critical energies are dense in a circle in the complex~$E$-plane. We therefore restrict our analysis to rational~$\\alpha$ or, equivalently, rational~$z$.\n\nWe have examined the principal sheet numerically for $(d,z) = (3,2)$,~$(3,3)$,~$(4,2)$, and~$(4,3)$. In each of these cases we find only the real contour. Turning now to secondary sheets, we will show that certain $z$ exhibit a special symmetry relating the principal sheet to a class of secondary sheets. This may be seen by choosing an integer $m$ and noting that the phase rotations\n\\be\n\\label{eq:rotations}\n\\rho \\to e^{2\\pi im\/\\alpha} \\rho, \\quad \\mathcal{E} \\to e^{-2\\pi im\/\\alpha} \\mathcal{E},\n\\ee\nact on the effective potential as~$\\widetilde{V}_\\mathrm{eff} \\to e^{4\\pi im\/\\alpha} \\widetilde{V}_\\mathrm{eff}$. Thus if~$\\rho^*$ is a root of~$\\widetilde{V}_\\mathrm{eff}$ at energy~$\\mathcal{E}$, then~$e^{2\\pi im\/\\alpha} \\rho^*$ is also a root of~$\\widetilde{V}_\\mathrm{eff}$ at energy~$e^{-2\\pi im\/\\alpha} \\mathcal{E}$.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[page=14]{Complex_surfaces_v10-pics.pdf}\n\\hspace{0.5cm}\n\\includegraphics[page=15]{Complex_surfaces_v10-pics.pdf}\n\\caption{Sample integration contours~$\\gamma$,~$\\gamma'$ in the complex~$\\rho$ plane for Schwarzschild-Lifshitz with~$d = 3$,~$z = 2$. The left panels shows the contour~$\\gamma$ which defines a real extremal surface for real~$E$. There are 4 real and two imaginary branch points, with $\\gamma$ encircling only the largest real branch point. The right panel is obtained from the left by ~\\eqref{eq:rotations}. The new contour contour~$\\gamma'$ defines a complex extremal surface that lies on a secondary sheet of~$\\Delta t$ and~$A_\\mathrm{ren}$.}\n\\label{fig:Lifshitzcuts}\n\\end{figure}\n\nConsider then any contour $\\gamma$ in the complex $\\rho$ plane that defines a real extremal surface. The contour~$\\gamma$ then runs along the real~$\\rho$ axis, coming in from~$\\rho = \\infty$ before turning around the largest real branch point~$\\rho_\\mathrm{turn}$ and returning to~$\\rho = \\infty$. The expressions for~$\\Delta t$ and~$A_\\mathrm{ren}$ can be written as\n\\begin{subequations}\n\\label{eqs:LifshitzdeltatA}\n\\bea\n\\Delta t &= \\frac{\\alpha\\beta}{2\\pi} \\int_{\\rho_\\mathrm{turn}}^\\infty \\frac{\\mathcal{E}}{\\rho^{z-1} \\left(\\rho^\\alpha-1\\right)\\sqrt{-\\widetilde{V}_\\mathrm{eff}(\\rho)}} \\, d\\rho, \\\\\nA_\\mathrm{ren} &= 2V_{d-2} \\ell \\, r_h^{d-2} \\lim_{\\epsilon \\to 0} \\left(\\int_{\\rho_\\mathrm{turn}}^{1\/\\epsilon} \\frac{\\rho^{d-2}}{\\sqrt{-\\widetilde{V}_\\mathrm{eff}(\\rho)}} \\, d\\rho - \\frac{1}{(d-2)\\epsilon^{d-2}}\\right), \\\\\n\t\t\t &= 2V_{d-2} \\ell \\, r_h^{d-2} \\left[\\int_{\\rho_\\mathrm{turn}}^\\infty \\left(\\frac{\\rho^{d-2}}{\\sqrt{-\\widetilde{V}_\\mathrm{eff}(\\rho)}} - \\rho^{d-3}\\right) \\, d\\rho - \\frac{\\rho_{\\mathrm{turn}}^{d-2}}{(d-2)}\\right],\n\\eea\n\\end{subequations}\nwhere we have conveniently reabsorbed the counterterm~$A_\\mathrm{ct}$ into the integral expression for~$A_\\mathrm{ren}$ in order to extend the integration out to~$\\rho = \\infty$.\n\nActing with~\\eqref{eq:rotations} takes the (real) turning point~$\\rho_\\mathrm{turn}$ to~$\\rho_\\mathrm{turn}' = e^{2\\pi i m\/\\alpha} \\rho_\\mathrm{turn}$. Consequently, the original contour~$\\gamma$ is taken to a new contour~$\\gamma'$ that runs from infinity to~$\\rho_\\mathrm{turn}'$ along a line of constant~$\\arg(\\rho) = 2\\pi i m\/\\alpha$. In particular, the contour~$\\gamma'$ does not approach~$\\rho = \\infty$ along the positive real axis, as we require of our allowed contours. But because both of the integrands in~\\eqref{eqs:LifshitzdeltatA} die off sufficiently fast at infinity,~$\\gamma'$ can be deformed to approach~$\\rho = \\infty$ along the positive real axis without changing~$\\Delta t$ and~$A_\\mathrm{ren}$. As a result, the new contour~$\\gamma'$ defines a secondary sheet of the Riemann surfaces for~$\\Delta t$ and~$A_\\mathrm{ren}$ which is related to the principal sheet by the transformations~\\eqref{eq:rotations}. Examples of~$\\gamma'$ for the special case~$d = 3$,~$z = 2$ are shown in figure~\\ref{fig:Lifshitzcuts}.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[page=16]{Complex_surfaces_v10-pics.pdf}\n\\hspace{0.5cm}\n\\includegraphics[page=17]{Complex_surfaces_v10-pics.pdf}\n\\hspace{0.5cm}\n\\includegraphics[page=18]{Complex_surfaces_v10-pics.pdf}\n\\caption{Three sheets of the Riemann surface for~$\\Delta t$ in~$d = 4$,~$z= 3$ Lifshitz. The left panel shows the principal sheet (generated by the contour~$\\gamma$ in figure \\ref{fig:Lifshitzcuts}) and the contour~$t_I = \\beta\/2$ corresponding to real extremal surfaces. The middle and right panels show the secondary sheets that are obtained from the principal sheet by acting with the transformations~\\eqref{eq:rotations}; each of these contains an image of the real contour.}\n\\label{fig:Lifshitzcontours}\n\\end{figure}\n\n\nIf~\\eqref{eq:rotations} preserve the condition $t_I = \\beta\/2$ (mod $\\beta$), then they will map the real $t_I = \\beta\/2$ (mod $\\beta$) contour to another on the secondary sheet defined by~$\\gamma'$; examples of these contours for the special case~$d = 4$,~$z = 3$ are shown in figure~\\ref{fig:Lifshitzcontours}. Setting ~$\\rho = e^{\\pi im\/\\alpha} \\rho'$,~$\\mathcal{E} = e^{-\\pi im\/\\alpha} \\mathcal{E}'$, we find\n\\begin{subequations}\n\\bea\n\\Delta t_\\gamma (\\mathcal{E}) &= e^{\\pi i(d-1)m\/\\alpha}\\frac{\\alpha\\beta}{2\\pi} \\int_{\\rho'_\\mathrm{turn}}^\\infty \\frac{\\mathcal{E}'}{(\\rho')^{z-1} \\left((\\rho')^\\alpha-1\\right)\\sqrt{-\\widetilde{V}_\\mathrm{eff}(\\rho')}} \\, d\\rho' \\\\\n &= e^{\\pi im(d-1)\/\\alpha} \\Delta t_{\\gamma '}\\mathrm(\\mathcal{E}').\n\\eea\n\\end{subequations}\nSo $t_I = \\beta\/2$ (mod $\\beta$) is preserved when $(d-1)m\/\\alpha$ is an integer.\n\nExamining the area, we find\n\\begin{subequations}\n\\bea\nA_{\\mathrm{ren},\\gamma}(\\mathcal{E}) &= e^{\\pi i m(d-2)\/\\alpha} 2 V_{d-2} \\ell \\, r_h^{d-2} \\left[\\int_{\\rho'_\\mathrm{turn}}^\\infty \\left(\\frac{(\\rho')^{d-2}}{\\sqrt{-\\widetilde{V}_\\mathrm{eff}(\\rho')}} - (\\rho')^{d-3}\\right) \\, d\\rho' - \\frac{(\\rho'_{\\mathrm{turn}})^{d-2}}{(d-2)}\\right], \\\\\n &= e^{\\pi im (d-1)\/\\alpha} e^{-\\pi i m \/\\alpha} A_{\\mathrm{ren},\\gamma'}(\\mathcal{E}').\n\\eea\n\\end{subequations}\nThus if~$e^{\\pi i m (d-1)\/\\alpha} = \\pm 1$ the behavior of~$A_{\\mathrm{ren}}$ on the secondary sheet will be related to its behavior on the principal branch by a rotation~$e^{\\pi i m\/\\alpha}$ in the complex~$\\mathcal{E}$-plane, \\textit{and} by a change of phase~$e^{-\\pi i m \/\\alpha}$. So since~$A_{\\mathrm{ren}}$ is real along the real contour, it acquires an imaginary part along these secondary contours. And since $\\cos \\theta \\le 1$, the real part $\\Re \\ A_{\\mathrm{ren}}$ is clearly smaller for the surfaces defined by $\\gamma'$ than for the original real extremal surfaces. However, if~$\\Re \\, e^{\\pi i m (d-1)\/\\alpha} e^{-\\pi i m\/\\alpha} < 0$, the real part of~$A$ along these secondary contours becomes large and \\textit{negative} at large times, in contrast with the physical behavior expected of the entanglement entropy. Thus the straw-man hypothesis of section \\ref{sec:interp} is inconsistent with the use of extremal surfaces on certain secondary contours though it is consistent with others.\n\nWe can be a bit more explicit as to when this occurs. Let us write $\\alpha = p\/q$ with $(p,q) =1$, where $(p,q)$ denotes the greatest common divisor of two integers $p,q$. We must satisfy the constraint $m(d-1)q\/p \\in {\\mathbb Z}$ for the above symmetry to preserve $t_I = \\beta\/2 \\ (\\mathrm{mod} \\, \\beta)$. But the map becomes trivial when $mq\/p$ is an even integer. If $p$ is a divisor of $m$, one can show that non-trivial solutions occur for any odd $q$ and that $\\mathrm{Re} \\ A_{\\mathrm{ren}}$ behaves as desired for even $d$, while for odd $d$ it has a global maximum at $t=0$ and is unbounded below at large $|t_b|$. When $p$ is not a divisor of $m$, non-trivial solutions occur when $(p, d-1) > 1$ and one can choose $m$ so that $A_{\\mathrm{ren}}$ behaves as desired for $(p,d-1) >2$; for $(p,d-1) =2$ one can choose $m$ so that $A_{\\mathrm{ren}}$ is purely imaginary. We thus find many cases where the dual CFT entropy may plausibly be controlled by complex surfaces instead of real extremal surfaces.\n\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\n\nThe above work considered the possible significance of complex extremal surfaces for the Ryu-Takayanagi and Hubeny-Rangamani-Takayanagi (HRT) holographic entanglement conjectures. As emphasized by the study of complex geodesics in $d > 4$ Schwarzschild-AdS$_{d+1}$ (section \\ref{subsec:geoSAdS}), this issue could in principle be as important for the static setting as for the time-dependent context. We began by discussing how the formula \\eqref{RT} might be modified if complex surfaces are indeed relevant. We reached no firm conclusions, but noted that a straw-man model replacing the renormalized area $A_{\\mathrm{ren}}$ by its real part is not without motivation.\n\nGiven the confusion surrounding how holographic entanglement conjectures might be extended to include codimension-2 surfaces with complex areas, one might have hoped that no such surfaces would meet the real conformal boundary in the manner that these conjectures require. But we showed that they do. Such complex surfaces exist in complexified spacetimes defined by analytic continuation of simple real solutions. For planar BTZ, or equivalently global AdS${}_3$, they are somewhat trivial copies of the real surfaces in which $A_{\\mathrm{ren}}$ differs from the real case only by a quantized purely imaginary offset. One might expect similar behavior for global AdS$_{d+1}$ for $d \\ge 3$. But for Schwarzschild-AdS${}_5$ we find many distinct families of surfaces with a rich structure; we suspect that this is the case in other dimensions as well. We also found interesting families for Schwarzschild-Lifshitz.\n\nGiven the existence of complex extremal surfaces, one might next have hoped that they would exhibit clearly pathological behavior so as to be excluded on physical grounds. But in all cases studied in depth we identified families of complex extremal surfaces consistent under the above straw-man proposal with basic physical expectations for the time-dependence of the entropy. Furthermore, these complex surfaces have $\\Re \\ A_{\\mathrm{ren}}$ smaller than (or sometimes equal to) that of corresponding real extremal surfaces. It is thus plausible at this level that the dual CFT entropy is indeed determined by such complex extremal surfaces and not by the real ones. \n\nNevertheless, one may contrast the situation here with that concerning the geodesic approximation for 2-point functions in Schwarzschild-AdS$_{d+1}$ for $d \\ge 3$. As shown in \\cite{Fidkowski:2003nf} for $d=4$ (and more generally in section \\ref{subsec:geoSAdS} for other $3 \\le d \\le 7$), use of the real geodesics in such cases would imply unphysical behavior for the two-point function. It is then clear that, if a geodesic approximation is to be maintained at all, the geodesics involved must be complex. On the other hand, at least in cases studied here the real codimension-2 extremal surfaces lead to no obvious unphysical behavior. Furthermore, one knows that entropies based on the real surfaces will satisfy strong subadditivity \\cite{Headrick:2007km,Wall:2012uf} -- a property we are unable to test using the complex surfaces found above since we considered only the entropy of a single boundary region at each time; see also related comments in footnote \\ref{foot:subadd}. On a similar note, recall that for Schwarzschild-AdS and Schwarzschild-Lifshitz we also find families where the behavior of $\\Re \\ A_{\\mathrm{ren}}$ does not match expectations for entropy in the dual CFT; this may indicate that the relevant path integral cannot generally be deformed to take advantage of such complex surfaces. So while the relevance of complex extremal surfaces is plausible, it is by no means assured.\n\nOur work studied planar black hole spacetimes and looked for surfaces as shown in figure \\ref{fig:wedges}, running from the left boundary to the right and intersecting each boundary on a plane (also of codimension-2 with respect to the boundary) at some given time $t_b$ in analogy with those studied in \\cite{Hartman:2013qma}\\footnote{If complex surfaces in the bulk do determine the dual CFT entropy, this would affect the detailed results of \\cite{Hartman:2013qma}. But the most plausible families of complex surfaces found above behave sufficiently similar to the real surfaces that this change would not alter their main conclusions.}. For extremal surfaces of this form the time difference $\\Delta t$ between the left and right ends defines an infinite-sheeted Riemann surface when expressed in terms of the conserved energy $E$. The same is true of the renormalized area $A_{\\mathrm{ren}}$. By definition, extremal surfaces in the real Lorentzian spacetime live on the principal sheet of this Riemann surface. In all cases studied, numerical investigation indicated that there are no further extremal surfaces on this sheet; all complex extremal surfaces mentioned above lie on secondary sheets. In addition to the spacetimes addressed in the main text, we have also checked that the hyperbolic AdS black hole\\footnote{In the hyperbolic black hole, the planar line element~$dx_{d-1}^2$ in~\\eqref{eq:general} is replaced by a metric of constant negative curvature, but otherwise the procedure is identical.}~\\cite{Emparan:1998he,Birmingham:1998nr,Emparan:1999gf} and planar Reissner-Nordstr\\\"om-AdS$_5$ are free of complex extremal surfaces on their primary sheets. In the latter case, the particular cases checked were~$T\/\\gamma\\mu \\approx 0.56$ and~$0.16$, where~$T$ and~$\\mu$ are the temperature and chemical potential of the black hole, and~$\\gamma \\equiv \\sqrt{3\/2} \\, g\\ell\/\\kappa$ is a dimensionless ratio of the Maxwell and gravitational couplings as in \\cite{Andrade:2013rra}.\n\nThe above discussion brings to the fore the issue of which extremal surfaces should actually contribute to \\eqref{RT} and the associated entanglement conjectures. Thinking of our surfaces as representing saddle points of a path integral suggests that the general answer may be difficult to determine. We refer the reader to the classic discussion of \\cite{Fidkowski:2003nf} in the perhaps-related context of geodesics in Schwarzschild-AdS${}_5$. But in typical cases one might expect saddles on the the principal sheet of our Riemann surface to be more accessible than those on secondary sheets. We therefore again remind the reader that, for codimension-2, the principal sheets studied here admit only real extremal surfaces. This may suggest that only such real surfaces are relevant to the entropies we consider.\n\nFor the geodesic approximation to the two-point function one can give a stronger argument \\cite{Andrade:2013rra} to exclude secondary sheets. The point is that, in that context, branch cuts are a clear artifact of taking what from the dual CFT perspective is the large-dimension limit of the operators involved. For any finite operator dimension, the actual two-point function resolves the branch cut into a discrete series of poles associated with bulk quasi-normal modes \\cite{Festuccia:2005pi,Festuccia:2008zx}. It follows that the geodesic approximation to two-point functions must break down whenever it involves geodesics on secondary sheets.\n\n\n\nThis last argument might perhaps be adapted to the present context using the fact that the Renyi entropies $S_n$ are given by correlators of twist operators \\cite{Holzhey:1994we}. In particular, one might argue that such correlators must again involve only poles (say, in the energy plane) and that branch cuts must be absent. But it is unclear what this would imply for the analytic structure of the von Neumann entropy whose construction requires the analytic continuation to general $n$ and taking the limit \\eqref{eq:RenyiTovN} as $n\\rightarrow 1$.\n\nIt would be interesting to determine whether the principal sheet remains free of complex extremal surfaces when one studies the entropy of other regions on the boundaries of these spacetimes (i.e., not just for the pair of half $(d-1)$-planes considered here). One might hope that the appearance of complex contours on the principal sheet is in fact forbidden by the null energy condition (NEC) so that this argument could be extended to truly general settings. However, in a forthcoming work~\\cite{Fischetti:2014} we describe spacetimes satisfying the NEC where complex extreme surfaces do indeed arise on the principal sheet.\n\n\nOur discussion of complex codimension-2 surfaces was in part motivated by analogy with the case of larger codimension $n> 2$. But comparison of figures \\ref{fig:geodesiccontours} and \\ref{fig:sheets} shows that, at least in practice, the $n=2$ setting behaves very differently. This is perhaps most clear on the principal sheet. While this may at first come as a surprise, one sees from e.g. \\cite{Wall:2012uf} that codimension-2 surfaces are subject to much tighter constraints than for $n > 2$. This occurs because $n=2$ surfaces define a pair of orthogonal null congruences (see e.g. \\cite{Wald:1984rg,Carroll:2004st}) and the extremality condition requires both to have vanishing expansions. The result is that properties of such extremal surfaces are dictated much more directly by the null energy condition than for $n > 2$. Some of the associated implications for real $n=2$ extremal surfaces were discussed in \\cite{Wall:2012uf,Engelhardt:2013tra}. It could be very useful to understand any ramifications for complex $n=2$ surfaces as well.\n\n\nWe conclude that there remain many open questions, and that the possible relevance of complex extremal surfaces to CFT entanglement remains mysterious. But the existence of physically-plausible contours for Schwarzschild-AdS and analogous results for Schwarzschild-Lifshitz makes it critical to understand this issue in detail. One would in particular like to find an independent calculation of the corresponding CFT entropy allowing quantitative comparison with figure \\ref{fig:contour2}. At least for this case such an analysis would definitively answer whether the CFT entropy is determined by real extremal surfaces, or instead by the complex surfaces found in this work.\n\n\n\\section*{Acknowledgements}\nWe thank Tom Hartman, Veronika Hubeny, Mukund Rangamani, Simon Ross, and Aron Wall for discussions related to various aspects of this work. This project was supported in part by the National Science Foundation under Grant No PHY11-25915, by FQXi grant FRP3-1338, and by funds from the University of California. We thank DAMTP, Cambridge U. for their hospitality during the time when this work was conceived.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}