diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzdzjs" "b/data_all_eng_slimpj/shuffled/split2/finalzzdzjs" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzdzjs" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\vglue0.4cm\n\nI am very happy to have the opportunity to speak about strong\/weak \ncoupling duality on\nthis occasion honoring the 60th birthday of Professor Keiji Kikkawa. \nHis own foundational work on T-duality \\cite{KY}, the \nworldsheet analogue of S-duality, \nwas in many ways instrumental in inspiring the recent \ndevelopments in nonperturbative string theory.\n\nStrong-weak coupling dualities now allow us to determine the strong coupling\ndynamics of string vacua with $N \\geq 4$ supersymmetry in four dimensions\n\\cite{Schwarz}.\nIt is natural to ask if this progress in our understanding of string theory\ncan be extended to the more physical vacua with less supersymmetry.\nFor N=2 theories in four dimensions, \nquantum corrections significantly modify the mathematical structure of\nthe moduli space of vacua, as well as the\nphysical interpretation of its apparent singularities. \nThis was beautifully demonstrated in the field theory case in \n\\cite{SeiWit} and it has more recently become possible to compute the\nexact quantum moduli spaces for N=2 string compactifications as \nwell \\cite{KV,FHSV}. \nThis constitutes the subject of the first part of my talk.\n\nOf course, the case of most physical interest is $N \\leq 1$ theories.\nIn the second part of my talk, I discuss examples of dual heterotic\/type II\nstring pairs\nwhere the heterotic theory is expected to exhibit nonperturbative dynamics\nwhich may fix the dilaton and break supersymmetry \\cite{KS}. The type II \ndual manages to reproduce the qualitative features expected of the\nheterotic side at tree level. It is to be hoped that further work\nalong similar lines will result in a better understanding of\nsupersymmetry breaking in string theory.\n\nThe first part of this talk is based on joint work with C. Vafa, and the\nsecond part of this talk is based on joint work with E. Silverstein.\n\n\\section{N=2 Gauge Theory and String Compactifications}\n\nRecall that the N=2 gauge theory with gauge group $SU(2)$ is the theory of\na single N=2 vector multiplet consisting of a vector $A^{\\mu}$, \ntwo Weyl fermions $\\lambda$ and $\\psi$,\nand a complex scalar field $\\phi$, all in the adjoint representation of \n$SU(2)$. \nIn N=1 language, this is a theory of an N=1 vector multiplet\n$(\\lambda, A^{\\mu})$ coupled to an N=1 chiral multiplet $(\\phi, \\psi)$.\nThe scalar potential of the theory is determined by supersymmetry to\nbe \n\\begin{equation}\nV(\\phi) = {1\\over g^{2}} [\\phi, \\phi^{+}]^{2}\n\\end{equation}\nWe see that $V$ vanishes as long as we take $\\phi = diag (a,-a)$, so there\nis a moduli space of classical vacua parameterized by the\ngauge invariant parameter $u = tr(\\phi^{2})$.\n\nAt generic points in this moduli space ${\\cal M}_{v}$ of vacua, there is\na massless N=2 U(1) vector multiplet $A$. The leading terms in its effective\nlagrangian are completely determined in terms of a single holomorphic\nfunction $F(A)$, the prepotential:\n\\begin{equation}\nL \\sim \\int d^{4}\\theta {\\partial F \\over {\\partial A}}\\bar A +\n\\int d^{2}\\theta {\\partial^{2} F \\over{\\partial A^{2}}}W_{\\alpha}W^{\\alpha}\n + c.c.\n\\end{equation} \nThe first term determines, in N=1 language, the Kahler potential (and hence\nthe metric on ${\\cal M}_{v}$) while the second term determines the\ngauge coupling as a function of moduli.\n\nIn \\cite{SeiWit} the exact form of $F$ including \ninstanton corrections was determined. \nIn addition, the masses of all of the BPS saturated particles were \ncomputed. This was reviewed in great detail in several other talks\nat this conference, so I will not repeat the solution here. It will\nsuffice to say that \nthe crucial insight is that the\nsingular \npoint $u=tr(\\phi^{2}) = 0$ where $SU(2)$ gauge symmetry is restored in\nthe classical theory splits, in the quantum theory, into two \nsingular points\n$u = \\pm \\Lambda^{2}$, where a monopole and a dyon become massless. \n\nIn this talk our interest \nis not really in $N=2$ gauge theories but in the string\ntheories which reduce to $N=2$ gauge theories in the infrared. \nThere are two particularly simple classes of $d=4, N=2$ supersymmetric\nstring compactifications. One obtains such theories from Type II (A or B)\nstrings on Calabi-Yau manifolds, and from heterotic strings on $K_{3}\n\\times T^{2}$ (with appropriate choices of instantons on the $K_3$). \nHere we briefly summarize some basic properties of these theories. \n\nType IIA strings on a Calabi-Yau threefold $M$ give rise to a four-dimensional\neffective theory with $n_v$ vector multiplets and $n_{h}$ hypermultiplets\nwhere \n\\begin{equation}\nn_{v} = h^{1,1}(M), ~~n_{h} = h^{2,1}(M) + 1 \n\\end{equation}\nThe $+1$ in $n_{h}$ corresponds to the fact that for such type II\nstring compactifications, the $\\it dilaton$ is in a hypermultiplet.\n\nThe vector fields in such a theory are Ramond-Ramond U(1)s, so there\nare no charged states in the perturbative string spectrum. Furthermore, \nbecause of the theorem of de Wit, Lauwers, and Van Proeyen \\cite{dLVP}\nwhich forbids couplings of vector multiplets to neutral hypermultiplets\nin N=2 effective lagrangians, the dilaton does not couple to the vector \nmoduli. This means that there are no perturbative or nonperturbative\ncorrections to the moduli space of vector multiplets.\nOn the other hand the moduli spaces of hypermultiplets are expected to\nreceive highly nontrivial corrections, including ``stringy'' corrections\nwith $e^{-1\/g}$ strength \\cite{BBS}.\n \nOne interesting feature of the moduli spaces of vector multiplets in\nsuch theories is the existence of conifold points at finite distance\nin the moduli space. At such points the low energy effective theory\nbecomes singular (e.g., the prepotential develops a logarithmic\nsingularity) \\cite{CDGP}. This phenomenon is reminiscent of \nthe singularities in the prepotential which occur at the ``massless\nmonopole'' points in the Seiberg-Witten solution of N=2 gauge theory,\nsingularities which are only present because one has integrated out a\ncharged field which is becoming massless. In the case at hand, in fact,\none can show that there are BPS saturated states (obtained by wrapping\n2-branes around collapsing 2-cycles) which become massless and which\nare charged under (some of) the Ramond-Ramond $U(1)$s \\cite{Strominger}. These\nexplain the singularity in the prepotential. In fact at special\nsuch points, where enough charged fields (charged under few enough\n$U(1)$s) become massless, one can give them VEVs consistent with\nD and F flatness. This results in new ``Higgs branches'' of the moduli\nspace. These new branches correspond to string compactifications on\ndifferent Calabi-Yau manifolds, topologically distinct from $M$\n\\cite{GMS}, and there is evidence that all Calabi-Yau \ncompactifications may be connected\nin this manner \\cite{Cornell,Texas}. \n\nThe other simple way of obtaining an N=2 theory in four dimensions from\nstring theory is to compactify the heterotic string (say $E_8\\times E_8$)\non $K_3 \\times T^2$. Because of the Bianchi identity\n\\begin{equation}\ndH = Tr(R\\wedge R) - Tr(F\\wedge F)\n\\end{equation} \none must embed 24 instantons in the $E_8 \\times E_8$ in order\nto obtain a consistent theory.\nAn $SU(N)$ k-instanton on $K_3$ comes with $Nk + 1 - N^2$ hypermultiplet moduli\n(where $k\\geq 2N$), and $K_3$ comes with 20 hypermultiplet moduli\nwhich determine its size and shape. \nEmbedding an $SU(N)$ instanton in $E_8$ breaks the observable low\nenergy gauge group to the maximal subgroup of $E_8$ which commutes with\n$SU(N)$ ($E_7$ for N=2, $E_6$ for N=3, and so forth). \n\nIn addition, there are three $U(1)$ vector multiplets associated with \nthe $T^2$. Their scalar components are the dilaton $S$ and the\ncomplex and kahler moduli $\\tau$ \nand $\\rho$ of the torus (both of which live on the upper half-plane\n$H$ mod $SL(2,Z)$). At special points in the moduli space\nthe $U(1)^2$ associated with $\\tau$ and $\\rho$ is enhanced to\na nonabelian gauge group:\n\\begin{equation}\n\\tau = \\rho \\rightarrow SU(2)\\times U(1),~~ \\tau=\\rho=i \\rightarrow SU(2)^2, \n~~\\tau = \\rho = 1\/2 + i{\\sqrt 3}\/2 \\rightarrow SU(3)\n\\end{equation}\n\n\nBecause the dilaton lives in a vector multiplet in such compactifications,\nthe moduli space of vectors is modified by quantum effects. On the other\nhand, the moduli space of hypermultiplets receives neither perturbative nor\nnonperturbative corrections. \n\nAn interesting feature of the heterotic ${\\cal M}_{v}$ \nis the existence of special points where the\nclassical theory exhibits an enhanced gauge symmetry (as described\nabove for the compactification on $T^2$).\nSometimes by \nappropriate passage to a Higgs or Coulomb phase, such enhanced gauge\nsymmetry points link moduli spaces of N=2 heterotic theories which\nhave different generic spectra (for some examples see\n\\cite{KV,AFIQ}). It is natural to conjecture that \nsuch transitions connect all heterotic N=2 models, in much the same way\nthat conifold transitions connect Calabi-Yau compactifications of type II\nstrings.\n\n\n\\section{N=2 String-String Duality}\n\n>From the brief description of heterotic and type II N=2 vacua in the\nprevious section, it is clear that a duality relating the two classes of\ntheories would be extremely powerful. If one were to find a model with dual\ndescriptions as a compactification of the Type IIA string on $M$ and \nthe heterotic string on $K_3 \\times T^{2}$, one could compute the exact\nprepotential for ${\\cal M}_{v}$ from the Type IIA side (summing up what from\nthe heterotic perspective would be an infinite series of instanton \ncorrections). Similarly, one would get exact results for ${\\cal M}_{h}$\nfrom the heterotic side -- this would effectively compute the $e^{-1\/g}$\ncorrections expected from the IIA perspective. In fact, such a duality \nhas been found to occur in several examples in \\cite{KV,FHSV}.\n\nOne of the simplest examples is as follows.\nConsider the heterotic string compactified to eight dimensions on $T^{2}$ \nwith $\\tau = \\rho$. Further compactify on a $K_{3}$, satisfying the\nBianchi identity for the $H$ field by embedding \n$c_{2}=10$ $SU(2)$ instantons in each $E_8$ and a $c_{2}=4$ $SU(2)$ instanton\ninto the ``enhanced'' $SU(2)$ arising from the $\\tau=\\rho$ torus.\nAfter Higgsing the remaining $E_7$ gauge groups one is left with a generic\nspectrum of 129 hypermultiplets and 2 vector multiplets.\nThe 2 vectors are $\\tau$ and the dilaton $S$ -- when $\\tau = i$, one expects\nan $SU(2)$ gauge symmetry to appear (the other $SU(2)$ factor that \nwould normally\nappear there has been broken in the compactification process). \n\n\n\nThis tells us that if there is a type IIA dual compactification on a \nCalabi-Yau $M$, then the Betti numbers of $M$ must be \n\\begin{equation} \nh_{11}(M) = 2, ~~h_{21}(M) = 128\n\\end{equation} \nThere is a known candidate manifold with these Betti numbers -- the\ndegree 12 hypersurface in $WP^{4}_{1,1,2,2,6}$ defined by the vanishing of\n$p$\n\\begin{equation}\np = z_{1}^{12} + z_{2}^{12} + z_{3}^{6} + z_{4}^{6} + z_{5}^{2} + ....\n\\end{equation}\nThis manifold has in fact been studied intensively as a simple example of\nmirror symmetry in \\cite{Hosono,Morrison}. \n\nThe mirror manifold $W$ has $h_{11}(W) = 128, h_{21}(W) = 2$. The conjecture\nthat IIA on $M$ is equivalent to the heterotic string described above \nimplies that IIB on $W$ is also equivalent to that heterotic\nstring. The structure of the moduli space of vector \nmultiplets of the heterotic string should be\n$\\it exactly$ given by the classical (in both sigma model and string\nperturbation theory) moduli space of complex structures of $W$. \n\nThe mirror manifold can be obtained by orbifolding\n$p=0$ by the maximal group of phase symmetries which preserves the\nholomorphic three-form \\cite{GP}. Then the two vector moduli are represented\nby $\\psi$ and $\\phi$ in the polynomial \n\\begin{equation}\np = z_{1}^{12} + z_{2}^{12} + z_{3}^{6} + z_{4}^{6} + z_{5}^{2} - 12 \\psi\nz_{1}z_{2}z_{3}z_{4}z_{5} - 2\\phi z_{1}^{6} z_{2}^{6} \n\\end{equation} \nIt is also useful, following \\cite{Hosono}, to introduce\n\\begin{equation}\nx = {-1\\over 864} {\\phi\\over \\psi^{6}}, ~~y = {1\\over \\phi^{2}}\n\\end{equation} \nThese are the convenient ``large complex structure'' coordinates on the\nmoduli space of vector multiplets for the IIB string. \n\nIn order to test our duality conjecture, we should start by checking\nthat the IIB string reproduces some qualitative features that we expect\nof the heterotic ${\\cal M}_{v}$. For example, $\\tau = i$ for weak coupling\n$S\\rightarrow \\infty$ is an $SU(2)$ point. There should therefore be a \nsingularity of ${\\cal M}_{v}$ at this point which splits, as one turns on \nthe string coupling, to $\\it two$ singular points (where monopoles\/dyons\nbecome massless), as in the case of pure $SU(2)$ gauge theory.\n\nThe ``discriminant locus'' where the IIB model becomes singular is given\nby\n\\begin{equation}\n(1-x)^{2} - x^{2} y = 0\n\\end{equation}\nSo we see that as a function of $y$ for $y \\neq 0$ there are two solutions\nfor $x$ and as $y \\rightarrow 0$ they merge to a single singular point\n$x=1$. This encourages us to identify $x=1, y=0$ with $\\tau =i, S \\rightarrow\n\\infty$ of the heterotic string -- the $SU(2)$ point. The metric on the\nmoduli space for $y$ at $y=0$ and $S$ at weak coupling also agree if one \nmakes the identification $y\\sim e^{-S}$.\n\nThere is also a remarkable observation in \n\\cite{Morrison} that the mirror map, restricted to $y=0$, is given by\n\\begin{equation}\nx = {j(i)\\over j(\\tau_{1})}\n\\end{equation}\nwhere $\\tau_{1}$ is one of the coordinates on the Kahler cone of $M$.\nHere $j$ is the elliptic j-function mapping $C$ onto $H\/SL(2,Z)$.\nThis tells us that the classical heterotic $\\tau$ moduli space,\nwhich is precisely $H\/SL(2,Z)$, is embedded in the moduli space of $M$\nat weak coupling precisely as expected from duality.\nIn fact using the uniqueness of special coordinates up to rotations,\none can find the exact formula expressing the IIB coordinates\n$(x,y)$ in terms of the heterotic coordinates $(\\tau,S)$. \n\nOf course with this map in hand there are now several additional things\none can check. The tests which have been performed in \n\\cite{KV,KLT,AGNT,KKLMV} include\n\\medskip\n\n\\noindent 1) A matching \nof the expected loop corrections to the heterotic prepotential\nwith the form of the tree-level exact Calabi-Yau prepotential.\n\\medskip\n\n\\noindent 2) A test that \nthe g-loop F-terms computed by the topological partition\nfunctions $F_g$ on the type II side (which include e.g. $R^{2}$ and other\nhigher derivative terms) are reproduced by appropriate (one-loop!)\ncomputations on the heterotic side.\n\\medskip\n\n\\noindent 3) A demonstration that in an appropriate double-scaling limit,\napproaching the $\\tau = i$, $S \\rightarrow \\infty$ point of the heterotic\nstring while taking $\\alpha^{'} \\rightarrow 0$, the IIB prepotential\nreproduces the exact prepotential of $SU(2)$ gauge theory \n(including Yang-Mills instanton effects) computed in \\cite{SeiWit}.\n\n\\medskip\nThese tests give very strong evidence in favor of the conjectured duality.\nGiven its veracity, what new physics does the duality bring into reach?\n\\medskip\n\n\\noindent $\\bullet$ One now has examples of four-dimensional theories\nwith exactly computable quantum gravity corrections. In the example\ndiscussed above, the\nSeiberg-Witten prepotential which one finds in an expansion\nabout $\\tau = i, S \\rightarrow \\infty$\nreceives gravitational corrections which\nare precisely computable as a power series in $\\alpha^{'}$. \n\\medskip\n\n\\noindent $\\bullet$ On a more conceptual level, the approximate duality of\n\\cite{SeiWit} between a microscopic $SU(2)$ theory (at certain points in its\nmoduli space) and a $U(1)$ monopole\/dyon theory is promoted to an\n$\\it exact$ duality, valid at all wavelengths, between heterotic and\ntype II strings. \n\\medskip\n\n\\noindent $\\bullet$ There is evidence that at strong heterotic coupling,\nnew gauge bosons and charged matter fields appear, sometimes giving rise\nto new branches of the moduli space \\cite{KMP,KM}.\n\\medskip\n\n\\noindent$\\bullet$ The $e^{-1\/g}$ corrections to the hypermultiplet moduli\nspace of type II strings are in\nprinciple exactly computable using duality (and may be of some mathematical\ninterest).\n\n\\medskip\nOne might wonder what is special about the Calabi-Yau manifolds which are\ndual to weakly coupled heterotic strings.\nIn fact it was soon realized that the examples of duality in \\cite{KV} involve\nCalabi-Yau manifolds which are $K_3$ fibrations \\cite{KLM}. That is, locally \nthe Calabi-Yau looks like $CP^{1}\\times K_{3}$.\nIn fact, one can prove that if the type IIA string on a Calabi-Yau $M$\n(at large radius) \nis dual to a weakly coupled heterotic string, then $M$ must be a $K_3$\nfibration \\cite{AL}.\n\nTo make this more concrete,\nin the example of the previous section, we saw $M$ was defined by the\nvanishing of\n\\begin{equation}\np = z_{1}^{12} + z_{2}^{12} + z_{3}^{6} + z_{4}^{6} + z_{5}^{2} + ...\n\\end{equation}\nin $WP^{4}_{1,1,2,2,6}$. Set \n$z_{1}=\\lambda z_{2}$ and define $y=z_{1}^{2}$ (which is an allowed\nchange of variables since an identification on the $WP^{4}$ takes \n$z_{1} \\rightarrow -z_{1}$ without acting on $z_{3,4,5}$). Then the\npolynomial becomes (after suitably rescaling to absorb $\\lambda$)\n\\begin{equation}\np = y^{6} + z_{3}^{6} + z_{4}^{6} + z_{5}^{2} + ... \n\\end{equation}\nwhich defines a $K_{3}$ surfaces in $WCP^{3}_{1,1,1,3}$. The choice\nof $\\lambda$ in $z_{1}=\\lambda z_{2}$ is a point on $CP^{1}$, and the\n$K_{3}$ for fixed choice of $\\lambda$ is the fiber. \n\nIt is not surprising that $K_3$ fibrations play a special role in \n4d N=2 heterotic\/type II duality. Indeed the most famous example of\nheterotic\/type II duality is the 6d duality between heterotic strings on \n$T^{4}$ and type IIA strings on $K_{3}$ \\cite{HT,Witten}. If \none compactifies the type IIA\nstring on a CY threefold which is a $K_3$ fibration, and simultaneously\ncompactifies the heterotic string on a $K_{3}\\times T^{2}$ where the $K_3$\nis an elliptic fibration, then locally one can imagine taking the bases\nof both fibrations to be large and obtaining in six dimensions an example\nof the well-understood 6d string-string duality \\cite{VW}. This picture is not\nquite precise because of the singularities in the $K_3$ fibration,\nbut it does provide an intuitive understanding of the special role of\n$K_3$ fibrations.\n\n\\section{N=1 Duality and Gaugino Condensation}\n \nStarting with an $N=2$ dual pair of the sort discussed above, one can try\nto obtain an $N=1$ dual pair by orbifolding both sides by freely acting\nsymmetries. This strategy was used in \\cite{VW,HLS} where several\nexamples with trivial infrared dynamics were obtained. Here we will\nfind that examples with highly nontrivial infrared dynamics can also\nbe constructed \\cite{KS}.\n\nOur starting point is \nan N=2 dual pair \n(IIA on a Calabi-Yau $M$ and heterotic on $K_{3}\\times T^{2}$)\nwhere the heterotic gauge group takes the form\n\\begin{equation}\nG ~=~E_{8}^{H} \\otimes E_{7}^{obs}\\otimes ...\n\\end{equation}\n$H$ denotes the hidden sector and $obs$ the observable sector.\nWe will first discuss the technical details of the $Z_2$ symmetry by \nwhich we can orbifold both sides\nto obtain an $N=1$ dual pair, and then we discuss the physics of the\nduality.\n\n\nOrbifold the heterotic side by the Enriques involution \nacting on $K_3$ \nand a total reflection on the $T^{2}$. This acts on the base of the elliptic\nfibration $(z_{1},z_{2})$ by\n\\begin{equation}\n(z_{1}, z_{2}) ~ \\rightarrow ~ (\\bar z_{2}, - \\bar z_{1})\n\\end{equation}\ntaking $CP^{1} \\rightarrow RP^{2}$.\nIn addition, we need to choose a lifting of the orbifold group to the gauge\ndegrees of freedom.\n\nWe do this as follows:\n\n\\noindent $\\bullet$ Put a modular invariant embedding into the ``observable''\npart of the gauge group alone.\n\n\\noindent $\\bullet$ Embed the translations which generate the $T^2$ into\n$E_{8}^{H}$, constrained by maintaining level-matching and the relations of\nthe space group. For example one could take Wilson lines $A_{1,2}$ \nalong the $a$\nand $b$ cycle of the $T^{2}$ given by\n\\begin{equation}\nA_{1} = {1\\over 2} (0,0,0,0,1,1,1,1),~~~A_{2} = {1\\over 2}(-2,0,0,0,0,0,0,0)\n\\end{equation}\nHere $A_{1,2} = {1\\over 2}L_{1,2}$ where $L_{1,2}$ are vectors in the \n$E_8$ root lattice. These Wilson lines break the $E_{8}^{H}$ gauge\nsymmetry to $SO(8)_{1} \\otimes SO(8)_{2}$. \n\n\\medskip\nHow does the $Z_2$ map over to the type II side?\n>From the action\n\\begin{equation}\n(z_{1},z_{2}) ~\\rightarrow ~(\\bar z_{2}, -\\bar z_{1})\n\\end{equation} \non the $CP^{1}$ base (which is common to both the heterotic and type II sides),\nwe infer that the $Z_2$ must be an antiholomorphic, orientation-reversing \nsymmetry of the Calabi-Yau manifold $M$.\nTo make this a symmetry of the type IIA string theory, we must simultaneously\nflip the worldsheet orientation, giving us an ``orientifold.'' \nIn such a string theory, one only includes maps $\\Phi$ of the worldsheet\n$\\Sigma$ to spacetime $M\/Z_{2}$ if they satisfy\n\\begin{equation}\n\\Phi^{*}(w_{1}(M\/Z_{2})) = w_{1}(\\Sigma)\n\\end{equation} \nwhere $w_1$ is the first Stieffel-Whitney class.\n\nWe know from 6d string-string duality that the Narain lattice $\\Gamma^{20,4}$\nof heterotic string compactification on $T^4$ maps to the integral cohomology\nlattice of the dual $K_{3}$. This means that we can infer from the action\nof the $Z_2$ on the heterotic gauge degrees of freedom, what the action of\nthe $Z_2$ must be on the integral cohomology of the $K_3$ fiber on the IIA\nside.\nSince we are frozen on the heterotic side at a point with $SO(8)^{2}$ gauge\nsymmetry in the hidden sector, the dual $K_3$ must be frozen at its singular\nenhanced gauge symmetry locus. \n\nThe $K_3$ dual to heterotic enhanced gauge symmetry $G$ has rational\ncurves $C_i$, $i = 1,...,rank(G)$ shrinking to zero area (with the \nassociated $\\theta_{i}=0$ too). It is easy to see, e.g. from Witten's\ngauged linear sigma model that in this situation the type II theory\nindeed exhibits an extra $Z_2$ symmetry. The bosonic potential of\nthe relevant gauged linear sigma model (for the case of a single\nshrinking curve) is given by \n$$V=\n{1\\over{2e^2}}\n\\sum_i\\biggl\\{\\biggl(\\bigl[\\sum_\\alpha Q_i^\\alpha(|\\phi^i_\\alpha|^2\n-|\\tilde\\phi^i_\\alpha|^2)\\bigr]-r_i^0\\biggr)^2$$\n$$+\\biggl(Re(\\sum_\\alpha\\phi^i_\\alpha\\tilde\\phi^i_\\alpha)-r_i^1\\biggr)^2\n+\\biggl(Im(\\sum_\\alpha\\phi^i_\\alpha\\tilde\\phi^i_\\alpha)-r_i^2\\biggr)^2\n\\biggr\\}$$ \n$$+{1\\over 2}\\sum_i\\bigl[\\sum_\\alpha Q^{\\alpha~2}_i\n(|\\phi^i_\\alpha|^2+|\\tilde\\phi^i_\\alpha|^2)\\bigr]|\\sigma_i|^2$$ \nHere the $\\phi$s represent the $K_3$ coordinates while $r$ parametrizes\nthe size of the curve and $\\sigma$ is the Kahler modulus.\nPrecisely when $\\vec r \\rightarrow 0$, the model has the $Z_2$ symmetry\n$\\phi \\rightarrow -\\tilde \\phi$, $\\sigma \\rightarrow - \\sigma$. \nOrbifolding\nby this $Z_2$ then freezes the $K_3$ at its enhanced gauge symmetry locus,\nas expected.\n\nWhat is the physics of the dual pairs that one constructs in this manner?\nIn the heterotic string, when there is a hidden sector pure gauge group\n\\begin{equation}\nG^{hidden} = \\Pi ~G^{b}\n\\end{equation}\none expects gaugino condensation to occur. This induces an effective\nsuperpotential\n\\begin{equation} \nW = \\sum ~h_{b}~ \\Lambda_{b}^{3}(S)\n\\end{equation}\nwhere $\\Lambda_{b}(S) \\sim e^{-\\alpha_{b} S}$ and $\\alpha_{b}$ is \nrelated to the\nbeta function for the running $G_b$ coupling. It was realized early on\n\\cite{Krasnikov,DKLP} that in such models (with more than one hidden factor)\none might expect both stabilization of the dilaton and supersymmetry \nbreaking.\nIt has remained a formidable problem to determine which (if any) such models\nactually do have a stable minimum at weak coupling with broken supersymmetry.\n\nFor now, we will be content to simply understand how the $\\it qualitative$\nstructure of the heterotic theory (e.g. the gaugino-condensation induced\neffective superpotential) is reproduced by the type II side.\nThis is mysterious because the type II N=2 theory we orientifolded had only\nabelian gauge fields in its spectrum, so we need to reproduce the strongly\ncoupled nonabelian dynamics of the heterotic string with an $\\it abelian$\ngauge theory on the type II side.\n\nThe heterotic orbifold indicates the spectrum of the string theory as\n$g_{het} \\rightarrow 0$. The\nheterotic dilaton $S$ maps to the radius $R$ of the $RP^{2}$ base of the\ntype II orientifold\n(recall one obtains the $RP^2$ by orbifolding the\nbase $P^1$ of the $K_3$ fibration)\n\\begin{equation}\nS_{het} \\leftrightarrow R_{RP^{2}}\n\\end{equation} \nThe purported stable vacuum of the heterotic theory should then be expected\nto lie at large radius for the base, and on the (orientifold of the) conifold\nlocus dual to enhanced gauge symmetry. There are two crucial features of this\nlocus:\n\\medskip\n\n\\noindent 1) The $RP^2$ base has $\\pi_{1}(RP^{2}) = Z_{2}$. So a state\nprojected out in orientifolding the N=2 theory will have a massive version\ninvariant under the $Z_2$. Say $\\beta \\in \\pi_{1}(RP^{2})$ \nis the nontrivial element. Take $x$ a coordinate along an appropriate\nrepresentative of $\\beta$ -- a representative can be obtained by taking \nthe image of a great circle on the original base $P^1$ after \norientifolding. \nThen if the original non-invariant vertex operator was $V$, a\nnew invariant vertex operator is given adiabatically by\n\\begin{equation}\nV^{\\prime} = e^{ix \\over R} V \n\\end{equation}\nThe $Z_2$ takes $x$ to $x + \\pi R$ and therefore $V^{\\prime}$ is invariant\nif $V$ was not. In particular this gives us massive versions of the\nscalars $a^{i}_{b,D}$ in the N=2 vector multiplets for $G^{b}$ with masses\n\\begin{equation}\nM_{a} \\sim {1\\over R^{2}}\n\\end{equation}\nEffectively, for very large $R$, one is restoring the original N=2 \nsupersymmetry. \n\\medskip \n\n\\noindent 2) The low energy theory for IIA at the conifold locus contains\nmassless $\\it solitonic$ states \\cite{Strominger}. One can see that\nthey survive the N=2 $\\rightarrow$ N=1 orientifolding by examining \nthe behavior of the gauge couplings \\cite{VW}. These extra solitonic\nstates play the role of the ``monopole hypermultiplets'' $M_{i}^{b}, \n\\tilde M_{i}^{b}$ of the N=2 theory. \n\n\\medskip\nThese two facts taken together imply that as $R \\rightarrow \\infty$ there\nis an effective superpotential\n\\begin{equation}\nW_{II} = \\sum_{b} \\left( m_{b}u^{b}_{2}(a^{i}_{b,D},R) +\n\\sum_{i=1}^{rank(b)} M_{i}^{b} a^{i}_{b,D}\\tilde M^{b}_{i} \\right)\n\\end{equation} \nwhere $u^{b}_{2}$ is the precise analogue of $u$ of \\S2 for $G^b$ and \nits functional dependence on $R$ can be found from the $N=2$ dual pair. \nAs we'll now discuss, this structure\n\\medskip \n\n\\noindent a) Allows us to reproduce the gaugino-condensation induced\neffective superpotential of the heterotic side. \n\\medskip\n\n\\noindent b) Implies $\\langle M \\rangle \\neq 0$, suggesting a geometrical\ndescription of the type II side by analogy with N=2 conifold transitions.\n\n\\medskip\nTo see a), recall how the physics of N=1 $SU(2)$ gauge theory is\nrecovered from the N=2 theory in \\cite{SeiWit}. One can obtain the\nN=1 theory by giving a bare mass to the adjoint scalar in the N=2\nvector multiplet and integrating it out. In the vicinity of the\nmonopole points this means there is an effective superpotential \n\\begin{equation}\nW = m u(a_{D}) + \\sqrt{2} a_{D} M \\tilde M\n\\end{equation}\nUsing the equations of motion and D-flatness, one finds\n\\begin{equation}\n\\vert \\langle M \\rangle \\vert = \n\\vert \\langle \\tilde M \\rangle \\vert \n= ( -mu^{\\prime}(0)\/\\sqrt{2})^{1\/2},~~a_{D} = 0 \n\\end{equation}\nThe monopoles condense and given a mass to the (dual) $U(1)$ gauge field\nby the Higgs mechanism, leaving a mass gap. Two vacua arise in this way --\none at each of the monopole\/dyon points -- in agreement with the \nWitten index computation for pure $SU(2)$ gauge theory. \n \nIn our case, we expect that condensation of the massless solitons will lead\nto the gaugino condensation induced superpotential (and, perhaps, supersymmetry\nbreaking). To see this we must expand $W_{II}$ in $a^{i}_{b,D}$ \nin anticipation of finding\na minimum at small $a_{D}$ (since the minimum was at $a_{D} = 0$ in the global\ncase). \nUsing\n\\begin{equation}\nu_{2}(a^{i}_{b,D},R)) = e^{i\\gamma^{b}}\\lambda_{b}^{2}(R) + \\cdots \n\\end{equation}\n(where $\\gamma^{b}$ comes from the phase of the gaugino condensate)\nas well as the matching condition\n\\begin{equation}\nm_{b}\\Lambda_{b,high}^{2} = \\Lambda_{b,low}^{3}\n\\end{equation}\none obtains from integrating out the massive adjoint scalar, one sees \n\\begin{equation}\nW_{II} = \\sum_{b} e^{i\\gamma^{b}}\\Lambda_{b}^{3}(S) + \\cdots\n\\end{equation} \nSimply minimizing the supergravity scalar potential\n\\begin{equation} \nV = e^{K}(D_{i}W G^{i\\bar j}D_{\\bar j}W - 3 \\vert W \\vert^{2}) + \n{1\\over 2}g^{2}D^{2}\n\\end{equation}\nwe also find that \n\\begin{equation}\n\\langle M_{i}^{b} \\tilde M_{i}^{b} \\rangle = -h_{b}m_{b}u_{2,i}^{b}(S) - K_{i}\nW\n\\end{equation}\nThat is, the ``wrapped two-branes'' which give us the massless monopoles have\ncondensed, in accord with the global result.\nSo integrating out the massive $M, \\tilde M$ and adjoint scalar degrees of\nfreedom yields the same form of bosonic potential that we expect from\ngaugino condensation on the heterotic side.\n\nIn summary, we have argued that the type II dual description of the effects\nof gaugino condensation involves a mass perturbation breaking N=2 \nsupersymmetry. One cannot add mass terms by hand in string theory: The\ntype II orientifold produces the requisite massive mode as a Kaluza-Klein\nexcitation of the original N=2 degrees of freedom that were projected out.\n\nOne intriguing feature of the IIA vacuum is the nonzero VEVs for the wrapped\ntwo-branes $M, \\tilde M$. In the N=2 context $\\langle M \\rangle \\neq 0 \n\\rightarrow$ conifold transition. There is a well known geometrical \ndescription\nof the conifold points. For example, in the IIB theory the conifold \nin vector multiplet moduli space is obtained by going to a point in \n${\\cal M}_{complex}$ where there is a cone over $S^{3}\\times S^2$\nin the Calabi-Yau. One can either ``deform the complex structure'' (return\nto the Coulomb phase, in physics language) by deforming the tip of the\ncone into an $S^3$, or one can do a ``small resolution'' and blow the tip\nof the cone into an $S^2$. The latter corresponds to moving to a new\nHiggs phase, in the N=2 examples \\cite{GMS}.\n\nIt was noted long ago by Candelas, De La Ossa, Green, and Parkes \\cite{CDGP}\nthat at a $\\it generic$ conifold singularity such a small resolution does\nnot produce a Kahler manifold. They speculated that such nonKahler resolutions\nmight correspond to supersymmetry breaking directions. It is natural\nto suggest that we might be seeing a realization of that idea by duality.\nThe analogy with N=2 conifold transitions suggests that\n$\\langle M \\rangle \\neq 0 \\rightarrow$ nonKahler resolution. One can hope\nthat this will provide a useful dual view of supersymmetry breaking in \nstring theory.\n\n\n\\vfill\\eject \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n \nWith recent advancements in VLSI technologies, integrated circuits (ICs) manufacturing involves multiple companies, introducing new security challenges at each stage of IC production. \nHardware Trojans (HTs) are defined as any undesired modification in an IC which can lead to erroneous outputs and or leak of information~\\cite{pan2021automated}. According to the adversarial model introduced by~\\cite{shakya2017benchmarking}, HTs can be inserted into the target IC in three different scenarios, namely intellectual properties (IPs) (processing cores, various I\/O components, and network-on-chip~\\cite{sarihi2021survey}), disgruntled employees at the integration stage, reverse-engineering by an untrusted foundry.\n\n\\begin{comment}\n\n \\begin{itemize}\n \\item The first scenario is through intellectual properties (IPs) (e.g., processing cores, memory modules, various I\/O components, and network-on-chip~\\cite{sarihi2021survey}) that are purchased from IP vendors to expedite the time-to-market of an IC and reduce design costs. As a result of integrating an infected third-party IP, HTs can be inserted into the IC. \n \\item Second, designs by engineers at a company can be attacked by compromised employees at the integration stage to inflict harm on the integrity of the IC. \n \\item Third, an untrusted foundry can reverse-engineer the design and insert HTs to infect chips at the fabrication stage.\n \\end{itemize}\n\n\\end{comment}\n\n\\begin{comment}\n\n\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[scale=.57]{figures\/Trigger_payload.pdf}\n \\%vspace{-3mm}\n \\caption{An HT with a trigger and payload. Whenever A=1, B=1, C =0, the trigger is activated (D=1) and the XOR payload inverts the value of E. }\n \\%vspace{-6mm}\n \\label{trigger_payload}\n\\end{figure}\n\n\\end{comment}\nTo study the behavior of HTs in digital circuits, researchers have been mostly using limited benchmarks ~\\cite{shakya2017benchmarking,salmani2013design} including a set of 91 trust benchmarks with different HT sizes and configurations (available at~\\url{trust-hub.org}). Over the past years, various HT detection approaches have been developed based on these benchmarks~\\cite{salmani2016cotd, sabri2021sat,hasegawa2017trojan, sebt2018circuit}. Despite the valuable effort to create such HT benchmarks, they are limited in size and variety necessary to push the detection tools. Another downside of using these existing benchmarks is the problem of dealing with fixed static trigger conditions~\\cite{cruz2018automated}, which stems from the human bias during HT insertions. As a result, HT detectors can be tuned in a way to enhance their detection accuracy while not truly being effective in complex and real-world modern ICs. These shortcomings emphasize the need for an automated HT-insertion tool free of human biases that can be used to create high-volume HT benchmarks. Such a tool will help implement HT benchmarks aligned with the fast growth of attack approaches and cater to the security needs in the realm of hardware design. By introducing new HTs into a design, one can create new benchmarks and push the capabilities of HT detection via introducing never-seen-before HTs.\n\nAlthough a few researchers have tried to address these problems by introducing tunable HT insertion toolsets~\\cite{cruz2018automated,yu2019improved}, these approaches have no concrete guideline for selecting trigger and payload nets; instead, triggering is done on an ad-hoc basis with little design space exploration capabilities. In this paper, we attempt to develop an HT-insertion tool, free of human biases, using a Reinforcement Learning (RL) agent that decides where to insert HTs through a trial and error method. Although machine learning techniques have been used to detect HTs in the past~\\cite{xue2020ten,salmani2016cotd,hasegawa2017trojan}, to the best of our knowledge, this work is the first that addresses HT insertion using a machine learning approach via design space exploration\n\nOur toolset translates each circuit to a graph representation in which different properties of each net, such as controllability, observability, and logical depth, are computed (so-called the SCOAP\\footnote{Sandia controllability and observability analysis program.} parameters ~\\cite{goldstein1980scoap}). The circuit graph is considered as an environment in which the RL agent tries to insert the HT to maximize the gained rewards.\nObtained results confirm that the inserted HTs are very hard to detect as the toolset maximizes the number of IC's inputs involved in the activation of the inserted HTs. We define a metric called the input coverage percentage (ICP) to determine the difficulty of HT activation.\n\\begin{comment}\nThe contributions of this work are the following:\n\\begin{itemize}\n \\item Development of an HT inserting toolset free of human biases using RL.\n \\item The automating of HT insertion; our toolset makes the selection for the trigger and payload nets.\n \\item The toolset can be tuned to allow different HT insertion goals.\n\\end{itemize}\n\\end{comment}\n\nThe paper is organized as follows: The mechanics of our proposed approach are presented in Section~\\ref{proposed}. Section~\\ref{results} demonstrates the experimental results and Section~\\ref{conclusion} concludes the paper.\n\n\\begin{comment}\n\n\n\\section{Background and Related work}\n\\label{background}\n\n\\subsection{Hardware Trojan Benchmarks}\n\\label{previous}\n\nThe first attempts to gather a benchmark with hard-to-activate HTs were made by Shakya \\etal{} and Salmani \\etal{}~\\cite{shakya2017benchmarking,salmani2013design}. A set of 91 trust benchmarks with different HT sizes and configurations are available at~\\url{trust-hub.org}. While these benchmarks have been a valuable contribution for researchers to assess detection techniques, they only represent a subset of possible HT insertion landscape in digital circuits. While the HTs are carefully inserted to seriously compromise the security, a more general approach is needed to explore more options and diversify the HT insertion process.\n\nCruz \\etal{}~\\cite{cruz2018automated} tried to address these shortcomings by presenting a toolset that is capable of inserting a variety of HTs based on the parameters passed to the toolset. Their software inserts HTs with the following configuration parameters: the number of trigger nodes, the number of rare nodes among the trigger nodes, the threshold of rare nodes computed with the SCOAP parameters, the number of the HT instances to be inserted, the HT effect, the activation method, its type, and the choice of payload.\nDespite increasing the variety of inserted HTs, there is no solution for finding the optimal trigger and payload nets.\n\nYu \\etal{}~\\cite{yu2019improved} considers a different criterion to identify rare nets in a circuit. The set of test vectors can lead to an inaccurate selection of rare nets and, subsequently, inefficient trigger nets. Instead, their approach is to use transition probability to represent the switching activity of the nets. The transition probability of each net is computed based on the time required for the value of each net to toggle, and it is modeled using geometric distribution. The HT insertion criteria are very similar to~\\cite{cruz2018automated}. In the end, the trigger and payload nets are selected randomly.\n\nIn an attempt to deceive machine learning HT detection approaches, Nozawa \\etal{}~\\cite{nozawa2021generating} have devised adversarial examples. Their proposed method replaces the HT instance with its logically equivalent circuit to make the classification algorithm erroneously overlook it. To design the best adversarial example, the authors have defined two parameters: Trojan-net concealment degree (TCD) is tuned in a way to maximize the loss function of the neural network in the detection process, and a modification evaluating value (MEV) that should be minimized to have the least impact on circuits. These two metrics help the attacker to look for more effective logical equivalents and limit the design of HTs to more effective ones. By doing so, the generated framework can decrease the detection accuracy significantly.\n\n\\%vspace{-2mm}\n\\subsection{Reinforcement Learning}\n\\label{Previous_RL}\n\nReinforcement Learning (RL) has recently attracted considerable attention as a powerful machine learning strategy for decision making through trial and error~\\cite{sutton2018reinforcement}. The training process is very similar to how humans and animals learn in the sense that good actions are rewarded positively, and bad actions are rewarded negatively. Figure~\\ref{RL} shows the typical flow of RL algorithms, which consist of five main components: \n\\begin{enumerate}[noitemsep,topsep=0pt]\n\\item{\\emph{Agent}: which interacts with the environment by taking actions. }\n\\item{\\emph{Action}: selecting from a set of possible decisions. }\n\\item{\\emph{Reward}: the feedback provided by the environment after the action has been done. }\n\\item{\\emph{Environment}: rewards the agent after receiving a new action. }\n\\item{\\emph{State}: the observation that the agent receives after every action. }\n\\end{enumerate}\nEvery agent starts from a reset condition where it takes action ($a_t$), based on the state ($s_t$) observed from the environment. The environment rewards the agent with new reward $r_{t+1}$ and updates the state to $s_{t+1}$. This cycle repeats until either a number of actions are reached or a terminal state is reached. The whole process is called an episode, e.g., one game of chess. During the training session, numerous episodes are played, and the agent's goal is to maximize the sum of collected rewards over all episodes. \n\n\n\n\\begin{figure}[!t]\n \\centering\n \\includegraphics[scale=0.77]{figures\/RL.pdf}\n \\%vspace{-8mm}\n \\caption{The main components of RL which consist of an agent, environment, action, state, and reward.}\n \\%vspace{-5mm}\n \\label{RL}\n\\end{figure}\n\n\n\n\n\nAlthough the first uses of RL were in the classic control domain, the HT community has also turned to RL recently. A study in~\\cite{pan2021automated} uses RL framework to detect HTs. The authors address the computation complexity and weak trigger coverage in large designs by using RL to find the best input test vectors. In this case, the advantage of using RL is the ability to solve problems with large and complex solution space. For each circuit, test vector bits are flipped to activate the most trigger nets and gain the highest summation of SCOAP parameters of these nets. The RL model significantly reduces the test generation time and is able to detect a variety of HTs in the ISCAS-85 and ISCAS-89 benchmarks. \n\n\\end{comment}\n\\section{RL-based HT insertion}\n\\label{proposed}\n\\begin{figure*}[ht]\n \\centering\n \\includegraphics[scale=.48]{figures\/Flowchart.pdf}\n \n \\caption{The proposed RL-based HT insertion tool flow.}\n \n \\label{Toolflow}\n\\end{figure*}\nFigure~\\ref{Toolflow} shows the flow of the proposed HT insertion tool. The first step to insert an HT into a circuit is to create a graph representation of the flattened netlist from the circuit. \nYosys Open Synthesis Suite~\\cite{wolf2013yosys} translates a Verilog file of the circuit into a JSON (JavaScript Object Notation)~\\cite{bassett2015introduction} netlist where the JSON file is used by a python script to parse the internal graph representation of the circuit. Next, the tool finds a set of rare nets to be used as HT trigger nets (this is described in detail in Subsection~\\ref{s1}). Finally, an RL agent uses the rare net information and attempts to insert an HT to maximize a reward function as described in Subsection~\\ref{s2}.\n\n\\subsection{Rare Nets Extraction}\n\\label{s1}\n\nAs discussed earlier, different circuit criteria have been used for trigger selection. In this work, we use the parameters introduced in \\cite{sebt2018circuit} where the trigger nets are selected based on functions of net \\emph{controllability} and \\emph{observability} \\cite{goldstein1980scoap}. \n\\begin{comment}\n\nControllability measures the difficulty of setting a particular net in a design to either \\emph{'0'} or \\emph{'1'}. Observability, on the other hand, is the difficulty of propagating the net to at least one of the circuit outputs.\n\n\n\\end{comment}\n\n\nThe first parameter is called the HT trigger susceptibility parameter, and it is derived from the fact that low switching nets have a high difference between their controllability values. Equation \\ref{HTS1} describes this parameter:\n\\begin{equation}\n HTS(Net_i)=\\frac{|CC1(Net_i)-CC0(Net_i)|}{Max(CC1(Net_i),CC0(Net_i))}\n \\label{HTS1}\n\\end{equation}\nwhere $HTS$ is the HT trigger susceptibility parameter of a net; $CC0(Net_i)$ and $CC1(Net_i)$ are the combinational controllability 0 and 1 of $Net_i$, respectively. The $HTS$ parameter ranges between $[0,1)$ such that higher values of $HTS$ are correlated with lower activity on the net. \n\nThe other used parameter is specified in Equation~\\ref{OCR} to measure the ratio of observability to controllability:\n\\begin{equation}\n OCR(Net_i)=\\frac{CO(Net_i)}{CC1(Net_i)+CC0(Net_i)}\n \\label{OCR}\n\\end{equation}\nwhere $OCR$ is the observability to controllability ratio of a net. This equation requires that the HT trigger nets must be very hard to control but not so hard to observe. Unlike the $HTS$ parameter, $OCR$ is not bounded, and it belongs to the interval of $[0,\\infty)$. We will specify thresholds (see Section~\\ref{s2}) for each parameter and use them as filters to populate the set of rarely-activated nets for our tool.\n\\begin{comment}\n\n\n\\begin{figure}[!t]\n \\centering\n \n \\includegraphics[scale=0.50]{figures\/Levels.pdf}\n \\%vspace{-4mm}\n \\caption{Levelizing a circuit. The output level of each digital gate is computed by $max(Level(in1),Level(in2))+1$.}\n \\%vspace{-3mm}\n \\label{level}\n\\end{figure}\n\\end{comment}\n\\subsection{RL-Based HT Insertion.}\n\\label{s2}\n\nAgent, Action, Environment, State, and Reward are the five main components of reinforcement learning. From an RL perspective, we define the environment as the circuit in which we are trying to insert HTs. The agent's action is the insertion of the HT. Note, we consider combinational HTs where trigger nets are ANDED, and the payload is an XOR gate. \n\nWe represent different HT insertions with a state vector in each circuit. To address this issue, we first levelize the circuit. The output level of an $m$-input gate is computed by equation~\\ref{eq1}:\n\\begin{equation}\n Level(out)=MAX(Level(in_1), Level(in_2), ... , Level(in_m))+1\n \\label{eq1}\n\\end{equation}\n\n\\begin{figure}[!t]\n \\centering\n \n \\includegraphics[width=3.3in, height=1.6in]{figures\/HT_insertion.pdf}\n \\caption{Obtaining the state vector in the presence of an HT.}\n \\label{State}\n\\end{figure}\n\nFigure~\\ref{State} depicts a 2-input HT (in yellow) where the XOR payload flips the value of the target net when the trigger is activated. For a given HT, the state vector is comprised of $s_t=[s_1,s_2, ...,s_{n-2},s_{n-1},s_{n}]$ where $s_1$ through $s_{n-2}$ are levels of the HT inputs and $s_{n-1}$ and $s_{n}$ are levels of the target net and the output of the XOR payload, respectively. As an example, the HT in Figure~\\ref{State} has the state vector $s_t=[2,1,3,4]$. The action space of the described HT agent is multi-discrete, i.e, each input of the HT can choose an action from a set of 5 available actions. These actions are:\n\n\\begin{itemize}\n \\item \\emph{\\textbf{Next level}}: the input of the HT moves to one of the nets that are one level higher than the current net level.\n \\item \\emph{\\textbf{Previous level}}: the input of the HT moves to one of the nets that are one level lower than the current net level.\n \\item \\emph{\\textbf{Same level up}}: the input of the HT will move to one of the nets at the same level as the current net level. The net is picked by pointing to the next net in the ascending list of net ids for the given level. \n \\item \\emph{\\textbf{Same level down}}: the input of the HT will move to one of the nets at the same level as the current net level. The net is picked by pointing to the previous net in the ascending list of nets for the given level. \n \\item \\emph{\\textbf{No action}}: the input of the HT will not move.\n\\end{itemize}\n\nIf an action leads the agent to step outside the circuit boundaries, it is substituted with a ``No action''.\n\nThe action space is also represented by a vector where its size is equal to the number of the HT inputs, and each action can be one of the five actions above, e.g., for the HT in Figure~\\ref{State}, the action space would be $a_t=[a_1,a_2]$ since it has two inputs. \n\nAs we explained in Section~\\ref{s1}, SCOAP parameters are first computed. We specify two thresholds $T_{HTS}$ and $T_{OCR}$ and require our algorithm to find nets that have higher $HTS$ values than $T_{HTS}$ and lower $OCR$ values than $T_{OCR}$. These nets are classified as suspicious nets. \n\nOur toolset utilizes an algorithm that consists of two conditional while loops that keep track of the terminal states and the elapsed timesteps. The first used function in the algorithm is called \\emph{reset\\_environment} which resets the environment before each episode. Upon reset, an HT is randomly inserted within the circuit according to the following set of rules.\n\\begin{itemize}\n \\item Rule 1) Trigger nets are selected randomly from the list of the total nets.\n \\item Rule 2) No trigger net is allowed to be fed from a previously used net.\n \\item Rule 3) Trigger nets cannot be assigned as the target.\n \\item Rule 4) The target net is selected considering the level of trigger nets. To prevent forming combinational loops, we specify that the level of the target net should be greater than that of the trigger nets.\n\\end{itemize}\n\nDuring the training process in each episode, we do not change the payload net to help the RL algorithm converge faster for possible solution\n. Unlike the manual payload selection, we allow the algorithm explore the environment during different episodes and decide which payload can more seriously compromise the security by collecting higher rewards. This solution addresses the problem of finding optimal payload selection.\n\n\nThe training process of the agent takes place in a loop where actions are being issued, rewards are collected, the state is updated, and eventually, the updated graph is returned. To evaluate the taken actions by the RL agent (meaning if the HT can be triggered with any input vector), we use PODEM (Path-Oriented Decision Making), an automatic test pattern generator~\\cite{bushnell2000essentials}. If the HT payload propagates through at least one of the circuit outputs, the action gains a reward proportional to the number of inputs of the circuit that are engaged in the activation of the HT (we call this feature input coverage). We believe that the number of inputs engaged in the HT activation could be viewed as a metric of how rarely the HT is activated (see Section \\ref{results}).\n\nIn PODEM, $input\\_stack$ is the list of circuit inputs and their values that activate the HT. The more inputs engaged, the higher the reward will be. For an action to get positively rewarded, at least one of the HT trigger nets must belong to the set of the suspicious nets. This type of rewarding encourages the agent to search in the vicinity of the suspicious nets (states) where rewards are more positive and hence result in stealthier HTs. The agent is rewarded -1 when the trigger nets do not belong to the set of suspicious nets. Our proposed RL rewarding scheme drives the agent towards inserting hard-to-active HTs and maximizing the input coverage.\nThe rewarding scheme is given in Equation \\ref{reward}\n\\begin{equation}\n reward+= 20*(size(input\\_stack)\/size(in\\_ports))\n \\label{reward}\n\\end{equation}\nwhere $in\\_ports$ is the total number of inputs.\nWe selected the coefficient 20 since it strikes a balance between the mostly '-1' rewards collected during training and the limited number of HTs found in each episode. One key benefit of using our tool is that the designer can adjust the reward scheme to achieve different goals. \n\n\nTo train the RL agent, we use the PPO (Proximal Policy Optimization) RL algorithm. PPO can train agents with muti-discrete action spaces in discrete or continuous spaces. The main idea of PPO is that the updated new policy (which is a set of actions to reach the goal) should not deviate too far from the old policy following an update in the algorithm. To avoid substantial updates, the algorithm uses a technique called clipping in the objective function~\\cite{schulman2017proximal}. At last, when the HTs are inserted, the toolset outputs Verilog gate-level netlist files that contain the malicious HTs.\n\n\n\\begin{comment}\n\n\n\\begin{algorithm}[t]\n \\caption{Training of the HT Reinforcement Learning Agent}\n \\begin{flushleft}\n \\hspace*{\\algorithmicindent}\\textbf{\\textit{Input: }}{Graph $G$, HTS Threshold $T_{HTS}$, OCR Threshold,}\\\\\n \\hspace*{\\algorithmicindent}{$T_{OCR}$, circuit inputs $in\\_ports$,state space $s_t$,}\\\\\n \\hspace*{\\algorithmicindent}{terminal state $TS$,circuit inputs $in\\_ports$,}\\\\\n \\hspace*{\\algorithmicindent}{total timesteps $j$;}\\\\\n \\hspace*{\\algorithmicindent}\\textbf{\\textit{Output: }}{Malicious design $T$;}\n \\end{flushleft}\n \n \\begin{algorithmic}[1]\n \\label{Alg1}\n \\STATE Compute SCOAP parameters:\\\\\n \\hspace*{\\algorithmicindent}{$=computeSCOAP(G)$};\n \\STATE Get the set of suspicious nets:\\\\\n \\hspace*{\\algorithmicindent}{$suspicious\\_nets=computeSuspiciousNets(G, T_{HTS}, T_{OCR});$}\\\\\n \n \n \\STATE $counter=0;$\n \\WHILE{($countert_2$, one has \n\\begin{eqnarray}\n\\left\\langle A\\left( t_{1}\\right) B\\left( t_{2}\\right) \\right\\rangle\n&=&{\\rm Tr}_{S\\otimes R}\\left[A\\left( t_{1}\\right) B\\left(\nt_{2}\\right) \\rho _{T}\\left( 0\\right) \\right]\n\\nonumber \\\\\n&=&{\\rm Tr}_{S\\otimes R}\\left[ A e^{-iHt\/\\hbar}B \\rho _{T}\\left( t_{2}\\right)\ne^{iHt\/\\hbar}\\right], \n\\label{AB12}\n\\end{eqnarray}\nwhere $t=t_1-t_2$, and $A=A(0)$ and $B=B(0)$ are system operators. \nLet $\\chi_T(0)=B \\rho _{T}\\left( t_{2}\\right)$. \nThen the two-time CF (\\ref{AB12}) becomes $\\left\\langle A\\left( t_{1}\\right) B\\left( t_{2}\\right) \\right\\rangle\n={\\rm Tr}_{S\\otimes R}\\left[ A\\chi_{T}\\left( t\\right) \\right]$. \nIt is then equivalent to\nthe expectation value of \nthe operator $A$\nwith respect to the effective density matrix operator \n$\\chi_T(t)=e^{-{iHt}\/{\\hbar}}B \\rho _{T}\\left( t_{2}\\right)e^{{iHt}\/{\\hbar} }$\nthat satisfies the same Liouville evolution equation as $\\rho_T(t)$\neven though $\\chi_T(t)$ may not be a proper density matrix \n(i.e., positive-definite trace-conservative operator).\nThe evolution equation of the two-time CF can be formally written as \n\\begin{eqnarray}\n{d\\langle A\\left( t_{1}\\right) B(t_2)\\rangle }\/{dt_{1}}\n&=&{\\rm Tr}_{S\\otimes R}\\left[ A ({d\\chi_T(t)}\/{dt})\\right] \n\\label{2time_evol}\\\\\n&=&{\\rm Tr}_{S}\\left[ A({d\\chi(t)}\/{dt})\\right],\n\\label{2time}\n\\end{eqnarray}\nwhere the relations of the reduced operator $\\chi(t)={\\rm Tr}_R[\\chi_T(t)]$\nand ${\\rm Tr}_R[{d\\chi_T(t)}\/{dt}]={d\\chi(t)}\/{dt}$\nhave been used.\nIf the reduced master equations \n${d\\chi(t)}\/{dt}$ and ${d\\rho(t)}\/{dt}$ had the same \noperator equation form, one might conclude that the structure and the form\nof the evolution equation of the two-time CF would be the same as those of the single-time evolution equation and thus the QRT would apply.\nIn fact, it has been shown that \nthe QRT is not valid in general \\cite{Ford96,Ford99}, but \nthe QRT or regression procedure is useful and correct for systems\nwhere the coupling to reservoirs is weak \nand the Markovian approximation holds \\cite{Carmichael99,Lax00,Ford00}. \nThe main purpose of the present paper is to derive \nthe non-Markovian finite-temperature evolution equation of the \ntwo-time system CF's using a quantum master equation approach, an\napproach different from those in Refs.~\\cite{Vega06,Alonso07}. \nOur equations, which are valid for both a Hermitian and a\nnon-Hermitian system coupling operators and thus generalize the\ncorresponding results in Refs.~\\cite{Vega06,Alonso07}, can be used to\ncalculate the two-time CF's for any factorized (separable) \nsystem-reservoir initial\nstate and for any arbitrary temperature as long as the approximation\nof the weak system-environment coupling still holds. \n\n\n\\subsection{Evolution equations in the weak system-environment coupling limit}\n\nLet us proceed to first derive perturbatively the explicit \nevolution equation of the\nsingle-time expectation values in the \nnon-Markovian case. \nHere we consider the second-order non-Markovian\ntime-convolutionless (time-local) evolution equation in our derivation. \nWe wish to\nobtain an evolution equation, $d\\rho_T(t)\/dt$, valid to second order\nin system-environment interaction Hamiltonian, to substitute into\nEq.~(\\ref{1time_evol}) for the single-time expectation values and into\nEq.~(\\ref{2time_evol}) for the two-time CF's. \nIt is convenient to first go to the interaction picture\nand obtain a time-local (time-convolutionless) evolution equation of\nthe density matrix valid to that order. This can be achieved by the\nsubstitution of $\\tilde{\\rho}_{T}(t')\\to \\tilde{\\rho}_{T}(t)$ in the\nsecond term on the right hand side of the equal sign of\nEq.~(\\ref{tilderho})\n\\cite{Breuer02,Paz01,Breuer99,Breuer01, Yan98,Schroder06,Ferraro09,Shibata77,Kleinekathofer04,Liu07,Sinayskiy09,Mogilevtsev09,Haikka10,Ali10,Chen11}.\nTo go back to the Schr\\\"{o}dinger picture, we substitute the resultant\nsecond-order equation obtained from Eq.~(\\ref{tilderho}) into\nEq.~(\\ref{rh0T_relation}) to obtain the evolution equation,\n$d\\rho_T(t)\/dt$. By substituting this equation $d\\rho_T(t)\/dt$ valid\nto second order in the interaction Hamiltonian into\nEq.~(\\ref{1time_evol}), the evolution equation of the single-time\nexpectation values then consists of three terms. The second term involves\nthe first term on the right hand side of the equal sign of\nEq.~(\\ref{tilderho}), and will vanish on the conditions that\n$\\rho_{T}(0)=\\tilde{\\rho}_{T}(0)=\\rho(0)\\otimes R_0$ and\n${\\rm Tr}_R[\\tilde{H}_{I}(t) R_0]=0$ [Eq.~(\\ref{traceless_1st_order})] \nare satisfied. \nAs a result, we obtain up to second order in the interaction Hamiltonian\n\\begin{eqnarray}\n\\frac{d\\left\\langle A\\left( t_{1}\\right) \\right\\rangle }{dt_{1}}\n&=&\\frac{i}{\\hbar }{\\rm Tr}_{S\\otimes R}\\left(\n[{H}_S , {A}] {\\rho}_{T}(t_1)\n\\right) \\nonumber \\\\\n&&+\\frac{1}{\\hbar^2 }\\int_0^{t_1}d\\tau {\\rm Tr}_{S\\otimes R} \\nonumber \\\\\n&&\\quad\n\\left(\\tilde{H}_I(\\tau-t_1)[A,H_I]\\rho_T(t_1) \\right.\n\\nonumber \\\\\n&&\\quad\\left.\n+[H_I,A]\\tilde{H}_I(\\tau-t_1)\\rho_T(t_1)\\right) \n\\label{1time_rhot}\n\\nonumber \\\\\n&=&({i}\/{\\hbar}){\\rm Tr}_{S\\otimes R}\\left( \\{[{H}_S, {A}] \\}(t_1) \n\\rho_{T}(0)\\right) \\nonumber \\\\\n&&+\\frac{1}{\\hbar^2 }\\int_0^{t_1}d\\tau {\\rm Tr}_{S\\otimes R} \\nonumber\\\\\n&&\\quad\n\\left( \\{\\tilde{H}_I(\\tau-t_1)[A,H_I]\\}(t_1)\\rho_T(0) \\right. \\nonumber\\\\\n&&\\quad\\left.\n+\\{[H_I,A]\\tilde{H}_I(\\tau-t_1)\\}(t_1)\\rho_T(0)\\right),\n\\label{1time_evol_eq}\n\\end{eqnarray}\nwhere we have transformed from the Schr\\\"{o}dinger picture to the\nHeisenberg picture in the second equal sign and \n$\\{AB\\}(t)\\equiv \\exp(iHt\/\\hbar) AB\\exp(-iHt\/\\hbar)$.\n\nSince $\\chi_T(t)$ and $\\rho_T(t)$ obey the same equations of \nEqs.~(\\ref{rh0T_relation}) and (\\ref{tilderho}),\nat first sight, one may think that the two-time evolution equations,\nEqs.~(\\ref{2time_evol}) and (\\ref{2time}), are similar to \nthe single-time evolution equations, Eqs.~(\\ref{1time_evol}) and\n(\\ref{1time}), and thus might be tempted to conclude that they have the\nsame form of the evolution equations. \nIndeed, by using Eqs.~(\\ref{2time_evol}), (\\ref{rh0T_relation}) \nand (\\ref{tilderho}), \nthe first and third terms of the resultant equation derived from\nEq.~(\\ref{2time_evol}) are similar to the right-hand side of the single-time \nevolution equation (\\ref{1time_evol_eq}) with the\nreplacement of \n$\\rho_T(0)\\to\\chi_T(-t_2)=B(t_2)\\rho_{T}(0)$ \nand with the change of the integration region from $[0,t_1]$ to\n$[t_2,t_1]$. Then we obtain \n\\begin{eqnarray}\n&&\\frac{i}{\\hbar }{\\rm Tr}_{S\\otimes R}\\left( \\{[{H}_S, {A}] \\}(t_1) \nB(t_2)\\rho_{T}(0)\\right) \\nonumber \\\\\n&&+\\frac{1}{\\hbar^2 }\\int_{t_2}^{t_1}d\\tau {\\rm Tr}_{S\\otimes R} \\nonumber\\\\\n&&\\quad\n\\left( \\{\\tilde{H}_I(\\tau-t_1)[A,H_I]\\}(t_1)B(t_2)\\rho_{T}(0) \\right. \\nonumber\\\\\n&&\\quad\\left.\n+\\{[H_I,A]\\tilde{H}_I(\\tau-t_1)\\}(t_1)B(t_2)\\rho_{T}(0)\\right).\n\\label{2time_evol_1_3}\n\\end{eqnarray}\n\nHowever, a significant difference is that the expectation values for the second term does not vanish, i.e., \n\\begin{equation}\n(-i\/\\hbar){\\rm Tr}_{S\\otimes R}\\left(Ae^{-iH_{0}t\/\\hbar}\\left[\\tilde{H}_{I}(t) ,\\tilde{\\chi}_{T}( 0)\\right]e^{iH_{0}t\/\\hbar}\\right)\\neq 0,\n\\label{non-Markovian_1st_order}\n\\end{equation} \nin the non-Markovian case, where $t=t_1-t_2$ \nin Eq.~(\\ref{non-Markovian_1st_order}). \nThe reason can be understood as follows.\nThe interaction Hamiltonian $\\tilde{H}_{I}(t_1-t_2)$ in\nEq.~(\\ref{non-Markovian_1st_order}) involves the\nenvironment operators in the time interval from $t_2$ to $t_1$, and \nthe effective density matrix operator $\\tilde{\\chi}_T(0)$ can be written as \n$\\tilde{\\chi}_T(0)=\\chi_T(0)=B\\rho_T(t_2)=BU(t_2,0)\\rho_T(0)U^{\\dagger}(t_2,0)$,\nwhere $U(t_2,0)=e^{-iHt_2\/\\hbar}$ is the Heisenberg evolution operator of the\ntotal Hamiltonian from time $0$ to $t_2$. \nIf the environment\nis Markovian where the environment operator CF's at two\ndifferent times are $\\delta$-correlated in time, then we may regard\nthat the environment operators in $\\tilde{H}_{I}(t_1-t_2)$ are not\ncorrelated with those in $U(t_2,0)$. So the trace over the environment degrees\nof freedom for operator $\\tilde{H}_{I}(t_1-t_2)$ and operator $U(t_2,0)$ can be\nperformed independently or separately. The trace of \n$\\rho_T(t_2)=U(t_2,0)\\rho_T(0)U^{\\dagger}(t_2,0)$ over the environment\ndegrees of freedom\nyields the reduced density matrix \n$\\rho(t_2)={\\rm Tr}_{R}[\\rho_T(t_2)]$, but the trace of\n$\\tilde{H}_{I}(t_1-t_2)$ vanishes, \ni.e., ${\\rm Tr}_{R}[\\tilde{H}_{I}(t_1-t_2)R_0]=0$, \nbecause of Eq.~(\\ref{traceless_1st_order}). \nThus Eq.~(\\ref{non-Markovian_1st_order}) vanishes in the Markovian limit.\nBut the situation differs for a\nnon-Markovian environment as the environment operator in\n $\\tilde{H}_{I}(t_1-t_2)$ may, in general, be correlated with that in \n$U(t_2,0)$.\nTherefore, the evolution from $\\rho_T(0)$ to $\\rho_T(t_2)$ \nunder the influence of interaction Hamiltonian \nin the presence of the reservoir needs to be taken into account \nbefore the trace over the environment is performed in \nEq.~(\\ref{non-Markovian_1st_order}). \nWe emphasize here that it is this nonlocal environment (bath) memory term,\nEq.~(\\ref{non-Markovian_1st_order}), that vanishes in the Markovian\ncase but makes the evolution equation of the two-time \nCF's of the system operators deviate from the QRT. \nAs we aim to obtain an evolution equation of the two-time CF's of the\nsystem operators, valid up to second order in the interaction\nHamiltonian, we need to find $\\rho_T(t_2)$ only up to \nfirst order in the interaction Hamiltonian. \nSo substituting \n$\\rho_{T}(t_2)= e^{-iH_{0}t_2\/\\hbar }\\tilde{\\rho}_{T}(t_2)e^{iH_{0}t_2\/\\hbar }$\nwith the expression\n\\begin{equation}\n\\tilde{\\rho}_{T}\\left( t_2\\right) \\approx \\tilde{\\rho}_{T}\\left( 0\\right) \n-\\frac{i}{\\hbar }\\int_{0}^{t_2}d\\tau\\left[ \\tilde{H}_{I}\\left(\n\\tau\\right) ,\\tilde{\\rho}_{T}\\left( t_2\\right) \\right] \n\\label{rhoT_t2}\n\\end{equation\nfor $\\tilde{\\chi}_T(0)=\\chi_T(0)=B\\rho_T(t_2)$ \nin Eq.~(\\ref{non-Markovian_1st_order}),\nwe then obtain up to second order in the interaction Hamiltonian (in\nthe system-environment coupling strength) \n\\begin{eqnarray}\n&&-\\frac{1}{\\hbar^2}\\int_0^{t_2}d\\tau\\, {\\rm Tr}_{S\\otimes R}\\left(Ae^{-iH_{0}t\/\\hbar}\\, \n\\left[ \\tilde{H}_{I}\\left(\nt\\right), \\right. \\right. \\nonumber\\\\\n&&\\quad \\quad\\left. \\left. B\\,e^{-iH_{0}t_2\/\\hbar} [\\tilde{H}_{I}(\\tau),\\tilde{\\rho}_{T}(t_2)]e^{iH_{0}t_2\/\\hbar} \\right]\\, e^{iH_{0}t\/\\hbar }\\right),\n\\nonumber \\\\\n&=& -\\frac{1}{\\hbar^2}\\int_0^{t_2}d\\tau\\, \n{\\rm Tr}_{S\\otimes R}\\left( A\n\\left[ H_{I},\\right. \\right. \\nonumber\\\\\n&&\\quad \\quad\\left. \\left. e^{-iH_{0}t\/\\hbar}\\,\nB[\\tilde{H}_{I}(\\tau-t_2),\\rho_{T}(t_2)] \\, e^{iH_{0}t\/\\hbar }\\right]\\right),\n\\label{extra1}\n\\end{eqnarray}\nwhere the first order term in the interaction Hamiltonian coming from $\\tilde{\\rho}_T(0)$ term in Eq.~(\\ref{rhoT_t2}) \nhas been dropped because of Eq.~(\\ref{traceless_1st_order}). \nSince the product of two $H_I$ already appear explicitly in Eq.~(\\ref{extra1}),\nwe may then transform Eq.~(\\ref{extra1}) into Heisenberg\nrepresentation with the evolution equation $\\exp(iHt\/\\hbar)\\approx\n\\exp(iH_0t\/\\hbar)+{\\cal O}(H_I)$. \nFurthermore, by writing out the commutators explicitly and rearranging the\nHeisenberg operator terms, \nthe resultant equation from Eq.~(\\ref{extra1}) then becomes\n\\begin{eqnarray}\n&& -\\frac{1}{\\hbar^2}\\int_0^{t_2}d\\tau\\, \n{\\rm Tr}_{S\\otimes R}\\left( \n\\{\\tilde{H}_I(\\tau-t_1)[H_I,A]\\}(t_1)B(t_2)\\rho_{T}(0) \\right. \\nonumber\\\\\n&&\\quad\\quad\\quad\n\\left. +\\{[A,H_I]\\}(t_1)\\{B\\tilde{H}_I(\\tau-t_2)\\}(t_2)\\rho_T(0) \\right).\n\\label{extra2}\n\\end{eqnarray}\nThe first term in Eq.~(\\ref{extra2}) is ready to combine with the second term in Eq.~(\\ref{2time_evol_1_3}) to extend the integration from $0$ to $t_1$. \nSimilarly, one may rewrite the last term in Eq.(\\ref{extra2}) using the relation $B\\tilde{H}_I(\\tau-t_2)=\\tilde{H}_I(\\tau-t_2)B+[B,\\tilde{H}_I(\\tau-t_2)]$ so that the first new term can be combined with last term in Eq.~(\\ref{2time_evol_1_3}) to extend the integration from $0$ to $t_1$.\n \nPutting all the resultant terms together, we obtain the evolution equation of the two-time CF's valid to second order in the interaction Hamiltonian as\n\\begin{eqnarray}\n&&{d\\left\\langle A(t_{1}) B(t_2)\\right\\rangle }\/{dt_{1}} \\nonumber \\\\\n&=&({i}\/{\\hbar}){\\rm Tr}_{S\\otimes R}\\left( \\{[{H}_S, {A}] \\}(t_1) \nB(t_2)\\rho_{T}(0)\\right) \\nonumber \\\\\n&&+\\frac{1}{\\hbar^2 }\\int_{0}^{t_1}d\\tau {\\rm Tr}_{S\\otimes R} \\nonumber\\\\\n&&\\quad\n\\left( \\{\\tilde{H}_I(\\tau-t_1)[A,H_I]\\}(t_1)B(t_2)\\rho_{T}(0) \\right. \\nonumber\\\\\n&&\\quad\n+\\left.\\{[H_I,A]\\tilde{H}_I(\\tau-t_1)\\}(t_1)B(t_2)\\rho_{T}(0) \\right)\n\\nonumber\\\\\n&&\n+\\frac{1}{\\hbar^2 }\\int_{0}^{t_2}d\\tau {\\rm Tr}_{S\\otimes R} \\nonumber\\\\\n&&\\quad \\left(\\{[H_I,A]\\}(t_1)\\{[B,\\tilde{H}_I(\\tau-t_2)]\\}(t_2)\\rho_T(0)\\right).\n\\label{2time_evol_eq}\n\\end{eqnarray}\nCompared to Eq.~(\\ref{1time_evol_eq}), it is the existence of the last\nterm in Eq.~(\\ref{2time_evol_eq}) that makes the QRT invalid. \nEquation (\\ref{2time_evol_eq}) is the main result of this paper. \nThe derivation is based on perturbative quantum master\nequation approach, so non-Markovian open quantum system models that are\nnot exactly solvable can use our derived evolution equation to \nobtain the time evolutions of \ntheir two-time CF's of system operators, valid to second order in\nthe system-environment interaction. \nIn the derivation of Eqs.~(\\ref{1time_evol_eq}) and (\\ref{2time_evol_eq}),\nwe have also used the assumption of a factorized initial system-bath state\n$\\rho_{T}(0)=\\tilde{\\rho}_{T}(0)=\\rho(0)\\otimes R_0$ and the\ncondition of \n${\\rm Tr}_R[\\tilde{H}_{I}(t) R_0]=0$, Eq.~(\\ref{traceless_1st_order}),\nto eliminate the first-order term.\nSince the form and nature of the Hamiltonians are not specified,\nEq.~(\\ref{2time_evol_eq}) can be used to calculate the two-time CF's \nfor non-Markovian open quantum systems with \nmulti-level discrete or continuous Hilbert\nspaces, interacting with bosonic and\/or\nfermionic environments. \nThe procedure and the degrees of difficulty \nto apply Eq.~(\\ref{2time_evol_eq}) to a open quantum system model\n(by taking into account \nnonlocal bath memory effects and \ntracing out the bath degrees of freedom for \nfactorized system-bath initial states) to obtain the two-time\nCF's of system operator \nare similar to those for the evaluation of the reduced density matrix\nof a second-order time-convolutionless non-Markovian \nquantum master equation [e.g., Eq.~(\\ref{time_convolutionless_ME})].\nWe will explicitly apply the evolution equation (\\ref{2time_evol_eq})\nto a general model of a quantum system coupled to a\nfinite-temperature bosonic environment in Sec.~\\ref{sec:bosonic}\nand a specific model of two-level system in Sec.~\\ref{sec:spin-boson}.\nOpen quantum systems coupled to fermionic reservoirs\n(environments) could, for example,\nbe quantum\ndots or other nanostructure systems coupled (connected) to\nnonequilibrium electron reservoirs (electrodes or leads) in the\nelectron transport problems \\cite{Kleinekathofer04,Welack06,Harbola06,\n Goan01, Li04, Utami04, Zedler09,Gudmundsson09,Jin10}. \nThe evolution equation\n(\\ref{2time_evol_eq}) can also be used to calculate the non-Markovian \ntwo-time CF's in such systems.\nIn summary, our evolution equation (\\ref{2time_evol_eq}) can be\napplied to a wide range \nof system-environment models with \nany factorized (separable) system-environment initial states (pure or mixed). \n \n\n\n\n\\section{Evolution equations for thermal bosonic bath models}\n\\label{sec:bosonic}\nTo make contact with Refs.~\\cite{Alonso05,Vega06,Alonso07}, we \nconsider a quantum system coupled to a\nbosonic environment with a general Hamiltonian of the form\n\\begin{eqnarray}\nH&=&H_{S}+\\sum_{\\lambda }\\hbar g_{\\lambda }\\left( L^{\\dagger} a_{\\lambda\n}+La_{\\lambda }^{\\dagger}\\right) +\\sum_{\\lambda }\\hbar\\omega_{\\lambda\n}a_{\\lambda }^{\\dagger}a_{\\lambda }, \n\\label{Hamiltonian_L}\n\\end{eqnarray}\nwhere the system coupling operator $L$ acts on the Hilbert space of\nthe system, \n$a_{\\lambda }$ and $a_{\\lambda }^{\\dagger}$ are the annihilation and creation\noperators on the bosonic environment Hilbert space, and $g_{\\lambda }$\nand $\\omega _{\\lambda }$ are respectively the coupling strength and the\nfrequency of the $\\lambda$th environmental oscillator.\n\n\nApplying Eq.~(\\ref{Hamiltonian_L}) to Eqs.~(\\ref{1time_evol_eq}) and\n(\\ref{2time_evol_eq}) and after tracing over the environmental degrees\nof freedom for factorized (separable) system-bath initial states, \nwe arrive at the second-order evolution equations of the single-time expectation values\n\\begin{eqnarray}\n&&\n{d\\left\\langle A\\left( t_{1}\\right) \\right\\rangle }\/{dt_{1}} \n\\nonumber \\\\\n&=&\n({i}\/{\\hbar }){\\rm Tr}_{S}\\left( \\{[H_{S},A]\\}(t_1) \\rho(0) \\right)\n\\nonumber \\\\\n&&+\n\\int_{0}^{t_{1}}d\\tau {\\rm Tr}_S\n\\nonumber \\\\\n&&\\quad \\left( \\alpha^{\\ast }( t_{1}-\\tau)\n\\{\\tilde{L}^{\\dagger}(\\tau -t_{1})[{A},{L}]\\}( t_{1}){\\rho}(0) \\right.\n\\nonumber \\\\\n&&\\quad\n+\\alpha( t_{1}-\\tau )\\{ [ {L}^{\\dagger},A] \\tilde{L}(\\tau -t_{1})\\}(t_{1}) \n\\rho( 0) \n \\nonumber \\\\\n&&\\quad+\n\\beta^{\\ast }(t_{1}-\\tau)\\{\\tilde{L}(\\tau -t_{1})[A,L^{\\dagger}]\\}(t_{1})\n\\rho(0) \n\\nonumber \\\\\n&&\n\\quad+\n\\left.\n\\beta(t_{1}-\\tau)\\{[L,A] \\tilde{L}^{\\dagger}(\\tau -t_{1})\\}(t_{1}) \n\\rho(0) \\right), \n\\label{1time_evol_eq_f}\n\\end{eqnarray}\nand of the two-time CF's\n\\begin{eqnarray}\n&&{d\\left\\langle A\\left( t_{1}\\right) B\\left( t_{2}\\right)\n\\right\\rangle }\/{dt_{1}} \\nonumber \\\\ \n&=&\n({i}\/{\\hbar}){\\rm Tr}_{S}\\left( \\{[{H}_{S}, A]\\}( t_{1}){B}( t_{2}) \n\\rho(0) \\right) \\nonumber \\\\\n&&+\n\\int_{0}^{t_{1}}d\\tau {\\rm Tr}_S\n\\nonumber \\\\\n&&\\quad\\left( \\alpha^{\\ast }( t_{1}-\\tau)\n\\{\\tilde{L}^{\\dagger}(\\tau -t_{1})[{A},{L}]\\}( t_{1}) \n{B}(t_{2}){\\rho}(0) \\right.\n\\nonumber \\\\\n&&\\quad+\n\\alpha( t_{1}-\\tau )\\{ [ {L}^{\\dagger},A] \\tilde{L}(\\tau -t_{1})\\} ( t_{1}) B( t_{2}) \n\\rho( 0) \n\\nonumber \\\\\n&&\\quad+\n\\beta^{\\ast }(t_{1}-\\tau)\\{\\tilde{L}(\\tau -t_{1})[A,L^{\\dagger}]\\}(t_{1}) B(t_{2}) \n\\rho(0) \n\\nonumber \\\\\n&&\\quad+\n\\left. \n\\beta(t_{1}-\\tau)\\{[L,A] \\tilde{L}^{\\dagger}(\\tau -t_{1})\\}(t_{1}) \nB(t_{2})\\rho(0) \n\\right) \n\\nonumber \\\\\n&&+\n\\int_{0}^{t_{2}}d\\tau \n{\\rm Tr}_S \\nonumber \\\\\n&&\\quad\\left( \\alpha(t_{1}-\\tau) \\{ [ L^{\\dagger},A]\\}(t_{1}) \n\\{ [B,\\tilde{L}(\\tau -t_{2})]\\}(t_{2}) \\rho(0) \\right.\n\\nonumber \\\\\n&&\\hspace{-0.3cm}+\n\\left. \\beta( t_{1}-\\tau)\n\\{ [L,A]\\}(t_{1})\\{[B,\\tilde{L}^{\\dagger}(\\tau -t_{2})]\\} (t_{2}) \n\\rho(0) \\right). \n\\label{2time_evol_eq_f}\n\\end{eqnarray}\nHere $\\tilde{L}(t)=\\exp(iH_St\/\\hbar)L\\exp(-iH_St\/\\hbar)$ is the system operator in the interaction picture with respect to $H_S$, and \n\\begin{eqnarray}\n\\alpha \\left( t_{1}-\\tau \\right) &=&\\sum_{\\lambda }\\left( \\bar{n}_{\\lambda }+1\\right)\n\\left\\vert g_{\\lambda }\\right\\vert ^{2}\ne^{-i\\omega _{\\lambda }\\left( t_{1}-\\tau \\right)},\n\\label{CFalpha}\\\\ \n\\beta \\left( t_{1}-\\tau \\right)\n&=&\\sum_{\\lambda }\\bar{n}_{\\lambda }\\left\\vert g_{\\lambda }\\right\\vert ^{2}e^{i\\omega\n_{\\lambda }\\left( t_{1}-\\tau \\right) }\n\\label{CFbeta}\n\\end{eqnarray}\nare the environment CF's with $\\alpha(t_1-\\tau)=\\left\\langle\n \\sum_{\\lambda}g_{\\lambda}\\tilde{a}_{\\lambda}(t_1)\\sum_{\\lambda'}g_{\\lambda'}\\tilde{a}^{\\dagger}_{\\lambda'}(\\tau)\\right\\rangle_R$\nand $\\beta(t_1-\\tau)=\\left\\langle\n \\sum_{\\lambda}g_{\\lambda}\\tilde{a}^{\\dagger}_{\\lambda}(t_1)\\sum_{\\lambda'}g_{\\lambda'}\\tilde{a}_{\\lambda'}(\\tau)\\right\\rangle_R$,\nwhere $\\tilde{a}_{\\lambda}(t_1)=a_{\\lambda}e^{-i\\omega_\\lambda t_1}$\nand\n$\\tilde{a}^{\\dagger}_{\\lambda}(t_1)=a^{\\dagger}_{\\lambda}e^{i\\omega_\\lambda\n t_1}$ are the environment operators in the interaction\npicture, and the symbol $\\langle \\cdots \\rangle_R$ denotes taking a trace with\nrespect to the density matrix of \nthe thermal bosonic reservoir (environment). \nThe thermal mean occupation number $\\bar{n}_\\lambda$ of the\nbosonic environment\noscillators in Eqs.~(\\ref{CFalpha}) and (\\ref{CFbeta}) is\n$\\bar{n}_\\lambda=(e^{\\hbar\\omega_\\lambda\/k_BT}-1)^{-1}$. \n\nThe evolution equations (\\ref{1time_evol_eq_f}) and (\\ref{2time_evol_eq_f})\nfor a non-Markovian bosonic environment have been presented in\nRef.~\\cite{Goan10} without any derivation. In this paper, the detailed\nderivation of the evolution equations is given.\nFurthermore, the two-time CF evolution equation, Eq.~(\\ref{2time_evol_eq}),\napplicable for both bosonic and fermionic environments and applicable\nfor more general form of system-environment interaction Hamiltonian \nhas not been published in the literature yet. \n\n\n\n\nAs mentioned, the two-time evolution equations derived \nin Refs.~\\cite{Alonso05,Vega06,Alonso07} is, strictly speaking,\napplicable only for a\nzero-temperature environment. However, these equations\nwere used to calculate the\ntwo-time CF's of system observables of dissipative spin-boson models\nat finite temperatures. \nThis is possible only for the dissipative spin-boson models \nwith Hermitian system coupling\noperators, $L=L^\\dagger$. We will discuss this point in details\nin subsection \\ref{sec:comparison}.\nIn contrast, our bosonic evolution equations,\nEqs.~(\\ref{1time_evol_eq_f}) and (\\ref{2time_evol_eq_f}), are valid\nfor both a Hermitian and a non-Hermitian system coupling operators and\ncan be used to calculate the two-time CF's for any factorized (separable)\nsystem-reservoir initial state and for any arbitrary temperature as\nlong as the assumption of the weak system-environment coupling still\nholds. \n\nIn Ref.~\\cite{Goan10}, we used Eqs.~(\\ref{1time_evol_eq_f}) and \n(\\ref{2time_evol_eq_f}) to calculate\nthe finite-temperature \nsingle-time expectation values and two-time CF's for a non-Markovian\npure-dephasing spin-boson model of \n\\begin{equation}\nH_S=(\\hbar\\omega_S\/2)\\sigma_z, \\quad\\quad\nL=\\sigma_z=L^{\\dagger}. \n\\label{SpinBoson} \n\\end{equation}\nSince the non-Markovian dynamics of this exactly solvable\npure-dephasing model can be cast into a time-local,\nconvolutionless form and $[L,H_S]=0$, the results obtained by\nour second-order evolution equations turn out to be\nexactly the same as the exact results obtained by the direct\noperator evaluation. \nHowever, these results significantly differ from the non-Markovian\ntwo-time CF's obtained by wrongly directly applying the quantum\nregression theorem (QRT). \nThis demonstrates the validity of\nthe evolution equations (\\ref{1time_evol_eq_f}) and \n(\\ref{2time_evol_eq_f}).\nBut the system coupling operators $L$ of this pure dephasing model\n\\cite{Goan10} and the examples calculated in\nRefs.~\\cite{Alonso05,Vega06,Alonso07} are all Hermitian, i.e., $L^{\\dagger}=L$. \nSo we will present in Sec.~\\ref{sec:spin-boson} the calculations of\none-time averages and two-time CF's for a thermal spin-boson model\nwith $L\\neq L^\\dagger$, for which only the evolution equations\n(\\ref{1time_evol_eq_f}) and (\\ref{2time_evol_eq_f}), rather than those\nin Refs.~\\cite{Alonso05,Vega06,Alonso07}, are applicable.\n\n\n\n\\subsection{Comparison and discussion} \n\\label{sec:comparison}\nWe discuss in the following the connection of our derived two-time \nevolution equation (\\ref{2time_evol_eq_f}) with those presented in \nRefs.~\\cite{Alonso05,Vega06,Alonso07}.\nIn Ref.~\\cite{Vega06}, a master equation conditioned on initial coherent states of the environment in Bargmann representation, $(z_0,z'_0)$, was derived in the weak system-environment coupling limit. Provided that the whole set of the initial conditions of the system of interest, $|\\psi(z^*_0)\\rangle$, and the statistical probability ${\\cal J}(z_0,z^*_0)$ for the member $|\\psi(z^*_0)\\rangle|z_0\\rangle$ of the statistical ensemble are known, this master equation \nwith $z'_0=z_0$ is capable of evaluating the evolution of \n{\\em single-time} expectation values for general initial conditions, including initially correlated mixed states between the system and environment. \nHowever, the evolution equations of {\\em two-time (multi-time)} CF's of system observables, Eq.~(6) in Ref.~\\cite{Alonso05}, Eq.~(31) in Ref.~\\cite{Vega06} and Eq.~(60) in Ref.~\\cite{Alonso07}, were derived for an initial vacuum state of the environment and an initial pure state of the system of interest. \nAs a result, these two-time evolution equations are valid only for a zero-temperature environment \n(if the system coupling operator is not Hermitian, i.e., $L\\neq L^\\dagger$; see discussions below). \nCompared with the corresponding zero-temperature two-time (multi-time) evolution equations \nderived in Refs.~\\cite{Alonso05,Vega06,Alonso07}, our finite-temperature \ntwo-time evolution equation \n(\\ref{2time_evol_eq_f}) is valid for any initial factorized (separable) states (pure or mixed) at finite temperatures and for both Hermitian and non-Hermitian system coupling operators. The extra terms containing the bath CF $\\beta(t_1-\\tau)$ or $\\beta^*(t_1-\\tau)$ are due to the finite-temperature thermal environment. If we take the zero-temperature limit by letting $\\bar{n}_\\lambda=0$ and thus $\\beta(t_1-\\tau)=\\beta^*(t_1-\\tau)=0$, as well as consider the initial pure-state case by letting ${\\rm Tr}_S[\\cdots\\rho(0)]\\to \\langle \\psi(0)|\\cdots |\\psi(0)\\rangle$, then Eqs.(\\ref{1time_evol_eq_f}) and (\\ref{2time_evol_eq_f}) reduce exactly to their corresponding zero-temperature pure-state evolution equations in Refs.~\\cite{Alonso05,Vega06,Alonso07}.\n\n\nHowever, calculations of the two-time CF's of system observables of\ndissipative spin-boson models in finite-temperature thermal baths (rather than zero-temperature baths) were presented in Refs.~\\cite{Alonso05,Vega06,Alonso07} \neven though in their derivations of the two-time (multi-time) evolution equations, the bath CF is given in its zero-temperature form, \n\\begin{equation}\n\\alpha_0(t-\\tau)=\\sum_{\\lambda }\n\\left\\vert g_{\\lambda }\\right\\vert ^{2}\ne^{-i\\omega _{\\lambda }\\left( t_{1}-\\tau \\right)}. \n\\label{CFalpha0}\n\\end{equation}\nThis is only possible due to the reason that the system coupling operator is Hermitian, $L=L^\\dagger$, in the thermal bath examples presented in Refs.~\\cite{Alonso05,Vega06,Alonso07}. \nOne may understand this as follows. \nIt is known that the finite-temperature density matrix operator of a thermal bath can be canonically mapped onto the zero-temperature density operator (the vacuum) of a larger (hypothetical) environment \\cite{Semenoff83,Yu04}. \nBy mapping the total Hamiltonian Eq.~(\\ref{Hamiltonian_L}) and an\ninitial thermal state to an extended total system with a vacuum state,\nthe finite-temperature problem can be reduced to a zero-temperature\nproblem, and the resultant pure state $\\psi_t=|\\psi_t(z^*,w^*)\\rangle$\nfor the system of interest satisfies the following linear\nfinite-temperature non-Markovian stochastic Schr\\\"odinger equation\nwith two independent noises $z^*_t$ and $w^*_t$ \\cite{Diosi98,Yu04}:\n\\begin{eqnarray} \n\\frac{\\partial}{\\partial t}\\psi_t&=&-iH_S\\psi_t+L z^*_t\\psi_t-L^\\dagger\\int_0^t d\\tau \\alpha(t-\\tau)\\frac{\\delta\\psi_t}{\\delta z^*_\\tau} \\nonumber \\\\\n&&+L^\\dagger w^*_t\\psi_t-L\\int_0^t d\\tau \\beta(t-\\tau)\\frac{\\delta\\psi_t}{\\delta w^*_\\tau}, \n\\label{SSEfiniteT}\n\\end{eqnarray}\nwhere \n\\begin{eqnarray} \nz^*_t&=&-i\\sum_\\lambda \\sqrt{\\bar{n}_{\\lambda }+1}\\, g^*_\\lambda z^*_\\lambda e^{i\\omega_\\lambda t},\n\\label{zt}\\\\\nw^*_t&=&-i\\sum_\\lambda \\sqrt{\\bar{n}_{\\lambda }}\\, g^*_\\lambda w^*_\\lambda e^{-i\\omega_\\lambda t}\n\\label{wt}\n\\end{eqnarray}\nare two independent, colored, complex Gaussian noises with zero mean \nand satisfy \n\\begin{eqnarray}\n{\\cal M}[z_tz_\\tau]={\\cal M}[z^*_tz^*_\\tau]=0,&&\\quad {\\cal M}[z^*_tz_\\tau]=\\alpha(t-\\tau);\n\\\\\n{\\cal M}[w_tw_\\tau]={\\cal M}[w^*_tw^*_\\tau]=0,&&\\quad {\\cal M}[w^*_tw_\\tau]=\\beta(t-\\tau).\n\\end{eqnarray}\nHere, $z^*_\\lambda$ and $w^*_\\lambda$ are coherent state variables of the extended environment in Bargmann representation, ${\\cal M}[\\cdots]$ denotes the statistical average over the Gaussian processes $z^*_t$ and $w^*_t$, and the bath CF's $\\alpha(t-\\tau)$ and $\\beta(t-\\tau)$ are defined in Eqs.~(\\ref{CFalpha}) and (\\ref{CFbeta}), respectively.\nIn the zero-temperature limit $T\\to 0$, the mean thermal occupation number of quanta in mode $\\lambda$ approaches zero, i.e., $\\bar{n}_{\\lambda } \\to 0$. \nand thus $\\beta(t-\\tau)\\to 0$ and $\\alpha(t-\\tau)\\to \\alpha_0(t-\\tau)$ \nthat is defined in Eq.~(\\ref{CFalpha0}). \nIn this case, the noises, Eqs.~(\\ref{zt}) and (\\ref{wt}), become\n $z^*_t=-i\\sum_\\lambda g^*_\\lambda z^*_\\lambda e^{i\\omega_\\lambda t}$\n and $w^*_t=0$, and \nthe finite-temperature equation (\\ref{SSEfiniteT}) reduces to the\nsimple zero-temperature equation \\cite{Diosi98,Yu04} \n\\begin{equation}\n\\frac{\\partial}{\\partial t}\\psi_t=-iH_S\\psi_t+L z^*_t\\psi_t-L^\\dagger\\int_0^t d\\tau \\alpha_0(t-\\tau)\\frac{\\delta\\psi_t}{\\delta z^*_\\tau}.\n\\label{SSEzeroT}\n\\end{equation} \n\n\nNow consider the case of a Hermitian system coupling operator $L=L^\\dagger$. \nThe finite-temperature equation (\\ref{SSEfiniteT}) can be simplified considerably by introducing the sum process $u^*_t=z^*_t+w^*_t$ that has a zero mean and satisfies\n\\begin{eqnarray}\n{\\cal M}[u_tu_\\tau]&=&{\\cal M}[u^*_tu^*_\\tau]=0;\\\\\n{\\cal M}[u^*_tu_\\tau]&=&\\alpha_{\\rm eff}(t-\\tau) \\nonumber\\\\\n&=&\\alpha(t-\\tau)+\\beta(t-\\tau) \\nonumber \\\\\n&=& \\sum_\\lambda |g_\\lambda|^2\n\\left\\{\\coth\\left({\\hbar\\omega_\\lambda}\/{2k_{B}T}\\right)\\cos[\\omega_\\lambda(t-\\tau)]\\right.\n\\nonumber\\\\\n&& \\hspace{1.5cm}\\left.-i\\sin[\\omega_\\lambda(t-\\tau)]\\right\\}. \n\\label{CFalpha_eff} \n\\end{eqnarray}\nIn terms of the single noise process $u^*_t$, the linear finite-temperature non-Markovian stochastic Schr\\\"odinger equation (\\ref{SSEfiniteT}) for the case of a Hermitian coupling operator $L=L^\\dagger$ takes the simple form of the zero-temperature equation (\\ref{SSEzeroT}) with the replacement of $z^*_t$ by $u^*_t$ and \n$\\alpha_0(t-\\tau)$ by $\\alpha_{\\rm eff}(t-\\tau)=\\alpha(t-\\tau)+\\beta(t-\\tau)$ \ndefined in Eq.~(\\ref{CFalpha_eff}). \nIt is for this reason of the Hermitian coupling operator\n$L=L^\\dagger=\\sigma_x$ in the dissipative spin-boson model with a\nthermal environment that the two-time CF's of the system observables\ncan be evaluated with the evolution equations derived in\nRefs.~\\cite{Alonso05,Vega06,Alonso07}. \nIn other words, if the system operator coupled to the environment is\nnot Hermitian $L\\neq L^\\dagger$, \nthe two-time (multi-time) differential evolution equations presented in \nRefs.~\\cite{Alonso05,Vega06,Alonso07} are valid only for a zero-temperature\nenvironment. \n\n\n\nIn contrast, our two-time evolution equation (\\ref{2time_evol_eq_f}) is valid for more general finite-temperature cases where the system coupling operator is not a Hermitian operator, i.e., $L\\neq L^\\dagger$. \nIn the case of $L\\neq L^\\dagger$, \nour two-time evolution equation contains additional finite-temperature $\\beta(t_1-\\tau)$ and $\\beta^*(t_1-\\tau)$ terms which can not be combined and simplified to a simpler form as the zero-temperature evolution equation derived in Refs.~\\cite{Alonso05,Vega06,Alonso07}.\nFor a Hermitian coupling operator $L=L^{\\dagger}$, one can see that besides the replacement of a more general system state with an initial pure system state by letting $\\langle \\psi(0)|\\cdots |\\psi(0)\\rangle \\to {\\rm Tr}_S[\\cdots\\rho(0)]$, the finite-temperature evolution equation (\\ref{2time_evol_eq_f}) reduces to its zero-temperature counterpart in Refs.~\\cite{Alonso05,Vega06,Alonso07} with the effective bath CF given by \n$\\alpha_{\\rm eff}=\\alpha(t_1-\\tau)+\\beta(t_1-\\tau)$ defined in Eq.~(\\ref{CFalpha_eff}). This demonstrates explicitly why the zero-temperature \ntwo-time evolution equations derived in Refs.~\\cite{Alonso05,Vega06,Alonso07} can be used to calculate the system operator CF's for a thermal spin-boson model with a Hermitian system coupling operator. \n\n\n\\subsection{Conditions for the QRT to hold} \n\\label{sec:QRT_hold}\n\nAs mentioned in subsection \\ref{sec:QRT},\nQRT is a very useful procedure that enables one\n(in certain circumstances) to calculate for two-time\n(multi-time) CF's of system operators from the knowledge of the\nevolution equations of \nsingle-time expectation values. \nOne can notice that if the last two terms in\nEq.~(\\ref{2time_evol_eq_f}) vanish, then the non-Markovian\nsingle-time and two-time evolution equations (\\ref{1time_evol_eq_f}) and (\\ref{2time_evol_eq_f}) will have the same form with the same evolution coefficients and thus the QRT can apply. As expected, these two terms vanish in the Markovian case since the time integration of the corresponding $\\delta$-correlated reservoir CF's, \n$\\alpha(t_1-\\tau)\\propto \\delta(t_1-\\tau)$ and \n$\\beta(t_1-\\tau)\\propto \\delta(t_1-\\tau)$, over the variable $\\tau$ in the domain from $0$ to $t_2$ is zero as $t_1>t_2$.\nSo from Eq.~(\\ref{2time_evol_eq_f}), in the weak system-environment\ncoupling case the QRT holds when (i) $[ L^{\\dagger},A]=0$ or\n$[B,\\tilde{L}(\\tau -t_{2})]=0$ at the zero temperature, (ii) at finite\ntemperatures, in addition to condition (i), the following condition\nalso needs to be satisfied: $[L,A]=0$ or $[B,\\tilde{L}^{\\dagger}(\\tau -t_{2})]=0$, (iii) in the Markovian case where the bath CF's are \n$\\delta$-correlated in time. \n\nNote that in some models, certain CF's, which formally obey the QRT but with evolution equations coupled with those of other CF's that do not obey the QRT,\nmay yield solutions different from those given by the QRT \\cite{Vega06}.\n\n\n\n\\section{Application to a thermal spin boson model with $L\\neq L^\\dagger$}\n\\label{sec:spin-boson}\nTo illustrate the usage of the equations we have derived, we apply them to the problem of a two-level system coupled to a thermal reservoir, in which $L\\neq L^{\\dagger}$. \nWe consider Hamiltonian, Eq.~(\\ref{Hamiltonian_L}), with $H_S=(\\hbar\\omega_A\/2)\\sigma_z$, a coupling operator $L=\\sigma_-$ and a system-environment interaction Hamiltonian whose magnitude is small enough to be considered as a perturbation. Since the coupling operator $L\\neq L^{\\dagger}$ is not Hermitian, \nthe two-time (multi-time) evolution equations derived in Refs.~\\cite{Alonso05,Vega06,Alonso07} are not applicable and \nour evolution equation (\\ref{2time_evol_eq_f}) should be employed to obtain the time evolutions of the two-time CF's. \n\n\\subsection{Single-time expectation values}\n\nBefore calculating the two-time CF's, it is instructive to\nobtain the master equation of the reduced system density matrix for\nthe model. \nTransferring from the interaction picture back to the Schr\\\"odinger\npicture for Eq.~(\\ref{time_convolutionless_ME}) \nand using the general Hamiltonian, Eq.~(\\ref{Hamiltonian_L}), we obtain \nthe second-order time-convolutionless non-Markovian master equation\nfor the reduced density matrix $\\rho(t)$ as\n\\begin{eqnarray}\n\\frac{d \\rho(t)}{dt} \n&=&\\frac{1}{i\\hbar}\\left[H_S,\\rho(t)\\right] \n\\nonumber \\\\\n&&\n\\hspace{-1cm}\n-\\int_0^t d\\tau \\{\\alpha(t-\\tau)[L^\\dagger\\tilde{L}(\\tau-t)\\rho(t)\n-\\tilde{L}(\\tau-t)\\rho(t)L^\\dagger] \\nonumber\\\\\n&&\n\\hspace{-0.5cm}\n +\\beta(t-\\tau)[L\\tilde{L}^\\dagger(\\tau-t)\\rho(t)\n-\\tilde{L}^\\dagger(\\tau-t)\\rho(t)L] \\nonumber \\\\\n&&\\hspace{-0.5cm} +{\\rm h.c.} \\},\n\\label{master_eq}\n\\end{eqnarray}\nwhere $\\alpha(t-\\tau)$ and $\\beta(t-\\tau)$ are defined in\nEqs.~(\\ref{CFalpha}) and (\\ref{CFbeta}) respectively, \nh.c. indicates the hermitian conjugate of previous terms, and an operator with a tilde on the top indicates that it is an operator in the interaction picture.\nThe only real assumption used to obtain Eq.~(\\ref{master_eq})\nvalid to second order in the system-environment coupling strength \nis the total density matrix\nfactorizing at the initial time $t=0$, $\\rho_T(0)=\\rho(0)\\otimes R_0$.\nTaking $L=\\sigma _{-}$, $L^{\\dagger }=\\sigma _{+}$, then\n$\\tilde{L}(t)=\\sigma _{-}e^{-i\\omega _{A}t}$, and\n$\\tilde{L}^{\\dagger}(t)=\\sigma _{+}e^{i\\omega _{A}t}$, and\nsubstituting them into Eq.~(\\ref{master_eq}), we obtain \n\\begin{eqnarray}\n\\frac{d\\rho(t)}{dt}&=&-i\\frac{\\omega_A}{2}\\left[\\sigma_z,\\rho(t)\\right]\\nonumber\\\\\n&&\n\\hspace{-1cm}\n-\\left\\{\\Gamma _{1}(t)\\left(\\sigma_+\\sigma_\n-\\rho(t)-\\sigma_-\\rho(t)\\sigma_+\\right)\\right.\n\\nonumber\\\\\n&&\n\\hspace{-1cm}\n\\left. +\\Gamma _{2}(t)\\left(\\sigma_-\\sigma_+\\rho(t)\n-\\sigma_+\\rho(t)\\sigma_-\\right)\n+h.c.\\right\\},\n\\label{master_eq_Schrodinger}\n\\end{eqnarray} \nwhere\n\\begin{eqnarray}\n\\Gamma _{1}(t) &=&\\int_{0}^{t}d\\tau \\alpha (t-\\tau)e^{+i\\omega _{A}\\left(t-\\tau \\right) }, \n\\label{Gamma_1} \\\\\n\\Gamma _{2}(t) &=&\\int_{0}^{t}d\\tau \\beta (t-\\tau)e^{-i\\omega _{A}\\left(t-\\tau \\right) }. \n\\label{Gamma_2}\n\\end{eqnarray}\nThe master equation (\\ref{master_eq_Schrodinger}) \nis a time-local and convolutionless differential equation.\nThe effect of the non-Markovian environment \non the second-order master equation (\\ref{master_eq_Schrodinger})\nis taken into account by the time-dependent coefficients $\\Gamma _{1}(t_{1})$ and $\\Gamma _{2}(t_{1})$ defined in Eqs.~(\\ref{Gamma_1}) and (\\ref{Gamma_2}) \ninstead of memory integrals. \nLikewise, the evolution equations of single-time expectation values and two-time CF's of system operators valid to the second order \nare also expected to be convolutionless.\nTaking again $L=\\sigma _{-}$, $L^{\\dagger }=\\sigma _{+}$, then\n$\\tilde{L}(t)=\\sigma _{-}e^{-i\\omega _{A}t}$, and\n$\\tilde{L}^{\\dagger}(t)=\\sigma _{+}e^{i\\omega _{A}t}$, and using the\ncommutation relation between the Pauli matrices, \nwe obtain \nstraightforwardly from Eq.~(\\ref{1time_evol_eq_f}) the following evolution\nequations of the single-time expectation values of system operators as \n\\begin{eqnarray}\nd\\langle \\sigma _{+}(t_{1})\\rangle \/dt_{1} &=&i\\omega _{A}\\langle \\sigma\n_{+}(t_{1})\\rangle \\nonumber \\\\\n&&-\\left[ \\Gamma _{1}^{\\ast }(t_{1})+\\Gamma _{2}(t_{1})\\right] \\langle\n\\sigma _{+}(t_{1})\\rangle, \\label{one_evol_+} \\\\\nd\\langle \\sigma _{-}(t_{1})\\rangle \/dt_{1} &=&-i\\omega _{A}\\langle \\sigma\n_{-}(t_{1})\\rangle \\nonumber \\\\\n&&-\\left[ \\Gamma _{1}(t_{1})+\\Gamma _{2}^{\\ast }(t_{1})\\right] \\langle\n\\sigma _{-}(t_{1})\\rangle, \\label{one_evol_-} \\\\\nd\\langle \\sigma _{z}(t_{1})\\rangle \/dt_{1} &=&-\\left[ \\Gamma\n_{1}(t_{1})+\\Gamma _{1}^{\\ast }(t_{1})+\\Gamma _{2}^{\\ast }(t_{1})+\\Gamma\n_{2}(t_{1})\\right] \\nonumber \\\\\n&&\\quad\\quad \\times \\langle \\sigma _{z}(t_{1})\\rangle \\nonumber \\\\\n&&-\\left[ \\Gamma _{1}(t_{1})+\\Gamma _{1}^{\\ast }(t_{1})-\\Gamma _{2}^{\\ast\n}(t_{1})-\\Gamma _{2}(t_{1})\\right]. \\nonumber \\\\\n\\label{one_evol_z}\n\\end{eqnarray}\nEquations (\\ref{one_evol_+})-(\\ref{one_evol_z}) \ncan also be obtained directly from the master equation\n(\\ref{master_eq_Schrodinger}) through ${d\\left\\langle A\\left( t_{1}\\right) \\right\\rangle }\/{dt_{1}}={\\rm Tr}_{S}\\left[ A({d\\rho(t_1)}\/{dt_1})\\right]$.\n\n\n\n\\subsection{Two-time CF's}\nFor the evaluations of the two-time CF's of the system observables, we consider the following four cases.\n\n{\\em {Case 1:}} $[L^{\\dagger },A]=0$ or $[B,\\tilde{L}(t)]=0$; and $[L,A]=0$\nor $[B,\\tilde{L}^{\\dagger }(t)]=0$. In this case, let $A=\\sigma _{i}$, \nB=\\sigma _{i}$ with $i=+,-$.\nApplying the commutation relations between the Pauli matrices and the\ndefinition of ${\\rm Tr}_S[\\sigma_i(t_1) \\sigma_i(t_2)\\rho(0)]=\\langle\n\\sigma_i(t_1) \\sigma_i(t_2) \\rangle$ to the right-hand side of the\ntwo-time evolution equation (\\ref{2time_evol_eq_f}), \nwe then obtain\n\\begin{eqnarray}\nd\\langle \\sigma _{+}(t_{1})\\sigma _{+}(t_{2})\\rangle \/dt_{1} &=&i\\omega\n_{A}\\langle \\sigma _{+}(t_{1})\\sigma _{+}(t_{2})\\rangle \\nonumber \\\\\n&&-\\left[ \\Gamma _{1}^{\\ast }(t_{1})+\\Gamma _{2}(t_{1})\\right] \\langle\n\\sigma _{+}(t_{1})\\sigma _{+}(t_{2})\\rangle , \\nonumber \\\\\n \\label{2time_evol_++} \\\\\nd\\langle \\sigma _{-}(t_{1})\\sigma _{-}(t_{2})\\rangle \/dt_{1} &=&-i\\omega\n_{A}\\langle \\sigma _{-}(t_{1})\\sigma _{-}(t_{2})\\rangle \\nonumber \\\\\n&&-\\left[ \\Gamma _{1}(t_{1})+\\Gamma _{2}^{\\ast }(t_{1})\\right] \\langle\n\\sigma _{-}(t_{1})\\sigma _{-}(t_{2})\\rangle , \\nonumber \\\\\n\\label{2time_evol_--}\n\\end{eqnarray\nwhere \n$\\Gamma _{1}(t_{1})$ and $\\Gamma _{2}(t_{1})$ are defined in Eqs.~(\\re\n{Gamma_1}) and (\\ref{Gamma_2}), respectively. One can see that the evolution\nequations of the two-time CF's, Eqs.~(\\ref{2time_evol_++}) and (\\re\n{2time_evol_--}), have the same forms as the evolution equations of the\nsingle-time expectation values $\\langle \\sigma _{+}(t_{1})\\rangle $ and \n\\langle \\sigma _{-}(t_{1})\\rangle $, Eqs.~(\\ref{one_evol_+}) and (\\re\n{one_evol_-}), respectively. Hence the QRT holds in this case.\n\n{\\em {Case 2:}} $[A,L]=0$ or $[B,\\tilde{L}(t)]=0$. In this case, using Eq.~\n\\ref{2time_evol_eq_f}), we obtain \n\\begin{eqnarray}\nd\\langle \\sigma _{-}(t_{1})\\sigma _{z}(t_{2})\\rangle \/dt_{1} &=&-i\\omega\n_{A}\\langle \\sigma _{-}(t_{1})\\sigma _{z}(t_{2})\\rangle \\nonumber \\\\\n&&-\\left[ \\Gamma _{1}(t_{1})+\\Gamma _{2}^{\\ast }(t_{1})\\right] \\langle\n\\sigma _{-}(t_{1})\\sigma _{z}(t_{2})\\rangle \\nonumber \\\\\n&&-2\\Gamma _{3}\\left( t_{1},t_{2}\\right) \\left\\langle \\sigma\n_{z}(t_{1})\\sigma _{-}(t_{2})\\right\\rangle , \\label{2time_evol_-z} \\\\\nd\\langle \\sigma _{z}(t_{1})\\sigma _{-}(t_{2})\\rangle \/dt_{1} &=&-\\left[\n\\Gamma _{1}(t_{1})+\\Gamma _{1}^{\\ast }(t_{1})+\\Gamma _{2}^{\\ast\n}(t_{1})+\\Gamma _{2}(t_{1})\\right] \\nonumber \\\\\n&&\\quad\\quad \\times \\langle \\sigma _{z}(t_{1})\\sigma\n_{-}(t_{2})\\rangle \\nonumber \\\\\n&&-\\left[ \\Gamma _{1}(t_{1})+\\Gamma _{1}^{\\ast }(t_{1})-\\Gamma _{2}^{\\ast\n}(t_{1})-\\Gamma _{2}(t_{1})\\right] \\nonumber \\\\\n&&\\quad\\quad \\times \\langle \\sigma\n_{-}(t_{2})\\rangle \\nonumber \\\\\n&&-2\\Gamma _{4}\\left( t_{1},t_{2}\\right) \\left\\langle \\sigma\n_{-}(t_{1})\\sigma _{z}(t_{2})\\right\\rangle , \\label{2time_evol_z-}\n\\end{eqnarray\nwhere \n\\begin{eqnarray}\n\\Gamma _{3}\\left( t_{1},t_{2}\\right) &=&\\int_{0}^{t_{2}}d\\tau \\alpha\n(t_{1}-\\tau )e^{i\\omega _{A}\\left( t_{2}-\\tau \\right) }, \\label{Gamma_3} \\\\\n\\Gamma _{4}\\left( t_{1},t_{2}\\right) &=&\\int_{0}^{t_{2}}d\\tau \\beta\n(t_{1}-\\tau )e^{-i\\omega _{A}\\left( t_{2}-\\tau \\right) }. \\label{Gamma_4}\n\\end{eqnarray\nWhen we obtain Eq.~(\\ref{2time_evol_-z}) from Eq.~(\\ref{2time_evol_eq_f}), the last term containing $\\Gamma _{3}(t_{1},t_{2})$ in Eq.~(\\ref{2time_evol_-z}) does not vanish since $[L^{\\dagger },A]\\neq 0$ and $[B,\\tilde{L}(\\tau -t_{2})]\\neq 0$. Similarly, when we obtain Eq.~(\\ref{2time_evol_z-}) from Eq.~(\\ref{2time_evol_eq_f}), because $[L,A]\\neq 0$ and \n$[B,\\tilde{L}^{\\dagger }(\\tau -t_{2})]\\neq 0$, the last term containing \n$\\Gamma _{4}(t_{1},t_{2})$ in Eq.~(\\ref{2time_evol_z-}) exists. Thus,\ncompared with single-time equations (\\ref{one_evol_-}) and (\\ref{one_evol_z}), \nthe two-time equations (\\ref{2time_evol_-z}) and (\\ref{2time_evol_z-})\nhave the extra last terms containing $\\Gamma _{3}(t_{1},t_{2})$ and \n$\\Gamma _{4}(t_{1},t_{2})$, respectively. As a result, the QRT does not hold\nin this case. It is also obvious from the sole appearance of the individual\ncoefficient of either $\\Gamma _{3}(t_{1},t_{2})$ in \nEq.~(\\ref{2time_evol_-z}) or $\\Gamma _{4}(t_{1},t_{2})$ \nin Eq.~(\\ref{2time_evol_z-})\nthat the finite-temperature bath CF's $\\alpha (t_{1}-\\tau )$ and $\\beta\n(t_{1}-\\tau )$ can not be combined into the single effective bath CF $\\alpha\n_{\\mathrm{eff}}(t_{1}-\\tau )=\\alpha (t_{1}-\\tau )+\\beta (t_{1}-\\tau )$ of\nEq.~(\\ref{CFalpha_eff}) as in the Hermitian coupling operator case.\n\n{\\em {Case 3:}} $[A,L^{\\dagger }]=0$ or $[B,\\tilde{L}^{\\dagger }(t)]=0$.\nUsing Eq.~(\\ref{2time_evol_eq_f}), we obtain for this case \n\\begin{eqnarray}\nd\\langle \\sigma _{+}(t_{1})\\sigma _{z}(t_{2})\\rangle \/dt_{1} &=&+i\\omega\n_{A}\\langle \\sigma _{+}(t_{1})\\sigma _{z}(t_{2})\\rangle \\nonumber \\\\\n&&-\\left[ \\Gamma _{1}^{\\ast }(t_{1})+\\Gamma _{2}(t_{1})\\right] \\langle\n\\sigma _{+}(t_{1})\\sigma _{z}(t_{2})\\rangle \\nonumber \\\\\n&&-2\\Gamma _{4}\\left( t_{1},t_{2}\\right) \\left\\langle \\sigma\n_{z}(t_{1})\\sigma _{+}(t_{2})\\right\\rangle , \\label{2time_evol_+z} \\\\\nd\\langle \\sigma _{z}(t_{1})\\sigma _{+}(t_{2})\\rangle \/dt_{1} &=&-\\left[\n\\Gamma _{1}(t_{1})+\\Gamma _{1}^{\\ast }(t_{1})+\\Gamma _{2}^{\\ast\n}(t_{1})+\\Gamma _{2}(t_{1})\\right] \\nonumber \\\\\n&&\\quad\\quad \\times \\langle \\sigma _{z}(t_{1})\\sigma\n_{+}(t_{2})\\rangle \\nonumber \\\\\n&&-\\left[ \\Gamma _{1}(t_{1})+\\Gamma _{1}^{\\ast }(t_{1})-\\Gamma _{2}^{\\ast\n}(t_{1})-\\Gamma _{2}(t_{1})\\right] \\nonumber \\\\\n&&\\quad\\quad \\times \\langle \\sigma\n_{+}(t_{2})\\rangle \\nonumber \\\\\n&&-2\\Gamma _{3}\\left( t_{1},t_{2}\\right) \\left\\langle \\sigma\n_{+}(t_{1})\\sigma _{z}(t_{2})\\right\\rangle. \n\\label{2time_evol_z+}\n\\end{eqnarray}\nSimilarly, compared with the single-time evolution equations \n(\\ref{one_evol_+}) and (\\ref{one_evol_z}), \nthe two-time evolution equations (\\ref{2time_evol_+z}) \nand (\\ref{2time_evol_z+}) have the extra last terms\ncontaining $\\Gamma _{4}^{\\ast }(t_{1})$ and $\\Gamma _{3}(t_{1})$,\nrespectively. As a result, the QRT also does not hold for these two CF's.\n\n{\\em {Case 4:}} $[A,L]\\neq 0$, $[A,L^{\\dagger }]\\neq 0$ and \n$[B,\\tilde{L}(t)]\\neq 0$ , $[B,\\tilde{L}^{\\dagger }(t)]\\neq 0$. \nIn this case, by using Eq.~(\\ref{2time_evol_eq_f}), we obtain\n the following equations \n\\begin{eqnarray}\nd\\langle \\sigma _{z}(t_{1})\\sigma _{z}(t_{2})\\rangle \/dt_{1} &=&-\\left[\n\\Gamma _{1}(t_{1})+\\Gamma _{1}^{\\dagger }(t_{1})+\\Gamma _{2}^{\\dagger\n}(t_{1})+\\Gamma _{2}(t_{1})\\right] \\nonumber \\\\\n&&\\quad\\quad \\times \\langle \\sigma _{z}(t_{1})\\sigma\n_{z}(t_{2})\\rangle \\nonumber \\\\\n&&-\\left[ \\Gamma _{1}(t_{1})+\\Gamma _{1}^{\\dagger }(t_{1})-\\Gamma\n_{2}^{\\dagger }(t_{1})-\\Gamma _{2}(t_{1})\\right] \\nonumber \\\\\n&&\\quad\\quad \\times \\langle \\sigma\n_{z}(t_{2})\\rangle \\nonumber \\\\\n&&+4\\Gamma _{3}\\left( t_{1},t_{2}\\right) \\left\\langle \\sigma\n_{+}(t_{1})\\sigma _{-}(t_{2})\\right\\rangle \\nonumber \\\\\n&&+4\\Gamma _{4}\\left( t_{1},t_{2}\\right) \\left\\langle \\sigma\n_{-}(t_{1})\\sigma _{+}(t_{2})\\right\\rangle. \n\\label{2time_evol_zz}\n\\end{eqnarray}\nThe evolution equation of the CF $\\langle \\sigma _{z}(t_{1})\\sigma\n_{z}(t_{2})\\rangle $, Eq.~(\\ref{2time_evol_zz}), is coupled with the evolution\nequations of the CF's $\\langle \\sigma _{+}(t_{1})\\sigma _{-}(t_{2})\\rangle $\nand $\\langle \\sigma _{-}(t_{1})\\sigma _{+}(t_{2})\\rangle $, which correspond\nto the CF's in Case 2 and Case 3, respectively. Their evolution equations,\nobtained from Eq.~(\\ref{2time_evol_eq_f}), are \n\\begin{eqnarray}\nd\\langle \\sigma _{-}(t_{1})\\sigma _{+}(t_{2})\\rangle \/dt_{1} &=&-i\\omega\n_{A}\\langle \\sigma _{-}(t_{1})\\sigma _{+}(t_{2})\\rangle \\nonumber \\\\\n&&-\\left[ \\Gamma _{1}(t_{1})+\\Gamma _{2}^{\\ast }(t_{1})\\right] \\langle\n\\sigma _{-}(t_{1})\\sigma _{+}(t_{2})\\rangle \\nonumber \\\\\n&&+\\Gamma _{3}\\left( t_{1},t_{2}\\right) \\left\\langle \\sigma\n_{z}(t_{1})\\sigma _{z}(t_{2})\\right\\rangle, \n\\label{2time_evol_-+} \\\\\nd\\langle \\sigma _{+}(t_{1})\\sigma _{-}(t_{2})\\rangle \/dt_{1} &=&i\\omega\n_{A}\\langle \\sigma _{+}(t_{1})\\sigma _{-}(t_{2})\\rangle \\nonumber \\\\\n&&-\\left[ \\Gamma _{1}^{\\ast }(t_{1})+\\Gamma _{2}(t_{1})\\right] \\langle\n\\sigma _{+}(t_{1})\\sigma _{-}(t_{2})\\rangle \\nonumber \\\\\n&&+\\Gamma _{4}\\left( t_{1},t_{2}\\right) \\left\\langle \\sigma\n_{z}(t_{1})\\sigma _{z}(t_{2})\\right\\rangle. \n\\label{2time_evol_+-}\n\\end{eqnarray}\nFrom Eqs.~(\\ref{2time_evol_zz}), (\\ref{2time_evol_-+}) and (\\re\n{2time_evol_+-}), it is obvious that the QRT also does not hold for CF's \n\\langle \\sigma _{z}(t_{1})\\sigma _{z}(t_{2})\\rangle $, $\\langle \\sigma\n_{+}(t_{1})\\sigma _{-}(t_{2})\\rangle $ and $\\langle \\sigma _{-}(t_{1})\\sigma\n_{+}(t_{2})\\rangle $.\n\n \n\n\\begin{figure}[tbp]\n\\includegraphics[width=0.95\\linewidth]{CF_noise.eps}\n\\caption{(Color online) (a) Real part of the time evolution and (b) Fourier spectrum $S(\\omega)$ of the system operator CF $\\langle\\sigma_+(t_1)\\sigma_-(t_2)\\rangle$, and (c) real part of \n$\\langle \\sigma _{z}(t_{1})\\sigma _{z}(t_{2})\\rangle $ for three different \ncases: Markovian using the QRT (blue dot-dashed line), non-Markovian\nusing the QRT (green dashed line) and non-Markovian (red solid line) using the\nevolution equation (\\ref{2time_evol_eq_f}). \nOther parameters used are $\\omega_A=3$, $(k_BT\/\\hbar)=1$, $\\Lambda=5$, $\\gamma=0.1$, and $t_{2}=1$.\n(d) Fourier spectrum $S(\\omega)$ of $\\langle\\sigma_+(t_1)\\sigma_-(t_2)\\rangle$ for a different parameter of $\\gamma =0.35$ and for an initial mixed system state.\nThe results of the non-Markovian QRT case and the non-Markovian\nevolution case in (c) become indistinguishable when $t=t_1-t_2$ is larger\nthan 1.5.\n}\n\\label{fig:CF}\n\\end{figure}\n\nWe may consider any spectral density for which the time-convolutionless\nperturbation theory is still valid to characterize the environment, but for simplicity we consider a spectral density $J(\\omega)=\\sum_\\lambda|g_\\lambda|^2\\delta(\\omega-\\omega_\\lambda)=\\gamma \\hbar\\omega (\\omega\/\\Lambda)^{n-1}\\exp(-\\omega^2\/\\Lambda^2)$ with $n=1$ (Ohmic), where $\\Lambda$ is the cut-off frequency and $\\gamma$ is a dimensionless constant characterizing the interaction strength \nto the environment. \nFigure \\ref{fig:CF} shows the real part of the time evolution \nof the system operator CF's $\\langle\\sigma_+(t_1)\\sigma_-(t_2)\\rangle$ and \n$\\langle \\sigma _{z}(t_{1})\\sigma _{z}(t_{2})\\rangle $, as well as \nthe Fourier spectrum $S(\\omega)$ of $\\langle\\sigma_+(t_1)\\sigma_-(t_2)\\rangle$. \nThe CF's are obtained in three different cases: the first is in the\nMarkovian case [i.e, taking the reservoir CF's \n$\\alpha(t_1-\\tau)$ and \n$\\beta(t_1-\\tau)$ in Eq.~(\\ref{2time_evol_eq_f}) to be $\\delta$-correlated in time, or equivalently\ntaking the coefficients of $\\Gamma_1$, $\\Gamma_1^\\dagger$, $\\Gamma_2$,\nand $\\Gamma_2^\\dagger$ to be time-independent and equal to their\nMarkovian long-time values and setting all $\\Gamma_3(t_1,t_2)$ and\n$\\Gamma_4(t_1,t_2)$ to be zero in Eqs.~(\\ref{2time_evol_zz}),\n(\\ref{2time_evol_-+}) and (\\ref{2time_evol_+-})], the second is in the\nnon-Markovian case with a finite cut-off frequency but wrongly\ndirectly using the QRT method [i.e., \nthe last two terms of Eq.~(\\ref{2time_evol_eq_f}) or equivalently the terms containing $\\Gamma_3(t_1,t_2)$ or $\\Gamma_4(t_1,t_2)$ in \nEqs.~(\\ref{2time_evol_zz}), (\\ref{2time_evol_-+}) and (\\ref{2time_evol_+-})\nbeing all neglected],\nand the third is in the non-Markovian case with a finite cut-off\nfrequency [i.e., using the evolution equation (\\ref{2time_evol_eq_f})\nor equivalently Eqs.~(\\ref{2time_evol_zz}), (\\ref{2time_evol_-+}) and\n(\\ref{2time_evol_+-}) derived\nin this paper]. \nThe initial environmental state is in the thermal state and the system state in Fig.~\\ref{fig:CF}(a)--(c) is arbitrarily chosen to be $\\vert \\Psi\\rangle =\\left( \\frac{\\sqrt{3}}{2}\\left\\vert e\\right\\rangle +\\frac{1}{\n}\\left\\vert g\\right\\rangle \\right)$. \nWe can see that there are considerable differences between the results \nobtained in the three different cases in Fig.~\\ref{fig:CF}(a) and (b), \nand more significant differences can be observed \nin Fig.~\\ref{fig:CF}(c) and (d). \nThe oscillations of the CF's are more pronounced in the non-Markovian cases. \nIn Fig.~\\ref{fig:CF}(b), the coherent peaks of the Fourier spectrum \ncentered at $\\omega=\\pm\\omega_A$\nare higher and the widths are narrower in the non-Markovian cases.\nThe CF $\\langle \\sigma _{z}(t_{1})\\sigma\n_{z}(t_{2})\\rangle $ of the non-Markovian evolution equation case (in\nred solid line) in\nFig.~\\ref{fig:CF}(c) \ndiffers more from the CF's of the other two cases (in green dashed\nline and in blue dot-dashed line) in the short-time regime \nthan the CF $\\langle\\sigma_+(t_1)\\sigma_-(t_2)\\rangle$ of the\nnon-Markovian evolution equation case (in red solid line) in\nFig.~\\ref{fig:CF}(a) does. \nThis is because as compared to the evolution equation of\n$\\langle\\sigma_+(t_1)\\sigma_-(t_2)\\rangle$ of\nEq.~(\\ref{2time_evol_+-}), the evolution equation of \n$\\langle \\sigma _{z}(t_{1})\\sigma_{z}(t_{2})\\rangle $ of\nEq.~(\\ref{2time_evol_zz}) has, in addition to a term proportional to\n$\\Gamma_4(t_1,t_2)$, an extra correction term \nproportional to $\\Gamma_3(t_1,t_2)$ over its QRT counterparts.\nIt is also found that generally the results of the non-Markovian QRT\ncase (in green dashed lines) approach those\nof the non-Markovian evolution equation case (in red solid lines)\nmore closely than the results of the Markovian QRT case (in blue\ndot-dashed lines) do.\nSimilar behaviors are also observed when the temperature is increased\nor when the cut-off frequency $\\Lambda$ is\nincreased.\nThe Markovian case can be recovered from the non-Markovian ones in the\nlimit when the cut-off frequency $\\Lambda\\to\\infty$, in which the\nthree results coincide. \nFor a \nlarger $\\gamma$ and for an initial mixed system state with the values of the off-diagonal density matrix elements being a quarter of those of the pure state $\\vert \\Psi\\rangle$, the peak heights of $S(\\omega)$ \nare lower as shown in Fig.~\\ref{fig:CF}(d).\nFurthermore in Fig.~\\ref{fig:CF}(d), \nthe two coherent peaks are still clearly visible in the non-Markovian cases, \nwhile the two peaks is barely visible in the Markovian case.\n \nFor the present spin-boson model with the system coupling operator\n$L\\neq L^\\dagger$, the self-Hamiltonian of the spin does not commute\nwith the system coupling operator, i.e, $[H_S,L]\\neq 0$, and the\nenvironment coupling operator also does not commute with the\nself-Hamiltonian of the environment, i.e., $[H_R, a_\\lambda]\\neq 0$. \nThus the exact non-Markovian finite-temperature two-time CF's of\nthe present spin-boson model are not directly available. \nBut in Ref.~\\cite{Goan10}, we evaluated the exact non-Markovian\nfinite-temperature two-time CF's of the system operators for an\nexactly solvable pure-dephasing spin-boson model in two ways, one by\nexact direct operator technique without any approximation and the\nother by the derived evolution equation (\\ref{2time_evol_eq_f})\nvalid to second order in the system-environment interaction Hamiltonian.\nThe perfect agreement of the results between the non-Markovian\nevolution equation case and the exact operator evaluation case, and\nthe significant difference between the non-Markovian evolution\nequation case and the case of wrongly applying non-Markovian QRT\n\\cite{Goan10}. \ndemonstrate clearly the validity of the derived evolution equation\n(\\ref{2time_evol_eq_f}).\nIt is thus believed that in the weak system-environment coupling\nlimit, the finite-temperature CF's calculated\nusing our evolution equation that takes into account the nonlocal\nenvironment memory term, Eq.~(\\ref{non-Markovian_1st_order}), for the\npresent spin-boson model would\nagree more closely with the exact \nresults than those in the non-Markoian QRT and Markovian QRT\ncases.\n\n\n\n\\section{Conclusion}\\label{sec:conclusion}\nIn summary, we have derived evolution equations of the single-time and\ntwo-time CF's of system operators, using a quantum master equation \ntechnique different from \nthose presented in Refs.~\\cite{Alonso05,Vega06,Alonso07}. \nThis quantum master equation approach allows us to explicitly point\nout an important nonlocal environment (bath) memory term that vanishes\nin the Markovian case but makes the evolution equation deviate from\nthe QRT in general cases. \nThe derived two-time \nequations are valid for thermal environments at any temperature with\nHermitian or non-Hermitian coupling operators and for any initially\nfactorized (separable) system-reservoir state (pure or mixed) as long as the \nassumption of Eq.~(\\ref{traceless_1st_order}) and the approximation \nof the weak system-environment coupling \nthat are used to derive the equations apply. \nIn contrast to the evolution equations presented in\nRefs.~\\cite{Alonso05,Vega06,Alonso07,Goan10} that are applicable for\nbosonic environments, Eq.~(\\ref{2time_evol_eq}) derived in this paper\ncan be used to calculate the two-time CF's for a wide range of \nsystem-environment models with\nbosonic and\/or fermionic environments.\nWe have also given conditions on which the QRT holds in the weak\nsystem-environment coupling case and have applied the derived equations to\na problem of a two-level system (atom) coupled to a\nfinite-temperature \nthermal bosonic environment (electromagnetic fields),\nin which the system coupling operator is not Hermitian, $L\\neq\nL^\\dagger$, and the evolution equations derived in\nRefs.~\\cite{Alonso05,Vega06,Alonso07} are not applicable. \nIt is easy to calculate the two-time CF's using the\nderived evolution equations. Other non-Markovian open quantum system\nmodels that are not exactly solvable can be proceeded in a similar way\nto obtain the time evolutions of their two-time system operator CF's\nvalid to second order in the system-environment interaction\nHamiltonian. This illustrates the practical usage of the evolution\nequations. \nTherefore, the derived evolution equations that\ngeneralize the QRT to the non-Markovian cases will\nhave broad applications in\nmany different branches of physics, such as quantum optics,\nstatistical physics, chemical physics, quantum\ntransport in nanostructure devices and so forth when the properties\nrelated to the two-time CF's are of interests. \n\n\\begin{acknowledgments}\nWe would like to acknowledge support from the National Science\nCouncil, Taiwan, under Grant No. 97-2112-M-002-012-MY3, \nsupport from the Frontier and Innovative Research Program \nof the National Taiwan University under Grants No. 99R80869 and \nNo. 99R80871, and support from the focus group program of the National Center for Theoretical Sciences, \nTaiwan. H.S.G. is grateful to the National Center for High-performance Computing, Taiwan, \nfor computer time and facilities.\n\\end{acknowledgments}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\n\\subsection{Astrophysical source targets} The O2 GstLAL search targeted GW\nsignals from merging binary compact objects with component masses between\n1\\,$\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$ and 399\\,$\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$. These include binary systems with two neutron\nstars (BNS), two black holes (BBH), or a neutron star and a black hole (NSBH).\nThis component mass region is known to be populated with compact objects\nproduced from the collapse of massive stars. With stellar evolution models,\nneutron stars can form in the mass range between 1\\,$\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$ and\n3\\,$\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$~\\cite{Rhoades1974, Kalogera1996, OzelNSLower, Lattimer2012,\nKiziltan2013} although there is only one observed neutron star with a mass\nlarger than 2\\,$\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$~\\cite{MassiveNS}, and those in binaries do not approach\n2\\,$\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$~\\cite{Ozel2016}. Stellar evolution models also predict that black\nholes may exist with a minimum mass down to 2\\,$\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$~\\cite{Shaughnessy2005}\nand a maximum mass up to 100\\,$\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$ or potentially\nhigher~\\cite{Belczynski2014, deMink2015}. Black holes with masses between\n$\\sim$100\\,$\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$ and $\\sim10^5$\\,$\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$ are classified as intermediate-mass\nblack holes and could have formed through hierarchical merging of lower mass\nblack holes~\\cite{Miller2004}. This search is also sensitive to GWs from\nbinaries of primordial black holes (PBH), formed from over-dense regions in the\nearly universe. However, distinguishing a PBH GW signal from a conventional\nstellar-evolution black hole GW signal would not be possible with this search\nand is instead pursued in a separate search of the sub-solar mass\nregion~\\cite{Subsolar2018}.\n\nWe also define different ranges of allowed angular momentum for component\nneutron stars and component black holes. We consider only the dimensionless\nspin $\\chi= c\\left| \\vec{S}\\right| \/Gm^2$ where $\\vec{S}$ is the angular\nmomentum and $m$ is the component mass. Observations of the fastest spinning\npulsar constrain $\\chi \\lesssim 0.4$~\\cite{Hessels2006} while pulsars in\nbinaries have $\\chi \\le 0.04$~\\cite{Kramer2009}. X-ray observations of\naccreting BHs indicate a broad distribution of BH spins~\\cite{Fabian2012,\nGou2011, Mcclintock2011}, while the relativistic Kerr bound $\\chi \\le 1$ gives\nthe theoretical constraint~\\cite{MisnerGrav}.\n\nThese observations and evolution models inform the ranges of parameters we\ndefine for our template banks. As shown in\nFig.~\\ref{fig:H1L1V1-GSTLAL_INSPIRAL_PLOTBANKS_bank_regions_imri-0-0.png}, we can\nsee the BNS, NSBH, and BBH populations represented in the O2 GstLAL offline search. We impose an additional\nconstraint on the component dimensionless spins of template waveforms by\nrequiring their orientations to be aligned or anti-aligned with the orbital\nangular momentum of the binary $\\hat{L}$. Then the dimensionless projections of\nthe component spins along $\\hat{L}$ are defined as $\\chi_i \\equiv c\\left|\n\\vec{S}_i \\cdot \\hat{L} \\right| \/Gm_i^2$. The region in green marks the BNS\ntemplates with component masses between 1\\,$\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$ and 2\\,$\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$ and\n(anti-)aligned dimensionless spin magnitudes with $ \\chi_{1,2}<0.05$. This\n$\\chi$ limit is motivated by the observational limit of $\\chi \\le 0.04$ but\nwith some added uncertainty. The region in blue marks the BBH templates with\ncomponent masses between 2\\,$\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$ and 399\\,$\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$ and (anti-)aligned\ndimensionless spin magnitudes with $\\chi_{1,2} <0.999$. This $\\chi$ limit is chosen to\nbe as close to the theoretical limit of 1 as possible with current waveform\napproximants, as described in Section~\\ref{sec:approximant}. The templates in red mark the NSBH\nrange with the neutron star mass between 1\\,$\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$ and 2\\,$\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$ and the\nblack hole mass between 2\\,$\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$ and 200\\,$\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$. For these systems, neutron\nstars have $ \\chi_{1,2} <0.05$ and black holes have $\\chi_{1,2} <0.999$.\n\n\\begin{figure}[hbt!]\n\\includegraphics[width=0.45\\textwidth]{H1L1V1-GSTLAL_INSPIRAL_PLOTBANKS_bank_regions_imri-0-0.png}\n\\caption{The template bank used by the O2 GstLAL offline search in component mass space.The templates representing the different astrophysical populations are shown in green for BNS, blue for BBH, and red for NSBH.}\n\\label{fig:H1L1V1-GSTLAL_INSPIRAL_PLOTBANKS_bank_regions_imri-0-0.png}\n\\end{figure}\n\nIn Fig.~\\ref{online_bank.png}, we can see the BNS, NSBH, and BBH populations represented in the O2 GstLAL online search.\nThe BNS templates cover the same component mass and dimensionless spin magnitude\nrange as the offline bank. However, a cut on total mass results\nin different mass ranges for NSBH and BBH templates. The maximum allowed total\nmass is 150\\,$\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$, to remove high-mass templates which correspond to short waveforms\nthat recover short transient noise fluctuations (glitches) at a high rate.\n\n\\begin{figure}[hbt!] \\includegraphics[width=0.45\\textwidth]{online_bank.png}\n\\caption{\\label{online_bank.png}\nThe template bank used by the O2 GstLAL online search in component\nmass space. The templates representing the different astrophysical populations\nare shown in green for BNS, blue for BBH, and red for NSBH.} \\end{figure}\n\n\\subsection{Construction of the O2 bank}\n\nThe construction of a template bank relies on a number of parameters, including the selection of\na representative noise power spectral density $S_n(f)$ and appropriate waveform models,\nthe waveform starting frequency $f_\\mathrm{low}$, the placement method, and a specified minimum\nfitting factor criteria~\\cite{privitera2014improving, fittingfactor, FFdef} for all templates in the bank.\n\nThe minimum fitting factor describes the effectualness of a template bank in recovering\nastrophysical sources. To define this quantity, we note that the matched filter output is maximized\nwhen a template waveform exactly overlaps the signal waveform. This optimization is impossible in practice,\nhowever, since the template bank samples the parameter space discretely while\nastrophysical sources arise from a continuum. Regardless, it is useful to\nquantify the degree to which two waveforms, $h_1$ and $h_2$, overlap. The\noverlap is defined as the noise-weighted inner product integral:\n\\begin{align} \\label{eq:overlap}\n(h_{1} | h_{2}) = 2 \\int^\\infty_{f_\\mathrm{low}} \\frac{\\tilde{h}_1(f)\\tilde{h}_2(f) + \\tilde{h}_1(f)\\tilde{h}_2(f)}{S_n(f)}df,\n\\end{align}\nwhere $f_\\mathrm{low}$ was set to 15\\,Hz, as motivated by the noise power spectral density described in\nSection~\\ref{sec:psd}.\n\nThe \\emph{match} between two waveforms is then defined as the maximization over\ncoalescence phase and time of the noise-weighted inner product:\n\n\\begin{align} \\label{eq:Match} M(h_{1}, h_{2}) = \\underset{\\phi_{c},\nt_{c}}{\\text{max}} (h_{1}|h_{2}(\\phi_{c},t_{c})) \\end{align}\nThis defines the percent of signal-to-noise ratio (SNR) retained when recovering waveform $h_2$\nwith the (non-identical) waveform $h_1$. Then, the fitting factor is the related quantity used\nin describing the effectualness of template banks:\n\\begin{align} \\label{eq:FF} FF(h_{s}) = \\underset{h \\in \\{h_{b}\\}}{\\text{max}} M(h_s, h) \\end{align}\nwhere $h_b$ is the set of templates in the bank and $h_s$ is a signal waveform\nwith parameters drawn from the continuum. The fitting factor describes the fraction of SNR\nretained for arbitrary signals in the parameter space covered by the bank. Typically,\ncompact binary coalescence searches have required a fitting factor of $97\\%$ to ensure\nthat no more than $\\sim10 \\%$ of possible astrophysical signals are lost due\nto the discrete nature of the bank. As described in Sect.~\\ref{sec:placement}, we\nuse a hierarchical set of fitting factor requirements to construct the bank.\n\n\\subsubsection{Modeling the detector noise}\\label{sec:psd}\nThe noise power spectral density (PSD) as shown in Fig.~\\ref{psd.pdf} was used to\ncompute the overlap integrals in the construction of the O2 template bank.\nThis projected O2 sensitivity curve was produced by combining some of the\nbest LIGO L1 sensitivities achieved before the start of O2. At low frequencies,\nbelow 100\\,Hz, the best sensitivity is taken from L1 measurements during commissioning\nin February 2016. At high frequencies, above 100\\,Hz, the best sensitivity is taken from\nL1 during O1, with projected improved shot noise due to slightly higher input power and\nimproved efficiency of the readout chain. Calculation of\nthis PSD has been documented in~\\cite{psd}.\n\n\\begin{figure}[hbt!]\n\\includegraphics[width=0.4\\textwidth]{psd_newcode.png}\n\\caption{\\label{psd.pdf}Representation of the model power spectral density of\ndetector noise. This was used to construct the O2 template bank. } \\end{figure}\n\t\t\n\\subsubsection{Waveform approximants} \\label{sec:approximant}\n Gravitational waveforms from compact binary mergers are described by\n 17 intrinsic and extrinsic parameters. However, as demonstrated in \\cite{ajith2014effectual}, for template\n placement purposes, we can parameterize these systems by three parameters composed of\ncomponent masses $m_i$ and the reduced-spin parameter $\\chi$ which is a function of\nthe dimensionless spin parameters $\\chi_i$ for $i=1,2$.\n\nAbove a total mass of 4\\,$\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$, the waveforms of the binary systems are\napproximated by an effective-one-body formalism (SEOBNRv4\\textunderscore\nROM)~\\cite{SEOBNRv4ROM}, combining results from the post-Newtonian approach,\nblack hole perturbation theory and numerical relativity to model the complete\ninspiral, merger and ringdown waveform. Below a total mass of 4\\,$\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$, the\nbinary systems are approximated by an inspiral waveform accurate to\nthird-and-a-half post-Newtonian order called the TaylorF2 waveform\nmodel~\\cite{TaylorF2,CBCcompanion}. The extent of the present parameter space\ncovered by the template bank is limited by the availability of waveform models\nand the sensitivity of the present search. We neglect the effect of precession\nand higher order modes in our templates.\n\n\\subsubsection{Template placement}\\label{sec:placement}\nBoth the O2 offline and online template banks were created in the same way, by constructing two sub-banks that were added together. For systems with total mass $2\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace \\le M \\le 4\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$ where the TaylorF2 approximant is used, the templates were first laid out using a\ngeometric metric technique~\\cite{geometric}. This geometric bank was used as a coarse seed bank for an additional stochastic method placement~\\cite{ajith2014effectual, stochastic} with a convergence threshold set to 97$\\%$. For systems with total mass greater than 4\\,$\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$, where the SEOBNRv4\\textunderscore ROM approximant is used, a coarse bank was first generated with the stochastic method but\nwith a very low convergence threshold. Again this stochastic bank was used as a coarse seed bank for an additional stochastic method placement with a convergence threshold set to 97$\\%$. Additionally, only waveforms with a duration longer than\n0.2\\,s were retained, to avoid recovering short transient noise glitches. The two sub-banks were added to form the full bank with a total of 661,335 templates.\n\n\\begin{figure}[hbt!]\n\\includegraphics[width=0.45\\textwidth]{gstlal_bank_m1m2.png}\n\\caption{\\label{gstlal_bank_m1m2.png}A visual representation of the original O2\nbank in the component mass space, containing a total of 661,335 templates\nplaced with a minimal match of 97$\\%$.} \\end{figure}\n\nThe original O2 offline bank, as shown in Fig.~\\ref{gstlal_bank_m1m2.png} aided in the discovery of\none of the earliest events detected during O2, GW170104 [3]. The higher density of\nthe bank at lower masses is expected because low mass systems have\nlonger waveforms and spend more time in the detectors' sensitive frequency band. This\nenables the matched-filter search to better distinguish between two different\nlow mass systems. This also means that more templates are required in the lower mass region of the bank for\nthe required fitting factor convergence. At the highest masses, the waveforms contain very few\ncycles and very few templates are required for coverage in this region.\n\nEarly in O2, short duration glitches were found to be particularly problematic for the online search, even with a duration cut of 0.2\\,s applied. Thus, to avoid delays in delivering low-latency gravitational-wave triggers, only waveforms with a total mass $<150\\,\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$ were retained in the online bank. This online bank, as shown in Fig.~\\ref{online_bank.png}, was used for the entirety of the O2 observing run.\n\n\\subsubsection{Overcoverage in the offline bank}\\label{sec:overcoverage}\nAs outlined in Section~\\ref{sec:implementation}, templates are grouped by the GstLAL search so that each\ngroup has the same number of templates with similar parameters and background noise\ncharacteristics~\\cite{O1Methods, O2Methods}. It was uncovered partway through O2 that the\nlower density of templates in the high mass part of the offline bank was resulting in templates with\nvery different background noise properties being grouped together. This led to incorrect averaging of\nnoise properties in the high mass groupings of templates and, in turn, resulted in incorrect estimation of the\nsignificance of loud coincident noise in time-shifted data from the two detectors ~\\cite{O2Methods}. This was\nnot an issue in the online bank due to the cut at total mass $>150\\,\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$.\n\nTwo different recourses were taken. The offline bank was\noverpopulated with extra templates in the higher mass region as outlined below. Additionally, the templates in\nthis part of the bank were grouped differently from those in the denser lower\nmass region such that templates with more similar noise characteristics can be\ngrouped together. Details of the template grouping methods are given in Ref.~\\cite{O2Methods}, which is to be published soon.\n\nRegarding the overcoverage, extra templates were added to the initial offline bank in the total mass range of\n$80\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace \\le M \\le 400\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$ using two methods. As a first step, the original offline bank was used as a seed\nfor an additional stochastic placement in the total mass range $80\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace \\le M \\le 400\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$ with an\nincreased convergence threshold of 98$\\%$. Additionally, no template duration threshold was used so as to not exclude the short\nwaveforms corresponding to the heavier mass systems. A total of 14,665\ntemplates, as shown in Fig.~\\ref{mtotalcut80_m1m2.png}, were added to the initial offline bank.\n\nDespite the increased convergence threshold, the high mass region of the bank remained sparsely\npopulated, as the overlap between high mass waveforms with few cycles are generally high. Thus, the convergence\nthreshold is already met, without the placement of additional templates. However, short duration glitches are also\nrecovered by a relatively few number of high mass templates, and if these few glitchy templates are\ngrouped together for background estimation with quieter templates, they can spoil the sensitivity over a\nbroad mass range. Thus, we chose to force the placement of additional templates at higher mass using a\nuniform grid placement in component mass space for the total mass range between $100\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace \\le M \\le 400\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$, with mass ratios between 1 and 97.989.\nA total of 1000 templates were placed without any limitations on the waveform duration. This gridded bank, as\nshown in Fig.~\\ref{uniformgrid_m1m2.png}, was then added to the offline bank produced in the previous step.\n\n\\begin{figure}[hbt!]\n\\includegraphics[width=0.45\\textwidth]{mtotalcut80_m1m2.png}\n\\caption{\\label{mtotalcut80_m1m2.png}The bank of extra 14,665 templates that\nwere added to the initial O2 bank, with a 98$\\%$ minimum match above a total\nmass of 80 $\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$ in the component mass space.} \\end{figure}\n\n\\begin{figure}[hbt!]\n\\includegraphics[width=0.45\\textwidth]{uniformgrid_m1m2.png}\n\\caption{\\label{uniformgrid_m1m2.png}The uniform grid bank with a 1000\ntemplates spanning 100-400 $\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$ in total mass .} \\end{figure}\n\nAll together, the final, improved O2 bank has a total of about 677,000 templates, as shown in\nFig.~\\ref{fig:H1L1V1-GSTLAL_INSPIRAL_PLOTBANKS_bank_regions_imri-0-0.png}.\n\n\\subsection{Implementation in the GstLAL pipeline}\\label{sec:implementation}\nThe GstLAL-based inspiral search is a matched-filtering pipeline. The noise-weighted inner\nproduct of each whitened template with the whitened data produces the signal-to-noise\nratio (SNR). Both signals and glitches can produce high SNR, thus a number of additional\nconsistency and coincidence checks are implemented in the pipeline, as detailed in Ref.~\\cite{O2Methods}.\nIn order to access the full waveform of systems up to 400\\,$\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$ that merge at lower frequencies,\nthe filtering frequency was reduced from 30\\,Hz in the O1 search to 15\\,Hz.\n\nFor the purpose of background estimation, templates are grouped together so that\neach group has the same number of templates with similar background noise characteristics.\nNoise properties are averaged separately for each group. Prior to O2, templates were\ngrouped according to two composite parameters that characterize the\nwaveform inspiral to leading order. These were the chirp mass of the binary $\\ensuremath{\\mathcal{M}}\\xspace$ and the\neffective spin parameter $\\chi_\\mathrm{eff}$. The chirp mass is \\begin{align}\n\\ensuremath{\\mathcal{M}} &= \\frac{(m_1 m_2)^{3\/5}}{(m_1 + m_2)^{1\/5}}. \\end{align} The effective\nspin parameter is defined as \\begin{align} \\chi_\\mathrm{eff} &\\equiv \\frac{m_1\n\\chi_1 + m_2 \\chi_2}{m_1 + m_2}, \\end{align} and acts as a mass-weighted\ncombination of the spin components (anti-)aligned with the total angular\nmomentum.\n\nHowever, as described in Sect.~\\ref{sec:overcoverage}, extra templates were placed in the\nhigh mass region of the offline bank, to better capture the properties of the noise in that regime.\nTemplates above a total mass of 80\\,$\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$ were then grouped by template duration from 15\\,Hz\nrather than the $\\ensuremath{\\mathcal{M}}\\xspace$ and $\\chi_\\mathrm{eff}$ binning used at lower masses. Template duration\nbetter characterizes the waveform merger and ringdown, the detectable part of the signal\nfor high mass systems.\n\n\\section{Effectualness} \\label{sec:effectual}\n\nTo assess the effectualness of this template bank, we compute again the\n$\\mathrm{FF}(h_{s})$ as defined in Eq.~\\ref{eq:FF} for a collection of simulated signals with\nparameters drawn randomly from the covered mass and spin space. The $FF$ depends on a number of parameters including\nmasses, spins, spin orientations and sky locations. Hence it is often\nrepresented and plotted as a function of two such parameters. In order to do so, the $FF$ is binned in the two parameters and the \naverage or mean $FF$ in each bin is plotted~\\cite{fittingfactor}:\n\\begin{equation} \\label{eq:FFav} FF_\\mathrm{mean} = \\langle FF \\rangle \\end{equation}\n\nWe selected simulated signals from various populations of BNS, NSBH, BBH, and\nIMBHB systems to check the effectualness of the bank. The\ndetails of the simulation sets are summarized in\ntable~\\ref{t:banksim_injections}. For each signal population,\n$10^{4}$ simulations were performed.\n\n\\begin{table*}[t]\n \\centering\n \\begin{tabular}{ lllll }\n\\hline\n\\hline\nPopulation & Mass($\\ensuremath{\\mathrm{M}_{\\odot}}\\xspace$) & Spin & & Waveform approximant\\\\\n\\hline \n\nBNS & $m_{1,2} \\in [1,3]$ & $\\chi_{1,2} \\in [-0.05,0.05]$, aligned & & TaylorF2~\\cite{TaylorF2} \\\\\n\nBNS & $m_{1,2} \\in [1,3]$ & $\\chi_{1,2} \\in [-0.4,0.4]$, precessing & & SpinTaylorT4~\\cite{SpinTaylorT4} \\\\\n\n\\multirow{2}*{NSBH} & $m_{1} \\in [1,3]$ & $\\chi_{1} \\in [-0.4,0.4]$, aligned & & SEOBNRv4\\_ROM~\\cite{SEOBNRv4ROM} \\\\\n & $m_{2} \\in [3,97]$ & $\\chi_{2} \\in [-0.989,0.989]$, aligned & & \\\\\n\nBBH & $m_{1,2} \\in [2,99]$ & $\\chi_{1,2} \\in [-0.99,0.99]$ aligned & & SEOBNRv4\\_ROM~\\cite{SEOBNRv4ROM} \\\\\n\nIMBHB & $m_{1,2} \\in [1,399]$ & $\\chi_{1,2} \\in [-0.998,0.998]$ aligned & & SEOBNRv4\\_ROM~\\cite{SEOBNRv4ROM} \\\\\n\nIMBHB & $m_{1,2} \\in [50,350]$ & Non-spinning & & EOBNRv2HM~\\cite{EOBNRv2HM} \\\\\n\n\\end{tabular}\n\\caption{\\label{t:banksim_injections}Description of different categories of\nastrophysical populations, from which random mass and spin parameters were drawn and used to generate\nwaveforms to check the effectualness of the template bank. Multiple simulation sets of the same population\nwere used, varying in the type of waveform, mass ranges covered and whether the\nspin is aligned to the orbital angular momentum.} \\end{table*}\n\nIn Fig.~\\ref{fig:bnsff}, we can see the fitting factors in the $M$-$\\chi_\\mathrm{eff}$ plane for BNS aligned-spin TaylorF2 waveform approximants~\\cite{TaylorF2} and precessing-spin SpinTaylorT4 waveform approximants~\\cite{SpinTaylorT4}. The majority of fitting factors are above 0.97, except along the low-mass edge of the bank at $M=2.0$ below which no templates are placed. The bank is constructed with aligned-spin TaylorF2 waveforms in this low mass region so fitting factors are expected to be at least as high as the required fitting factor of 0.97 to ensure that no more than $\\sim10 \\%$ of possible astrophysical signals are lost due to the discrete nature of the bank. We can also see that the majority of fitting factors for precessing-spin SpinTaylorT4 waveform approximants are also above 0.9 although sensitivity falls off rapidly outside $-0.05 < \\chi_\\mathrm{eff} < 0.05$ for systems with NS component mass less than 2 \\ensuremath{\\mathrm{M}_{\\odot}}\\xspace. There are no templates placed in this region so the fall off in fitting factor is expected. This also demonstrates that a search based on an aligned-spin template bank can recover precessing-spin signals.\n\n\\begin{figure*}[t] \n\\centering\n\\begin{minipage}[b]{0.47\\textwidth}\n\\centering\n \\includegraphics[width=\\textwidth]{ban_sim_plots\/bns_TaylorF2_chiM_hexbin.png}\n \\end{minipage}\n \\begin{minipage}[b]{0.47\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{ban_sim_plots\/bns_SpinTaylorT4_chiM_hexbin.png}\n \\end{minipage}\n\\caption{\\label{fig:bnsff} Fitting factors in $M$-$\\chi_\\mathrm{eff}$ plane for BNS aligned-spin TaylorF2 waveform approximants~\\cite{TaylorF2} ({\\it left}) and precessing-spin SpinTaylorT4 waveform approximants~\\cite{SpinTaylorT4} ({\\it right}). ({\\it Left}) The majority of fitting factors are above 0.97, except along the low-mass edge of the bank at $M=2.0$ where the fitting factor starts to fall off. The bank is constructed with TaylorF2 waveforms so fitting factors are expected to be at least as high as the required fitting factor of 0.97 to ensure that no more than $\\sim10 \\%$ of possible astrophysical signals are lost due to the discrete nature of the bank. ({\\it Right}) The majority of fitting factors are above 0.9 although sensitivity falls off rapidly outside $-0.05 < \\chi_\\mathrm{eff} < 0.05$ for systems with NS component mass less than 2 \\ensuremath{\\mathrm{M}_{\\odot}}\\xspace. There are no templates placed in this region so the fall off in fitting factor is expected. This also demonstrates that a search based on an aligned-spin template bank can recover precessing-spin signals.}\n\\end{figure*}\n\nIn Fig.~\\ref{fig:nsbhff}, we can see the fitting factors in the $M$-$\\chi_\\mathrm{eff}$ plane and as a function of mass ratio for NSBH aligned-spin SEOBNRv4\\_ROM waveform approximants~\\cite{SEOBNRv4ROM}. The majority of fitting factors are above 0.97 although fitting factors down to 0.885 are present across this region. Lower fitting factors occur for systems with more extreme mass ratios indicating that the bank is not optimized to recover signals from highly asymmetric systems. A dedicated search in this region may be required to find signals from systems with extreme mass ratios.\n\n\\begin{figure*}[t]\n\\centering\n\\begin{minipage}[b]{0.47\\textwidth}\n\\centering\n \\includegraphics[width=\\textwidth]{ban_sim_plots\/nsbh_highmass_chiM_hexbin.png}\n \\end{minipage}\n \\begin{minipage}[b]{0.47\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{ban_sim_plots\/nsbh_highmass_qVsFF.png}\n \\end{minipage}\n\\caption{\\label{fig:nsbhff} Fitting factors in $M$-$\\chi_\\mathrm{eff}$ plane ({\\it left}) and as a function of mass ratio ({\\it right}) for NSBH aligned-spin SEOBNRv4\\_ROM waveform approximants~\\cite{SEOBNRv4ROM}. ({\\it Left}) The majority of fitting factors are above 0.97 although fitting factors down to 0.885 are present across this region. ({\\it Right}) We can see that the low fitting factors occur for systems with more extreme mass ratios. This indicates that the bank is not optimized to recover signals from highly asymmetric systems.}\n\\end{figure*}\n\nIn Figures~\\ref{fig:bbhff} and~\\ref{fig:imbhff}, we can see the fitting factors in $M$-$\\chi_\\mathrm{eff}$ plane for BBH and IMBH aligned-spin SEOBNRv4\\_ROM waveform approximants~\\cite{SEOBNRv4ROM} and as a function of mass ratio for IMBH non-spinning EOBNRv2HM waveform approximants~\\cite{EOBNRv2HM}. For the recovery of aligned-spin SEOBNRv4\\_ROM waveform approximants, the majority of fitting factors are above 0.97. The bank is constructed with SEOBNRv4\\_ROM waveforms in the high mass region so fitting factors are expected to be at least as high as the required fitting factor of 0.97. We note that fitting factors fall off for high $\\chi_\\mathrm{eff}$ although coverage in this region is still high. Non-spinning EOBNRv2HM waveform approximants with higher-order modes can also be recovered by the search in the IMBHB region, despite template waveforms not including higher-order mode effects. The fitting factors have a dependency on mass ratios, as higher-order modes become more significant at higher mass ratios. Higher-order modes have higher frequency content and will be within the sensitive frequency band of LIGO and Virgo for IMBH signals. Hence it will become important to include templates containing higher-order modes in their waveforms, in order to increase the sensitivity of the search towards heavier mass systems~\\cite{IMBHHM}.\n\n\\begin{figure}[hbt!]\n\\includegraphics[width=0.45\\textwidth]{ban_sim_plots\/bbh_new_chiM_hexbin.png}\n\\caption{\\label{fig:bbhff} Fitting factors in $M$-$\\chi_\\mathrm{eff}$ plane for BBH aligned-spin SEOBNRv4\\_ROM waveform approximants~\\cite{SEOBNRv4ROM}.} \\end{figure}\n\n\\begin{figure*}[t]\n\\centering\n\\begin{minipage}[b]{0.47\\textwidth}\n\\centering\n \\includegraphics[width=\\textwidth]{ban_sim_plots\/imbh_noHM_chiM_hexbin.png}\n \\end{minipage}\n \\begin{minipage}[b]{0.47\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{ban_sim_plots\/imbh_HM_qVsFF.png}\n \\end{minipage}\n\\caption{\\label{fig:imbhff} Fitting factors in $M$-$\\chi_\\mathrm{eff}$ plane for IMBH aligned-spin SEOBNRv4\\_ROM waveform approximants~\\cite{SEOBNRv4ROM} ({\\it left}) and as a function of mass ratio for IMBH non-spinning EOBNRv2HM waveform approximants~\\cite{EOBNRv2HM} ({\\it right}). ({\\it Left}) The majority of fitting factors are above 0.97 for the recovery of aligned-spin SEOBNRv4\\_ROM waveform approximants. The bank is constructed with SEOBNRv4\\_ROM waveforms in the high mass region so fitting factors are expected to be at least as high as the required fitting factor of 0.97 to ensure that no more than $\\sim10 \\%$ of possible astrophysical signals are lost due to the discrete nature of the bank. We note that fitting factors fall off for high $\\chi_\\mathrm{eff}$ although coverage in this region is still high. ({\\it Right}) Non-spinning EOBNRv2HM waveform approximants with higher-order modes can also be recovered by the search in the IMBHB region, despite template waveforms not including higher-order mode effects.}\n\\end{figure*}\n\n\n\n\\section{Introduction}\\label{sec:intro}\n\t\\input{intro.tex}\n\n\\section{Design and construction of the O2 bank} \\label{sec:design}\n\t\\input{design.tex}\n\n\\section{Conclusion} \\label{sec:conclusion}\n\t\\input{conclusion.tex}\t\n\n\\section{Acknowledgements} \\label{sec:ack}\n\nWe thank the LIGO-Virgo Scientific Collaboration for access to data. LIGO was\nconstructed by the California Institute of Technology and Massachusetts\nInstitute of Technology with funding from the National Science Foundation (NSF)\nand operates under cooperative agreement PHY-0757058. We thank Satya Mohapatra for the helpful comments and suggestions. We also thank Graham Woan for helping with the review of the template bank used for O2. We gratefully acknowledge the support by NSF grant PHY-1626190 for the UWM computer cluster and PHY-1607585 for JC, PB, DC and DM. SC is supported by the research programme of the Netherlands Organisation for Scientific Research (NWO). SS was supported in part by the Eberly Research Funds of Penn State, The Pennsylvania State University, University Park, PA 16802, USA. HF was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). CH was supported in part by the NSF through PHY-1454389. Funding for this project was provided by the Charles E. Kaufman Foundation of The Pittsburgh Foundation. TGFL was partially supported by a grant from the Research Grants Council of the Hong Kong (Project No. CUHK 14310816 and CUHK 24304317) and the Direct Grant for Research from the Research Committee of the Chinese University of Hong Kong. MW was supported by NSF grant PHY-1607178. This paper carries LIGO Document Number LIGO-P1700412. \n\n\\input{hyperbank.bbl}\n\n\\end{document}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nData flows from new and planned astronomical survey telescopes are steadily increasing. This shows no sign of stopping, with LSST starting operations in \\raise.17ex\\hbox{$\\scriptstyle\\mathtt{\\sim}$} 2020. There is clearly a need for accurate, fast, automated classification of photometric lightcurves to maximise the scientific returns from these surveys. Even when later spectroscopic followup is required, finding which targets to prioritise is a necessary first step. \n\nThe literature contains multiple examples of such classification, using a wide variety of techniques. These include a variety of supervised machine learning applications \\citep[e.g.][]{Eyer:2005ce,Mahabal:2008if,Blomme:2010bq,Debosscher:2011kz,Brink:2013hv,Nun:2014kv}. Recently Random Forests (RF) have begun to gain popularity, due to their robustness and applicability to different sets of data, extracted lightcurve properties, and classification schemes \\citep[e.g.][]{Richards:2011ji,Richards:2012ea,Masci:2014bk}. Several improvements have been proposed, in areas such as parametrising lightcurves with maximal information retention \\citep{Kugler:2015jq}, and adjusting for training set deficiencies \\citep{Richards:2011bn}. One method of \\emph{unsupervised} machine learning is a Kohonen Self-Organising-Map \\citep[SOM, ][]{Kohonen:1990fd} demonstrated by \\citet{Brett:2004cr} in an astronomical context. Here we adopt a novel technique based on a combination of SOM and RF machine learning. SOMs can efficiently parametrise lightcurve shapes without resorting to specific lightcurve features, and RFs are capable of placing objects into classes.\n\n\nIn this work we apply these techniques to data from the K2 mission, the repurposed \\emph{Kepler} satellite \\citep{Borucki:2010dn}. K2 and its predecessor \\emph{Kepler} have left a lasting mark in studies of variable stars, showing that most $\\delta$ Scuti and $\\gamma$ Dor stars show pulsations in both the p-mode and g-mode frequency regimes \\citep{Grigahcene:2010kd}. Many studies have been performed on \\emph{Kepler} variable stars \\citep[e.g.][]{Blomme:2010bq,Balona:2011kw,Balona:2011gn,Debosscher:2011kz,Uytterhoeven:2011jv,Tkachenko:2013jr,Bradley:2015ep}, but few so far on K2. \\citet{Balona:2015jh} studied B star variability in \\emph{Kepler} and K2, and found that K2 data presented some new challenges from the original mission. Despite these, it has for example discovered the several RR Lyrae stars known outside our own Galaxy \\citep{Molnar:2015tr}. \\citet{LaCourse:2015jr} have also produced a catalogue of eclipsing binary stars in K2 field 0.\n\nThe initial version of this catalogue \\citep{Armstrong:2015bn} classified several thousand K2 variable stars in K2 fields 0 and 1. This classification was based on an interpretation of lightcurve periodicity, and split objects into Periodic, Quasiperiodic, and Aperiodic variables. Here we improve on this initial work, by applying an automated technique to classify variables into more usual classes. We extend the classification to K2 fields 0--4, and will release updates as more K2 fields become available.\n\n\\section{Data}\n\\subsection{Source}\nData are taken from the K2 satellite \\citep{Howell:2014ju}. K2 is the repurposed \\emph{Kepler} mission, and provides lightcurve flux measurements at a 30 minute `long' cadence continuously for 80 days per target. Targets are organised into campaigns, with each campaign spanning an \\raise.17ex\\hbox{$\\scriptstyle\\mathtt{\\sim}$} 80 day period and covering several thousand objects. A much smaller number of targets (a few tens per campaign) are available at the `short' cadence of \\raise.17ex\\hbox{$\\scriptstyle\\mathtt{\\sim}$} 1 minutes. For the purposes of this work, we restrict ourselves to long cadence data only, to preserve uniformity in the data. At the time of writing, 5 campaigns had been released to the public (covering fields 0--4), with more due as the mission continues. Four of these campaigns cover \\mytilde80 days, with the first campaign 0 covering \\mytilde40 days. We take data for these campaigns from the Michulski Archive for Space Telescopes (MAST) website\\footnote{https:\/\/archive.stsci.edu\/k2\/}, limiting ourselves to objects classified as stars in the MAST catalogue. This cut primarily removes a small number of solar system bodies and extended sources from the analysis. At this point, we have 68910 object lightcurves.\n\nFor the purposes of training the classifier, we also use data from the original \\emph{Kepler} mission. In these cases a single quarter of long cadence \\emph{Kepler} data is randomly selected. This covers \\raise.17ex\\hbox{$\\scriptstyle\\mathtt{\\sim}$} 90 days, and hence is similar to a single K2 campaign in duration and cadence. \\emph{Kepler} does however have different noise properties than K2, particularly in regards to the \\raise.17ex\\hbox{$\\scriptstyle\\mathtt{\\sim}$} 6 hour thruster firing, which is present in K2 but not in \\emph{Kepler}. \\emph{Kepler} data was also downloaded from MAST, and the Presearch-Data-Conditioning (PDC) detrended lightcurves \\citep{Stumpe:2012bj,Smith:2012ji} used.\n\n\\subsection{Extraction and Detrending}\n\\label{sectExtDet}\nK2 data shows instrumental artefacts not previously seen in the original \\emph{Kepler} mission. The strongest of these is a signal at \\raise.17ex\\hbox{$\\scriptstyle\\mathtt{\\sim}$} 6 hours, which is the timescale on which the satellite thrusters are fired to adjust the spacecraft pointing. This pointing adjustment is necessary due to drift associated with the new mode of operations, and is explained fully in the K2 mission papers. It has the unfortunate effect of causing systematic noise, due to aperture losses and inter-pixel sensitivity changes. A number of techniques have been put forward for removing this noise \\citep{Vanderburg:2014bi,Aigrain:2015ew,Lund:2015cs}, including one in the previous version of this catalogue \\citep{Armstrong:2015bn}. Each has advantages and disadvantages; our experience has been that while overall most techniques perform comparably, for individual objects the differences can be large. We use an updated version of our own extraction and detrending method here, which is fully described in \\citet{Armstrong:2015bn}. The only change from that publication is the performing of a polynomial fit to the lightcurve, prior to detrending. This fit is performed by considering successive 0.3 day long regions of the lightcurve, and fitting third degree polynomials to 4 day regions centred on these. Outlier points more than $10\\sigma$ from the initial fit are masked, and the fit redone without these points. The $10\\sigma$ masking and refitting is repeated for 10 iterations. Masked points are not cut from the final lightcurves. The final fit is removed, detrending is performed, and the fit then added back in. This step was added to improve preservation of variability signals, a notable improvement on the first method. Lightcurves detrended using this method are publicly available at the MAST website.\n\nIt is important to note that, as described in \\citet{Armstrong:2015bn}, our detrending method works best when performed separately on each half of the lightcurve (the exact split can be a few days from the precise halfway time). This is due to a change in the pointing characteristics of the spacecraft near the middle of each campaign, possibly the result of a change in orientation to the Sun. The precise times used to split the data are given in Table \\ref{tabsplittimes}. Before conducting the analysis presented later in this work, we normalise each lightcurve half by performing a linear fit.\n\n\\begin{table}\n\\caption{Times of pointing characteristic change, used to split the K2 data before detrending}\n\\label{tabsplittimes}\n\\begin{tabular}{lr}\n\\hline\nCampaign & Split Time \\\\\n & BJD - 2454833 \\\\\n\\hline\n0 & N\/A \\\\\n1 & 2016.0\\\\\n2 & 2101.41\\\\\n3 & N\/A \\\\\n4 & 2273.0\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nWith the release of campaign 3, the K2 mission team began to release its own detrended lightcurves (these are not available for earlier campaigns at the time of writing). Similarly to the other methods, we find that these perform well overall but are by no means the best choice for every object. We will apply the classifier to both our lightcurves (hereafter the `Warwick' set) and the K2 team lightcurves (hereafter the `PDC' set) for campaigns 3--4. The comparison is complicated by the fact that the above mentioned change in pointing characteristics does not occur in the usual way for these campaigns. Rather than change once in the middle of the campaign, in campaign 3 the change occurs twice, at roughly one third intervals. We do not adjust our detrending method for this, as introducing the option for another split adds an additional layer of complexity, and reduces the number of points available in each section (a risky option, as these points form the base surface used to decorrelate flux from pointing). Instead we perform the detrending with no split at all. For campaign 4, we split at time 2273 (BJD-2454833, as given in the K2 data files), and cut points up to the first change in pointing at 2240.5. This shortens each campaign 4 lightcurve by 11 days, but results in improved detrending. We do not perform such an adjustment for campaign 3 as even more data would need to be cut.\n\n\n\n\\section{Classification}\n\n\\subsection{Methodology}\nWe employ a classification scheme using two distinct components. These are Self-Organising-Maps (SOMs), otherwise known as Kohonen maps, and a Random Forest (RF) classifier. Each is described below.\n\n\\subsubsection{SOM}\n\nSOMs have been tested in an astrophysical context before \\citep{CarrascoKind:2014gb,Torniainen:2008cc,Brett:2004cr}, but are rarely to date applied in astronomy in practice. As such we outline their methodology here.\n\nA SOM is a form of dimensionality reduction; data consisting of multiple pieces of information can be condensed into a pre-defined number of dimensions, and is grouped together according to similarity. In our case, the SOM takes phase folded lightcurve shapes and groups similar shapes into clusters, in one or two dimensions. The great strength of a SOM is in the unsupervised nature of its clustering algorithm. The user need not specify what groups or labels to look for; any set of similar input data, including for example previously unseen variability classes, will form a cluster in the resulting map. Similar clusters will lie near each other, those that are the same according to the input data will overlap. Furthermore, the input parameters for the algorithm are quite insensitive to small variations, making the clustering process robust \\citep{Brett:2004cr}.\n\nThe key component of a SOM is the Kohonen layer. This can be N-dimensional, but we will consider 2D layers here for clarity. The layer consists of pixels, each of which represents a template against which the input data is compared. The size of the layer is unimportant as long as it is sufficiently large to express the variation present in the input data. Once trained on a set of data, the Kohonen layer becomes a set of templates, representing the observed data features that it was trained on. These templates can be examined to spot interesting features in the data set, such as variation within an already known class. Further data (or the original data itself) can be compared to the trained layer and the closest matching template found. In this way, an object is placed onto the map. \n\nThe specific implementation of SOMs used here is described in Section \\ref{sectSOMtraining}, with an example of their use shown. The result is a map against which any input K2 phase-folded lightcurve can be compared. The location of the lightcurve on the map gives us its similarity to certain shapes, such as the distinctive lightcurve of an eclipsing binary star.\n\n\\subsubsection{Random Forest}\nThe SOM allows us to classify and study the \\emph{shape} of a given phase curve, and the sets of similar shapes found within a dataset. It does not place an object into a specific variability class. For that we utilise a RF classifier \\citep{Breiman:fb}. These have been used in a number of previous variable star studies cited above. To use a RF classifier the lightcurve must be broken down into specific features, which represent the data (see Section \\ref{sectdatafeatures} for those used here). These features are then paired with known classes in a training set of known variables, and the classifier fit to this set. For a given object, the RF classifier can then map sets of features to probabilities for class membership, giving the likelihood for an unclassified object to be in each class.\n\nRF classifiers are ensemble methods, in that they give results based on a large sample of simple estimators, in this case decision trees. In this way they can reduce bias in estimation. The core components of an RF are these decision trees. See \\citet{Richards:2011ji} for a concise discussion of the underlying trees and how they are constructed. The specific parameters and implementation used here are discussed in Section \\ref{sectRFimplement}.\n\n\\subsection{Automated period finding}\n\\label{sectautoper}\nOur classification methodology relies heavily on the phase-folded lightcurves of our targets. This requires knowledge of the target's dominant period. Such knowledge is available for some known variables, but not for the general K2 sample at the time of writing. As such we use the K2 photometry to determine frequencies for each target.\n\nThere are a number of methods popularly used for determining lightcurve frequencies. The most common is the Lomb-Scargle (LS) periodogram \\citep{Lomb:1976bo,Scargle:1982eu}, which performs a fit of sinusoids at a series of test frequencies. Other available methods include the autocorrelation function \\citep[ACF, see e.g.][]{McQuillan:2014gp} and wavelet analyses \\citep{Torrence:1998wk}. We use LS here, due to its provenance and simplicity of implementation. The same arguments can be made for the ACF, which for stellar rotation periods has been shown to be more resilient than LS at detecting dominant frequencies \\citep{McQuillan:2013df}. However we find removing unwanted power from frequencies and harmonics, and detecting multiple frequencies from the same lightcurve, to be simpler for the LS method, at least in the implementations that we had available. In future utilising the ACF alone or in combination with the LS may be possible.\n\nWe use the fast LS method of \\citet{Press:1989hb}, with an oversampling factor of 20 run up to our Nyquist frequency of 24.5\\ d$^{-1}$. To avoid excessive human interference (and maintain the `automated' status of this classification), the dominant frequencies for a target must be found without supervision. To avoid frequencies commonly associated with thruster firing noise in K2 (see Section \\ref{sectExtDet}) we remove frequencies within 5\\% of $4.0850$d$^{-1}$ and their $1\/2$, 1st, 2nd, 3rd and 4th harmonics from the periodogram, by removing the best fitting sinusoid of form\n\n\\begin{equation}\n\\label{eqnLSmodel}\n z = a\\sin(2\\pi ft)+b\\cos(2\\pi ft)+c\n\\end{equation}\n\n\\noindent at each of these frequencies. In this model $f$ represents the frequency being removed, $t$ and $z$ the time and flux data, and $a$, $b$ and $c$ free parameters of the model. We then cut these frequencies altogether before extracting the dominant period. We also remove frequencies associated with the data cadence which commonly show power in our periodogram ($48.94355819$d$^{-1}$ and $20.394709$d)$^{-1}$) and their $1\/2$ frequency harmonics, by similarly fitting and removing a sinusoid at these frequencies and then cutting the frequencies from the periodogram. We did not find it necessary to remove other harmonics of the data cadence frequencies, as doing so provided little improvement. Finally periods above 20\\ d (10\\ d in campaign 0) are cut, as the data baseline is not long enough to reliably determine them without the introduction of spurious noise related frequencies. At this point the most significant peak in the LS periodogram is taken.\n\nTo extract other significant frequencies, we remove the dominant frequency using a fit of the model of Equation \\ref{eqnLSmodel}, then recalculate the LS periodogram, again ignoring thruster firing and cadence related frequencies as above. The remaining most significant peak is taken. To compare the power of different peaks, we calculate their amplitude $A$ using $A=\\left(a^2+b^2\\right)^{\\frac{1}{2}}$. This is used to produce the frequency amplitude ratios used later in this work. We repeat this process to extract a total of 3 frequencies from each lightcurve.\n\nA common weakness in period-finding algorithms occurs for eclipsing binary stars, a significant variability class. The LS periodogram often gives its highest power for half the true binary orbital period (i.e. when the primary and secondary eclipses occur at the same phase). This error is simple to spot by inspection, but harder to correct automatically. We account for this potential error source by introducing a check into the automated period finder. This phase folds each lightcurve at double its LS-determined dominant period. The phase folded lightcurve is then binned into 64 bins, and the bin values at the minimum bin and the lowest bin value between phases 0.45 and 0.55 from this minimum found. We perform two checks on these two bin values. If the initial period is correct, they should be the same. We first check for an absolute difference between the two, finding that 0.0025 in relative flux works well as a threshold. We further test that the difference between them is greater than 3\\% of the range of the un-phasefolded lightcurve. We calculate this range by taking the difference between the median of the largest 30 and median of the lowest 30 flux points in the lightcurve, to avoid unwanted outlier effects. If the difference between the two tested bin values is greater than both of these thresholds, the object period is doubled. If the doubled period would be over the 20\\ d upper period limit already applied (10\\ d for campaign 0), the doubling is not allowed. Similar adjustments have been made in previous variability studies \\citep[e.g.][]{Richards:2012ea}. Only the dominant extracted period may be adjusted in this way.\n\nTo test the efficacy of our automated period finding software, we trial it against a known sample of variable stars from the \\emph{Kepler} data. See Section \\ref{secttrainingset} for a full description of this set, which is also used a training set for our classifier. We use one randomly selected quarter of \\emph{Kepler} data, to give data with a similar baseline and cadence to a single K2 campaign. There are 2128 training objects with previously determined periods (after removing objects with periods below our Nyquist period of 0.0408d). Figure \\ref{figpercomp} shows the comparison between our dominant determined periods and the previously known ones. The acceptance rate is 70.3\\%, rising to 82.2\\% if half and double periods are included. In 90.8\\% of the sample, one of our 3 determined periods finds either the previously known period or its half or double harmonic. In the remaining lightcurves, we find that either the noise obscures the known period (due possibly to different quarters with differing noise properties being used by us and previous studies) or that the dominant period has changed.\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{periodcomparev2.pdf}}\n\\caption{Periods determined using our method compared to previously known period, for a set of known variables in \\emph{Kepler}. An acceptance rate of 70.3\\% is obtained, 82.2\\% if half and double periods are included, and 90.8\\% if second and third detected frequencies are included. Variables lying at the correct, half or double frequency are plotted as red stars.}\n\\label{figpercomp}\n\\end{figure}\n\n\n\n\n\\subsubsection{Phase curve template preparation}\nThe SOM element of our classifier requires phased lightcurve shapes to function. We create these using the periods determined in Section \\ref{sectautoper}. Each lightcurve is phase-folded on this period. For known training set objects (see Section \\ref{secttrainingset}), the literature period is used. Once phase-folded, the lightcurve is binned into 64 equal width bins, and the mean of each bin used to form the phase curve that will be passed to the classifier. The exact number of bins is unimportant, as long as it gives enough resolution to see any variability in the phase curve. \\citet{Brett:2004cr} used 32 bins and found satisfactory results, we use 64 as the performance decrease is small and it reduces the chances of missing rapidly changing variability such as eclipses.\n\nIt is essential that the phase curves be on the same scale and aligned, so that the classifier can spot similarities between them (see next Section). As such we normalise each phase curve to span between 0 and 1, and shift it so that the minimum bin is at phase 0. Each phase curve then consists of 64 elements, with the first being at (0,0).\n\n\n\\subsection{Training the SOM}\n\\label{sectSOMtraining}\nThere are variations in the literature on how precisely to train the SOM. Here we run through the procedure followed for this work. The input parameters are the initial learning rate, $\\alpha_0$, which influences the rate at which pixels in the Kohonen layer are adjusted, and the initial learning radius, $\\sigma_0$, which affects the size of groups. Initially each pixel is randomised so that each of its 64 elements lies between 0 and 1, as our phase curves have been scaled to this range. For each of a series of iterations, each input phase curve is compared to the Kohonen layer. The best matching pixel in the layer is found, via minimising the difference between the pixel elements and the phase curve. Each element in each pixel in the layer is then updated according to the expression\n\n\\begin{equation}\nm_{xy,k,new} = \\alpha e^\\frac{-d_{xy}^2}{2\\sigma^2} \\left(s_k - m_{xy,k,old}\\right)\n\\end{equation}\n\n\\noindent where $m_{xy,k}$ is the value $m$ of the pixel at coordinates $x$,$y$ and element $k$ in the phase curve, $d_{xy}$ is the euclidean distance of that pixel from the best matching pixel in the layer, and $s_k$ is the kth element of the considered input phase curve. This expression is specific to 2-dimensional SOMs, but can be easily adapted for 1-dimension by setting the size of the second dimension to be 1. Note that distances are continued across the Kohonen layer boundaries, i.e. they are periodic. Once this has been performed for each phase curve, $\\alpha$ and $\\sigma$ are updated according to\n\n\\begin{equation}\n\\label{eqnsigmadecay}\n\\sigma = \\sigma_0 e^{\\left(\\frac{-i*log(r)}{n_\\textrm{iter}}\\right)}\n\\end{equation}\n\\begin{equation}\n\\label{eqnalphadecay}\n\\alpha = \\alpha_0 \\left( 1 - \\frac{i}{n_\\textrm{iter}} \\right)\n\\end{equation}\n\n\\noindent where $i$ is the current iteration, and $r$ is the size of the largest dimension of the Kohonen layer. This is then repeated for $n_\\textrm{iter}$ iterations. \n\nIt is possible to use different functional forms for the evolution of $\\alpha$ and $\\sigma$; typically a linear or exponential decay is used. \\citet{Brett:2004cr} found that the performance of the SOM was largely unimpeded by the choice of form or initial value, as long as the learning rate does not drop too quickly. We find satisfactory results for the expressions above and values of $\\alpha_0=0.1$ and $\\sigma_0=r$, as can be seen in the below example. The code used in this study was initially adapted from the \\texttt{SOM} module of the open source \\texttt{PyMVPA} package\\footnote{http:\/\/www.pymvpa.org}\\citep{Hanke:2009bm}, and has now been contributed as an update to that package by the authors. As such any readers wishing to use this code should look to the given reference. Note that the functional form of Equations \\ref{eqnsigmadecay} and \\ref{eqnalphadecay} are slightly different in the online version of the code, to preserve compatibility with older versions of the module. The formulae described here are the ones used in this work.\n\nAs an example we train a SOM on the K2 data from campaigns 0-2, as well as \\emph{Kepler} data used for training the classifier (see Section \\ref{secttrainingset} for a full description of the data set). We use a 40x40 Kohonen Layer. K2 data was only used if the range of variation in the phase curve before normalisation was greater than 1.5 times the overall mean of the standard deviations of points falling in each phase bin (see previous Section). This cut was imposed to avoid essentially flat lightcurves from impacting the SOM, removing \\raise.17ex\\hbox{$\\scriptstyle\\mathtt{\\sim}$} 40\\% of the K2 lightcurves. The majority of these were classified as 'Noise' or 'AP' in \\citet{Armstrong:2015bn}, showing that we are not removing many periodically varying sources. We note that the SOM is robust enough to work without this cut, and it is imposed only to increase the purity of the training set.\n\nWe take the known \\emph{Kepler} variables, along with `OTHPER' other periodic and quasi-periodic objects from K2, and plot them on the resulting SOM in Figure \\ref{figsommap}. Clear groups can be seen, with eclipsing binary types well differentiated but bordering each other, as would be expected. RR Lyraes are very well grouped, and $\\delta$ Scuti variables cluster but more weakly. Example templates from the Kohonen layer are shown in Figure \\ref{figsomtemplates}, representing the major clusters seen. Note that the size of a group is determined by a number of factors, including the number of input objects matching it, and the extent of small variations within the group. As there are many more sinusoidal variables than eclipsing binaries or RR Lyraes, the $\\delta$ Scuti, $\\gamma$ Doradus and `OTHPER' groups fill most of the map. Different regions within these groups show for example slight skews from a pure sinusoid, and may represent interesting intra-class differences. $\\delta$ Scutis lying near the eclipsing binary groups have likely been mapped using double their true period, and so look similar to a contact binary star. They may also have been previously misclassified. It is also interesting to see that $\\delta$ Scutis and `OTHPER' objects overlap, as would be expected given that their phase curve shapes are not particularly distinctive to their respective classes. `OTHPER' objects also overlap with the RR Lyrae cluster, and likely mark out newly discovered RR Lyrae stars.\n\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{SOMmapv2.pdf}}\n\\caption{Known variables placed onto a SOM. Random jitter within each pixel has been added for clarity. Green triangles = 'EA' (detached eclipsing binaries), red crosses = 'EB' (semi-detached and contact eclipsing binaries), pink stars = 'RRab' (ab-type fundamental mode RR Lyraes), blue circles = 'DSCUT' ($\\delta$ Scuti variables), black dots = 'GDOR' ($\\gamma$ Dor variables) and yellow pluses = `OTHPER' (other periodic and quasi-periodic objects). See Section \\ref{sectclassscheme} for more detail on these variability classes.}\n\\label{figsommap}\n\\end{figure}\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{SOMtemplates.pdf}}\n\\caption{Template phase curves from the Kohonen layer of the SOM in Figure \\ref{figsommap}. Clockwise from top left, templates are for pixel [13,34] (EA), [6,32] (EB), [37,35] (RRab) and [25,19] (DSCUT). See Section \\ref{sectclassscheme} for a description of the classes. Note that templates do not have to span the range 0-1, even if the input phase curves do. Note also that all these templates were found from initially random pixels without any human guidance or input.}\n\\label{figsomtemplates}\n\\end{figure}\n\nThe SOM used for final classification is the same as that described above, but using only one dimension of 1600 pixels. This produces the same clustering results, but is less useful for visualisation. We use only one dimension so that the other part of our classifier (the RF) can more easily make use of the information contained within the SOM.\n\n\n\n\\subsection{Data Features}\n\\label{sectdatafeatures}\nFor the classification of variables into classes, we use a number of specific features of each lightcurve. This is common practice in general classification problems \\citep[e.g.][]{Richards:2011ji}. However, there is a subjective element to selecting features, and it can be desirable to minimise this if possible \\citep[see e.g.][]{Kugler:2015jq}. We do so through the use of the SOM. This encodes the shape of the phase curve into one parameter (the location of the closest pixel in the SOM to the lightcurve in question), rather than a series of features, none of which may capture the desired shape properties.\n\nThere are however other features which are useful and which are uninformed by the SOM. A key example is the dominant (most significant) period of the lightcurve. Other significant frequencies can also be used, and in some cases many more have been studied. We only use the three most significant periods here.\n\nThe full range of features used is described in Table \\ref{tabfeatures}. These features are incorporated largely to separate out lightcurves which show purely noise, something which is generally uninformed by the SOM, as well as those without one particularly dominant frequency. We take the potentially controversial step of adjusting some of the noise related features between the \\emph{Kepler} and K2 datasets, due to the differing noise properties between each set. This is unavoidable here, as the scatter and increased noise in K2 causes catastrophic errors in the classifier if \\emph{Kepler} lightcurves are used as they come. In this case the general result is that the vast majority of K2 objects are classified as Noise. This problem is solved by multiplying the marked features in Table \\ref{tabfeatures} by a factor to align their median values with those of K2. These features are those driven primarily by dataset noise, rather than those associated with periodicity (noise-related periodicity is assumed to have been removed by the procedure in Section \\ref{sectautoper}). As the \\emph{Kepler} data used all comes from known variable stars, the median of the features is not strictly comparable to K2, where the data comes from the whole target list. As such we set the multiplication factor so that the median of the non-eclipsing binary \\emph{Kepler} data features is increased to equal the median of the `OTHPER' K2 data features. Eclipsing binaries are left alone, as their features are in our case dominated by the binary eclipses.\n\nA similar problem arises when studying the PDC lightcurves. These have different characteristics to the Warwick lightcurves. Assuming that the intrinsic distribution of stellar variability should be the same across fields, this difference is due to the differing detrending methods. We adjust for it in the same way and to the same features as above, marked in Table \\ref{tabfeatures}. As we do not have prior classifications for fields 3--4, the factor is applied to the whole dataset, and set so as to match the medians of these features between the PDC campaigns 3 and 4 and the Warwick campaigns 0--2. Each PDC campaign is adjusted separately.\n\nIt would be desirable to use colour information as a feature to aid classification of variability types connected to specific stellar spectral types. However, colours are not uniformly available for the K2 sample, although some can be found through a cross-match with the TESS input catalog \\citep{Stassun:2014wz}. As such we do not use them, as doing so would mean large fractions of the K2 targets would need to be disregarded. This has consequences for the variability classes we use, see Section \\ref{sectclassscheme}.\n\n\n\\begin{table}\n\\caption{Data Features}\n\\label{tabfeatures}\n\\begin{tabular}{ll}\n\\hline\nFeature Name & Description \\\\\n\\hline\nperiod & Most significant period (Section \\ref{sectautoper})\\\\\namplitude & Max - min of phase curve\t\t\t\\\\\nperiod\\_2 & Second detected period (Section \\ref{sectautoper}) \\\\\nperiod\\_3 & Third detected period (Section \\ref{sectautoper}) \\\\\nampratio\\_21 & period\\_2 to period amplitude ratio \\\\\nampratio\\_31 & period\\_3 to period amplitude ratio \\\\\nSOM\\_index & Index of closest pixel in 1D SOM\t\t\\\\\nSOM\\_distance & Euclidean distance to closest pixel \t\t\t\\\\\n & in 1D SOM \\\\\np2p\\_98perc $^a$& 98th percentile of point to point scatter\\\\\n & in lightcurve\t\t\t\\\\\t\np2p\\_mean $^a$& Mean of point to point scatter in lightcurve \\\\\nphase\\_p2p\\_max & Maximum point to point scatter in binned\\\\\n & phase curve\t\t\t\\\\\nphase\\_p2p\\_mean & Mean of point to point scatter in binned \\\\\n & phase curve\t\t\t\\\\\nstd\\_ov\\_err $^a$& Whole lightcurve standard deviation over \\\\\n &mean point error\t\t\t\\\\\n\\hline\n\\multicolumn{2}{l}{$^a$ adjusted between datasets, see text.}\n\\end{tabular}\n\\end{table}\n\n\\subsection{Classification Scheme}\n\\label{sectclassscheme}\nAn important decision is in which variability classes to use. We experimented with classifying RR Lyrae (subtype ab), $\\delta$ Scuti, eclipsing binary (split into detached, subtype EA, and semi-detached or contact, subtype EB), $\\gamma$ Dor, and so-called ROT variables, a class applying to likely rotationally modulated lightcurves seen in \\citet{Bradley:2015ep}. We also attempted to split the $\\gamma$ Dor class into symmetric, asymmetric, and 'MULT' classes, as defined in \\citet{Balona:2011kw}. This approach had varied success; RR Lyrae ab, $\\delta$ Scuti, $\\gamma$ Dor and eclipsing binary classes performed well, but we found that the $\\gamma$ Dor subtypes were not well constrained by our available features. This may be because we lack sufficient training objects to reliably map the range of features offered by these subtypes. This problem could be navigable when an increased sample of objects is available through K2, and we plan to address this in later work. \n\nSimilarly, we found that the 'ROT' class was not very coherent - the classifier struggled to identify regions in parameter space corresponding to these variables. This likely arises due to the tendency of this class to have an indistinct cluster of low frequency peaks rather than one clear signal \\citep{Bradley:2015ep}. Rather than use the ROT class by itself, we make use of the previous version of this catalogue, which contained a `QP' quasiperiodic variable class. This class contains a number of variable types, but is characterised by periodic variability that is not strictly sinusoidal, and changes in amplitude and\/or period. We use this as a variable classification, to catch interesting variables of astrophysical origin which are not one of the five other classes (RR Lyrae ab, EA, EB, $\\delta$ Scuti, $\\gamma$ Dor). It is likely dominated by spot-modulated stars, but also contains other variables such as Cepheids. We rename this class to `OTHPER' for `other periodic' to avoid confusion, as variables which are strictly periodic but not in another class can be classified by this group.\n\nWe considered including other variable classes, such as Cepheids, the other RR Lyrae subtypes (first-overtone or multimode RR Lyraes), and Mira variables. We could not find sufficient training set objects in any of these classes (less than 20 in each case). While it is possible to attempt classification with small training sets, rather than present a weak or unreliable classification for these classes we prefer to wait for more K2 data. As more fields are observed, more training set objects will become available. We intend to include more classes in future versions of this catalogue.\n\nFinally, we include 'Noise', non-variable lightcurves, as a class label. This leave 7 classes, DSCUT ($\\delta$ Scuti), GDOR ($\\gamma$ Doradus), EA (detached eclipsing binaries), EB (semi-detached and contact eclipsing binaries), OTHPER (other periodic and quasi-periodic variables), RRab (RR Lyrae ab type) and Noise. It is important to note that as we do not have colour information, there will be degeneracy in the DSCUT class between true $\\delta$ Scutis and $\\beta$ Ceph variables, as in \\citet{Debosscher:2011kz}. This is also true for slowly pulsating B stars, which are degenerate with $\\gamma$ Dor variables.\n\n\\subsection{Training Set}\n\\label{secttrainingset}\nAlthough the SOM described is unsupervised and so requires no training set, the RF classifier we use for final classification does. An ideal training set would consist of a set of known variable stars from the K2 mission, to which we can fit the classifier. Some previous classification work on K2 has been done (for B stars \\citep{Balona:2015jh}, for eclipsing binaries \\citep{LaCourse:2015jr}, and in the previous version of this catalogue). These sources however suffer from either small numbers, only being applicable to a few variable types, or in the \\citet{Armstrong:2015bn} case using variability classes derived from the lightcurves rather than externally recognised types. We cross matched the observed K2 targets in fields 0--3 (4 was not available at that time) with catalogues of known variable stars, including those from AAVSO\\footnote{www.aavso.org}, GCVS \\citep{Samus:2009tf} and ASAS \\citep{Richards:2012ea}. This led to a small number of targets (a few tens of each class at best), not enough for a full training set. As such, we turned to the original \\emph{Kepler} mission. Much classification work has been done on the \\emph{Kepler} lightcurves. The data has differing noise properties to K2 data, but the same cadence, instrument, and if only one 90 day quarter of data is used a similar baseline to a K2 campaign.\n\nAlthough multiple works are available offering classified variable stars in \\emph{Kepler}, we limit ourselves to a small number of relatively large scale catalogues, in order to maintain homogeneity among classification methods and simplify the process. We began by taking the EA, EB, DSCUT classes from \\citet{Bradley:2015ep} We also took ROT, SPOTM and SPOTV, low frequency variables likely due to rotational modulation, reclassifying these objects as OTHPER. We supplemented the DSCUT set with those from \\citet{Uytterhoeven:2011jv}. The bulk of our eclipsing binary training set come from the Kepler Eclipsing Binary Catalogue \\citep{Prsa:2011dx,Slawson:2011fg}. We removed all heartbeat binaries \\citep{Thompson:2012ca} and those where the primary eclipse depth was less than 1\\%. A threshold of 1\\% was implemented in order to avoid shallow, likely blended binary eclipses from being included in the training set and hence increase training set purity. This also avoids the problem of noisy lightcurves with instrumental systematics of order a percent being misclassified as eclipsing binaries. Binaries were then classified as EA or EB based on a morphology threshold of 0.5 (see \\citet{Matijevic:2012di} for a discussion of morphology in this context). For RR Lyrae stars we use the list in \\citet{Nemec:2013bp}. Fundamental mode subtype ab stars were labelled RRab, and the first-overtone subtype c stars classified as OTHPER. To increase this relatively small RR Lyrae sample we used the results from the K2 AAVSO cross-match, taking fundamental mode RR Lyraes and adding them to the RRab training set. The B-star catalogue of \\citet{Balona:2015jh} was also used, with the SPB class reclassified as GDOR (given the degeneracy between GDOR and SPB present without temperature information) and the ROT class being reclassified as OTHPER.\n\nFor the OTHPER and Noise classes, we also use our previous catalogue. This contained 5445 OTHPER (QP in the original catalogue) and 29228 Noise objects in fields 0--2, with labels assigned by human eyeballing. To avoid having an excessive disparity between training set classes, we downsample this set to 1000 of each class, selected randomly, which are then added on to the \\emph{Kepler} OTHPER set above. This also makes the results on fields 0--2 more independent, as we can compare previously classified OTHPERs (the majority of which are now not in the training sample) with newly found ones. To reduce the impact of potential mistakes in the previous catalogue, we removed the small number of objects in the OTHPER training set which were in an initial run of this classifier reclassified as another class. Objects with a probability of being in the RRab class of greater than 0.2 were also removed, as the probabilities for the RRab class are not well calibrated (see Section \\ref{sectprobcal}). These cuts caught \\raise.17ex\\hbox{$\\scriptstyle\\mathtt{\\sim}$} 50 objects misclassified as OTHPER and \\raise.17ex\\hbox{$\\scriptstyle\\mathtt{\\sim}$} 30 objects misclassified as Noise out of the 1000 each initially selected. \n\nThe final classes and number of objects in each training set are shown in Table \\ref{tabtrainingset}.\n\n\n\n\\begin{table}\n\\caption{Training Set}\n\\label{tabtrainingset}\n\\begin{tabular}{lr}\n\\hline\nClass & N objects \\\\\n\\hline\nRRab & 91\\\\\nDSCUT & \t278\t\\\\\nGDOR & 233 \\\\\nEA & \t694\t\\\\\nEB & 759\t\\\\\nOTHPER & 1992 \\\\\nNoise & 976 \\\\\n\\hline\n\n\\end{tabular}\n\\end{table}\n\n\n\\subsection{Random Forest Implementation}\n\\label{sectRFimplement}\nWe use the implementation of RFs in the \\texttt{scikit-learn} \\texttt{Python} module\\footnote{http:\/\/scikit-learn.org\/stable\/}. There are several input parameters for an RF classifier. The key ones are the number of estimators, the maximum features considered at each branch in the component decision trees, and the minimum number of samples required to split a node on the tree, which controls how far each tree is extended. In a typical case, increasing the number of estimators always leads to improvement in performance but with decreasing returns and increasing computation time. The theoretical optimum maximum features for a classification problem is the square root of the total number of features, in our case 3. We optimise the parameters using the 'out-of-bag' score of the RF. When training, the classifier uses a random subset of the total data sample given to it for each tree, to reduce the chance of bias. The left out data is then used to test the performance of the tree -- its known class is compared to the predicted class, giving a performance metric between 0 (for absolute failure) and 1 (for perfect classification). Maximising this metric allows us to optimise the parameters. We find the best results for 300 estimators, a maximum of 3 features, and 5 samples to split a node. These parameters are used for classification. Additionally we apply weights to the training set, so that each class is inversely weighted according to its frequency in the training set (input option class\\_weight=`auto'). This makes sure that classes with more members (such as OTHPER and Noise) do not drown out other classes, and in effect imposes a uniform prior on the class probabilities.\n\nThere are several random elements in our method. These are the selection of the OTHPER and Noise training sets, as well as certain elements of the RF. Random subsets of training objects and features are selected for each decision tree as part of the RF method, to avoid bias. To minimise any effects of this randomness (especially the OTHPER and Noise selection), we train 50 classifiers with the above parameters and repeat the selection for each, applying each classifier to the K2 dataset. The average class probability across the classifiers gives the final result.\n \nTo explore the power of the SOM method, we trial the RF on only the SOM map location (SOM\\_index). The classifier is cross-validated by taking one training set member and training the classifier on the remaining members (so-called leave-one-out cross validation). The left out object is then tested on the classifier, and the process repeated for each member. The performance of the classifier is best described by a `confusion matrix', shown in Figure \\ref{figconfmatrixsom}. This shows what proportion of training members in each class were assigned to which other classes. In the ideal case each object is predicted correctly. Here we can see clearly which classes are well-informed by the SOM. RRab, EA, and EB classes are strongly recovered, as expected from their strong localisation in Figure \\ref{figsommap}. The DSCUT class is also recovered although less so. On the other side, OTHPER and Noise classes are found more weakly, and GDOR barely at all, due to the often multiple pulsation frequencies in this class combining to produce no distinctive phase curve shape. This demonstrates the power of the SOM alone to classify certain classes of variable stars. \n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{confmatrix_somonlyv2.pdf}}\n\\caption{Confusion matrix for a RF considering only SOM map location, generated using leave-one-out cross validation. Text shows the percentage of each sample which was classified into the relevant box. Correct classification lies on the diagonal.}\n\\label{figconfmatrixsom}\n\\end{figure}\n\n\nMoving on to the full classification scheme, we test the RF in a similar manner. All 7 classes are used, and the classifier cross-validated as before. The resulting confusion matrix is shown in Figure \\ref{figconfmatrix}. It highlights some interesting cases. Firstly, the classifier works well, with an overall success rate of 92.0\\%. There is some porosity between the two eclipsing binary classes, with objects of one class being placed into the other. As there is no rigid boundary in lightcurve shape between them, this is to be expected. Similarly there is some spread between OTHPER and Noise. This is not desirable, but the numbers involved are low, and represent objects with either variability only just emerging above the noise or objects with unusual noise properties. The biggest misclassification occurs between the GDOR and OTHPER classes. This arises due to the less distinct nature of the OTHPER class - it acts as a `catch-all' class to find any periodic or quasi-periodic variables which do not fit the other classes. GDOR objects can in some circumstances present similar lightcurve features to for example fast rotating stars, leading to some confusion between the classes.\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{confmatrixv2.pdf}}\n\\caption{Confusion matrix for a RF considering all features and classes, generated using leave-one-out cross validation. Text shows the percentage of each sample which was classified into the relevant box. Correct classification lies on the diagonal.}\n\\label{figconfmatrix}\n\\end{figure}\n\nOne advantage of RF classifiers is the ability to estimate feature importance. The classifier naturally measures which features have more descriptive power, through for example how often those features are used in the decision trees, or through the reduction in performance that would be observed is a feature was replaced by a randomly sampled distribution. This allows for model refinement, and is of great use in developing a classifier. We plot the importance of our features in Figure \\ref{figfeatimportance}. These are found through training the classifier 100 times, and extracting the mean and standard deviation of the feature importances for each classifier.\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{featimportancev2.pdf}}\n\\caption{Relative importance of features to the RF. Values and errors arise from the mean and standard deviation of the feature importances extracted from 100 trained classifiers.}\n\\label{figfeatimportance}\n\\end{figure}\n\n\n\n\\subsection{Class posterior probability calibration}\n\\label{sectprobcal}\nThe RF classifier automatically generates class probabilities (through the proportion of estimators classifying an object into each class). These probabilities are not necessarily accurate. Although it is true that higher class probability means more likelihood of an object being in that class, the probabilities can need calibrating to ensure that they are true posterior probabilities. This is where, if a set of objects have probability p that they are in a certain class, the same proportion of them actually are of that class.\n\nInitially we test the calibration of our `raw' class probabilities. Figure \\ref{figprobcal} shows the class probabilities found from the cross validated training set data created as described in Section \\ref{sectRFimplement}. This allows the predicted class probabilities for each training set object to be compared to their known classes. They are clearly not true posterior probabilities, especially for the RRab class, where essentially every object with class probability $>0.5$ is a true class member. For the other classes the given probabilities are closer, but still show some departure from the ideal case.\n\nOne common way of testing classifier performance in this way is the Brier score \\citep{BRIER:1950hg}. Our raw probabilities have a Brier score of 0.1336. We attempted a number of methods of calibrating them (and so reducing this score). The most usual methods are sigmoid and isotonic regression, which fit certain functions to the calibration curve to transform the probabilities. Similarly to \\citet{Richards:2012ea}, we find that these methods are not effective in our case. We attempted the method of \\citet{Bostrom:2008dt} to transform the initial class probabilities, but also found the results to be unsatisfactory. Rather than present an incomplete calibration, we give the class probabilities as they are. Users should be aware of this, and avoid interpreting class probabilities as true posterior probabilities.\n\n\n\n\n\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{probcalv2.pdf}}\n\\caption{Overall classifier predicted probability against true probability for the RRab class (crosses) and the average of all other classes (dots). The straight black dashed line represents the ideal case.}\n\\label{figprobcal}\n\\end{figure}\n\nAs the training set will not be representative of the true K2 distribution, biases may exist. As the priors are not well known, and the distribution of training sources by no means matches the underlying distribution of variables in K2, true posterior probabilities are impossible to create. Hence the given class probabilities, even if calibrated, would only be posterior probabilities under the assumption that each class has a uniform probability of arising.\n\n\n\\section{Catalogue}\n\n\\subsection{Overview}\nThe full catalogue for K2 fields 0--4 inclusive is given in Table \\ref{tabcatalogue}. This Table contains classifications using the Warwick lightcurves, as described in Section \\ref{sectExtDet}. The features used to classify these objects are given in Table \\ref{tabcatfeatures}. We also run the classifier on the PDC lightcurves produced by the K2 mission team. These were only available for campaigns 3--4. The resulting classifications are given in Table \\ref{tabcatalogueKTeam}, and their associated features in Table \\ref{tabcatfeaturesKTeam}.\n\n\\begin{landscape}\n\\begin{table}\n\\caption{Catalogue table for our Warwick detrended lightcurves. Fields 0--4 are included. Only an extract is shown here for guidance in form. The full table is available online.}\n\\label{tabcatalogue}\n\\begin{tabular}{lllllllllll}\n\\hline\nK2 ID & Campaign & Class & \\multicolumn{7}{c}{Class Probabilities} & Anomaly \\\\\n & & & DSCUT & EA & EB & GDOR & Noise & OTHPER & RRab & \\\\\n\\hline\n202059070 & 0 & Noise & 0.004195 & 0.120507 & 0.016615 & 0.005925 & 0.604636 & 0.246088 & 0.002034 & 0.023891\\\\\n .&.&.&.&.& . &.&.&.&.&. \\\\\n\\hline\n\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\caption{Data features for our Warwick detrended lightcurves. Fields 0--4 are included. Only an extract is shown here for guidance in form. The full table is available online.}\n\\label{tabcatfeatures}\n\\begin{tabular}{llllllllllll}\n\\hline\nK2 ID & Campaign & SOM\\_index & period & period\\_2 & period\\_3 & SOM\\_distance & phase\\_p2p\\_mean & phase\\_p2p\\_max & amplitude & ampratio\\_21 & ampratio\\_31 \\\\\n & & & d & d & d & & rel. flux & rel. flux & rel. flux & & \\\\\n\\hline\n202059070 & 0 & 1544 & 4.764370 & 1.241680 & 0.174448 & 1.180831 & 0.003801 & 0.487419 & 0.042283 & 0.629987 & 0.548721 \\\\\n .&.&.&.&.& . &.&.&.&.&. & . \\\\\n\\hline\np2p\\_mean & p2p\\_98perc & std\\_ov\\_err&&&&&&&&& \\\\\nrel. flux & rel. flux &&&&&&&&&& \\\\\n\\hline\n0.016326 & 0.047548 & 1.310764&&&&&&&&& \\\\\n\n .&.&.&.&.& . &.&.&.&.&. & . \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\caption{Catalogue table for PDC detrended lightcurves. Fields 3--4 only. Only an extract is shown here. The full table is available online.}\n\\label{tabcatalogueKTeam}\n\\begin{tabular}{lllllllllll}\n\\hline\nK2 ID & Campaign & Class & \\multicolumn{7}{c}{Class Probabilities} & Anomaly \\\\\n & & & DSCUT & EA & EB & GDOR & Noise & OTHPER & RRab & \\\\\n\\hline\n205889250 & 3 & Noise & 0.000067 & 0.000000 & 0.000000 & 0.000030 & 0.966544 & 0.033359 & 0.000000 & 0.000000\\\\\n\n .&.&.&.&.& . &.&.&.&.&. \\\\\n\\hline\n\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\caption{Data features for PDC detrended lightcurves. Fields 3--4 only. Only an extract is shown here. The full table is available online.}\n\\label{tabcatfeaturesKTeam}\n\\begin{tabular}{llllllllllll}\n\\hline\nK2 ID & Campaign & SOM\\_index & period & period\\_2 & period\\_3 & SOM\\_distance & phase\\_p2p\\_mean & phase\\_p2p\\_max & amplitude & ampratio\\_21 & ampratio\\_31 \\\\\n & & & d & d & d & & rel. flux & rel. flux & rel. flux & & \\\\\n\\hline\n205889250 & 3 & 0630 & 19.754572 & 12.803889 & 2.281881 & 1.179035 & 0.003795 & 0.421976 & 0.008715 & 0.741302 & 0.592596\\\\\n\n .&.&.&.&.& . &.&.&.&.&. & . \\\\\n\\hline\np2p\\_mean & p2p\\_98perc & std\\_ov\\_err&&&&&&&&& \\\\\nrel. flux & rel. flux &&&&&&&&&& \\\\\n\\hline\n0.005249 & 0.017133 & 1.371857&&&&&&&&& \\\\\n\n .&.&.&.&.& . &.&.&.&.&. & . \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\end{landscape}\n\nThe total number of objects found in each class is given in Table \\ref{tabnclass}, at various probability cuts. Note that for RRab class objects in particular, most objects with class probability $>0.5$ are real classifications. In the other cases the probability calibration is better, but these probabilities should still not be interpreted as posterior probabilities.\n\n\\begin{table}\n\\caption{Total objects in each class.}\n\\label{tabnclass}\n\\begin{tabular}{lllll}\n\\hline\nClass & Total & Prob $>0.5$ & Prob $> 0.7$ & Prob $> 0.9$ \\\\\n\\hline\nRRab & 248 & 154 & 72 & 25 \\\\\nDSCUT & 750 & 562 &377 & 166\t\\\\\nGDOR & 451 & 264 & 133 & 37\t\\\\\nEA & 607 & 308 & 183 & 99 \t\\\\\nEB & 463 & 392 & 290 & 186 \t\t\\\\\nOTHPER & 22428 & 18698 & 9399 & 3547 \\\\\nNoise & 43963 & 38609 & 21210 & 6018 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\nWe find that the classifier works well on all fields. The RRab class performs well throughout, due to the distinctive shape of their phasecurves. These are well characterised by the SOM. There are however some distinct features unique to fields 3 and 4. The EA class has a tendency to pick up noise dominated lightcurves in these fields, primarily because their point to point scatter is much higher than in fields 0--2. In these cases the class probability, although highest for EA, is still relatively low however. Similarly for DSCUT objects, there are a higher proportion of objects in these fields with many anomalous points, possibly due to flaring or instrumental noise. These points can cause biases in the phase curve, resulting in an artificial sinusoid, which when combined with a short period results in a DSCUT classification. Again these noise objects have a lower probability than real DSCUT lightcurves. One final interesting property is the split between OTHPER and Noise lightcurves. This is good for fields 0--2. In fields 3 and 4, while OTHPER lightcurves are recognised, several Noise lightcurves can be classified as OTHPER. Probability cuts remove the worst of these, but there is no way to distinguish between quasi-periodic instrumental noise and astrophysical variability in this scheme. These issues all lead to the conclusion that the classifier has more trouble with fields 3--4, due to a pattern of increased noise. We expect this issue to improve as K2 detrending methods become more robust.\n\n\\subsection{Detrending method comparison}\n\n\\begin{table*}\n\\caption{Total objects in each class in fields 3--4, split by detrending method (W=Warwick, PDC=K2 Team released lightcurves).}\n\\label{tabdetcomp}\n\\begin{tabular}{lllllll}\n\\hline\nClass & Total W & Total PDC & Prob $> 0.5$ W & Prob $> 0.5$ PDC &Prob $> 0.7$ W & Prob $> 0.7$ PDC \\\\\n\\hline\nRRab & 141 & 152 & 95 &115 & 48 & 83 \\\\\nDSCUT & 280 & 266 & 180 & 201 & 116 & 148\\\\\nGDOR & 198 & 382 & 122 & 238 & 61 & 101\\\\\nEA & 255 & 413 & 97 & 223 & 54 & 102 \t\\\\\nEB & 168 & 150 & 140 & 131 & 106 & 105\t\\\\\nOTHPER & 11402 & 9102 & 8709 & 8034 & 3522 & 4565 \\\\\nNoise & 17143 & 19126 &13012 & 17919 & 3625 & 11566 \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\nTable \\ref{tabdetcomp} shows the numbers of variable stars found using each dataset. At first glance the numbers in Table \\ref{tabdetcomp} seem to imply significant differences between detrending methods. The discrepancy in RRab numbers is largely a result of differing probability calibration - the same stars are found in both datasets, but those in the Warwick set given lower probabilities (although still higher than all other classes). Other major discrepancies are in the GDOR and EA classes. For GDOR, we find that the PDC set gives better results. Several GDOR lightcurves are misclassified in the Warwick set due to poor detrending masking the true variability. In some cases the PDC GDOR classification is inaccurate, but this is rare for the class probability $>0.7$ objects. For the EA objects, the reverse is true. Several PDC lightcurves are misclassified as EA due to a higher number of lightcurves in the PDC set with very significant remnant outliers. These lead to a high point-to-point scatter, which is interpreted by the classifier as an eclipse. Here the Warwick set is more reliable. The largest absolute difference in the variable classes is in the OTHPER objects, where \\raise.17ex\\hbox{$\\scriptstyle\\mathtt{\\sim}$} 1000 lightcurves extra pass the high probability cut for the PDC set. This is partly a result of a similar effect as for the RRab objects, where similarly classified objects are given lower probabilities in the Warwick set. However, there are also several objects found in the PDC set which are missed in the Warwick set, due to increased noise levels. The converse is also true, with some lightcurves found in the Warwick set but missed by the PDC. Overall, the two detrending methods perform comparably well, and can be used to reinforce each other when studying variable classes.\n\n\n\\subsection{Anomaly detection}\nDue to the limited classification scheme used, it is inevitable that some objects will not fit any of the given classes \\citep{Protopapas:2006br}. Due to the inclusion of Noise and OTHPER as classes, this is not a large problem as each class is quite broad. However it is worth noting any particular anomalies. One way of doing this is already intrinsic to the SOM -- the Euclidean distance of a phase curve to its nearest matching pixel template. However this metric only works for periodic sources, and can flag high for noisy sources. We perform a check for anomalies following the method of \\citet{Richards:2012ea}. This works by extracting the proximity measure, $\\rho_{ij}$ between each tested object $i$ and each object $j$ in the training set. The proximity measure is the proportion of trees in the classifier for which each object ends at the same final classification. It is close to unity for similar objects, and close to zero for dissimilar ones. From the proximity the discrepancy $d$ is calculated, via\n\n\\begin{equation}\nd_{ij} = \\frac{1-\\rho_{ij}}{\\rho_{ij}}\n\\end{equation}\n\nThe anomaly score is then given by the second smallest discrepancy of an object to the training set. High anomaly scores represent objects which are not well explained by any object in the training set, and are hence outliers.\n\nWe find that in this case, the highest few percentiles of anomalous objects are a mixture of noise-dominated lightcurves, unusual eclipsing binaries and variability which does not fit into the used classification scheme. We leave a full analysis of these unusual lightcurves to future work.\n\n\n\n\n\n\n\n\n\\subsection{Eclipsing Binaries}\nEncouragingly we identify 139 (96 at class probability $>0.7$) of the 165 EPIC, non-M35 eclipsing binaries identified by \\citet{LaCourse:2015jr} in field 0 as either 'EA' or 'EB' type, despite automating the process and not focusing on exclusively eclipsing binaries. The majority of the remainder are identified as 'OTHPER' or 'DSCUT', and are discussed below. We further identify an additional 61 EPIC, non-M35 objects in field 0 as 'EA' or 'EB' at class probability $>0.7$, although as our identification is automated rather than visual some of these may be misidentified by the classifier. Many more eclipsing binaries are found in the other fields.\n\nThe previously labelled, but not identified by our classifier, eclipsing binaries fall into three main groups. The first show near-sinusoidal short period lightcurves, and are generally identified as 'DSCUT'. In these cases it is difficult to reliably assign a class with the information available. These objects may be actual $\\delta$ Scuti stars, or contact eclipsing binaries. The other and largest group, with 14 members, are identified as 'OTHPER', and show pulsations or spot-modulation in addition to the known eclipses. We note that the classifier will assign a class based largely on the dominant period and phasecurve at this period, hence performs as expected in these cases. Pulsating stars in eclipsing binaries are useful objects, and so while a detailed study of these objects is beyond the scope of this paper we provide a list of such objects in Table \\ref{tabqpebs}. These are eclipsing binaries identified by a visual check of the lightcurves performed ourselves (as the \\citet{LaCourse:2015jr} catalogue only covered field 0), which are classified as `OTHPER' by our classifier. Some may be blended signals, and hence the pulsator or spot-modulated star may not be a member of the eclipsing binary system.\n\n\\begin{table}\n\\caption{EPIC IDs for 29 visually identified eclipsing binaries classified as `OTHPER' by our classifier, from fields 0--4.}\n\\label{tabqpebs}\n\\begin{tabular}{lllll}\n\\hline\n201158453 &201173390 &201569483 & 201584594 \\\\\n 201638314 &202072962 &202137580 & 203371239\\\\\n203476597 &203637922 &204043888 &204193529\\\\\n204328391 &204411840 &205510143 &205919993 \\\\\n 205985357 &205990339 &\n206047297 &\n206060972 \\\\\n206066862 &\n206226010 &\n206311743 &\n206500801 \\\\\n210350446 &\n210501149 &\n210766835 &\n210945342 \\\\\n211093684 &\n211135350 &\n \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\subsection{$\\delta$ Scuti Stars}\nWe have a sample of 377 $\\delta$ Scuti candidates, using a class probability cut of 0.7. The majority of these candidates were previously unknown. It is interesting to study their frequency and amplitude distribution. Note that here we use amplitude defined as in the max-min of the binned phase curve, and semi-amplitude as half this value. The distribution of amplitudes for the 377 $\\delta$ Scuti candidates is shown in Figure \\ref{figdscutampdist}. We see a number of HADS (high amplitude $\\delta$ Scutis). Using an amplitude threshold of $10^4$ ppm as used by \\citet{Bradley:2015ep}, 104 of our candidates are HADS. Included in this sample are 11 candidates with an amplitude greater than $10^5$ ppm. The period distribution of the whole sample is shown in Figure \\ref{figdscutperdist}, and covers the expected range for $\\delta$ Scuti variables, limited by our Nyquist sampling frequency.\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{dscut_ampdistv2.pdf}}\n\\caption{The distribution of phase curve amplitude for DSCUT classified objects. Several high amplitude candidates are visible.}\n\\label{figdscutampdist}\n\\end{figure}\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{dscut_perdistv2.pdf}}\n\\caption{The distribution of pulsation periods for DSCUT classified objects. The cutoff at the low period end is imposed by our Nyquist sampling frequency.}\n\\label{figdscutperdist}\n\\end{figure}\n\nAs has been mentioned, the DSCUT classified objects are degenerate with $\\beta$ Ceph variables due to the lack of colour information available. There is a catalogue of estimated K2 temperatures available for some objects \\citep{Stassun:2014wz} which could be used to make probable distinctions if necessary.\n\n\\subsection{$\\gamma$ Doradus Stars}\nWe have a sample of 133 $\\gamma$ Doradus candidates, using a class probability cut of 0.7. We plot the amplitude and period distributions in Figures \\ref{figgdorampdist} and \\ref{figgdorperdist}, following the same definition of amplitude as for the $\\delta$ Scuti sample. Note that this amplitude is only for the dominant period phase curve, and so does not include the other significant frequencies often present in $\\gamma$ Doradus lightcurves. The period distribution covers the expected range for $\\gamma$ Doradus variables. Due to the lack of colour information available, $\\gamma$ Doradus objects are degenerate with slowly pulsating B stars.\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{gdor_ampdistv2.pdf}}\n\\caption{The distribution of phase curve amplitude for GDOR classified objects.}\n\\label{figgdorampdist}\n\\end{figure}\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{gdor_perdistv2.pdf}}\n\\caption{The distribution of pulsation periods for GDOR classified objects.}\n\\label{figgdorperdist}\n\\end{figure}\n\n\\subsection{RR Lyrae ab-type Stars}\nAs the RRab class has less well calibrated probability (almost all candidates with Prob(RRab) greater than 0.5 seem to be real) we use an adjusted class probability threshold of 0.5 to study this class. This leaves 154 candidates. Their amplitude distribution is shown in Figure \\ref{figrrabampdist}, and peaks at significantly higher amplitude than that of the DSCUT and GDOR candidates as would be expected. Most of these candidates are previously known; we find that 129 of them are in K2 proposals focused on RR Lyrae stars. These proposals contain both known and candidate RR Lyraes; in the candidate cases our classification provides some support for them truly being RR Lyrae variables. Assuming these proposals were comprehensive (reasonable, given the multiple teams involved), this leaves 25 candidates as potential new discoveries by this catalogue. However, as these objects are those not in the proposals, there is a selection effect in favour of misclassified non-RR Lyrae objects. We performed a visual examination of each of these 25 lightcurves, which resulted in 8 of the 25 being confirmed as real RR Lyrae candidates (the others being either misclassified outbursting stars or particularly high amplitude noise). An additional 3 candidates were found by using the PDC lightcurve set and checking objects in both sets with class probability between 0.4 and 0.5, resulting in 10 total new candidates. These objects may still be blends of true RR Lyraes, hence the candidate designation. We plot the phase folded lightcurves for two new discoveries and two known RR Lyrae stars in Figure \\ref{fignewrrab}. Some amplitude modulation can be seen, due to some of these targets exhibiting the Blazhko effect \\citep{1907AN....175..325B}. RR Lyraes are immensely useful objects, allowing studies of the evolution of stellar populations throughout the Galaxy and in other nearby galaxies. Due to an absolute magnitude-metallicity relation \\citep{Sandage:1981ja} it is possible to use them for distance estimation.\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{rrab_ampdistv2.pdf}}\n\\caption{The distribution of phase curve amplitude for RRab classified objects.}\n\\label{figrrabampdist}\n\\end{figure}\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{newrrabs.pdf}}\n\\caption{Four phase folded RRAB classified lightcurves. Clockwise from top-left, the EPIC IDs are 210830646, 206409426, 211069540 and 203692906.}\n\\label{fignewrrab}\n\\end{figure}\n\t\n\\section{Conclusion}\nWe have implemented a novel combined machine learning algorithm, using both Self Organising Maps and Random Forests to classify variable stars in the K2 data. We consider fields 0--4, and intend to update the catalogue as more fields are released. As more data builds up, it may become possible to implement new variability classes, and study the effect of different detrending methods on the catalogue performance. We obtain a success rate of 92\\% using out of bag estimates on the training set.\n\nWe train the classifier on a set of Kepler and some K2 data from fields 0--2. As such it is applied completely independently to the majority of the K2 data, and the whole of fields 3--4. That we obtain good results for fields 3--4 bodes well for application of the classifier to future data. \n\nAlgorithms like this will become an increasingly important step in processing the data volumes expected from future astronomical surveys. To maximise scientific return it is critical to select interesting candidates, and do so rapidly and with minimal input. We hope that this method will contribute to the growing body of work attempting to address this issue. \n\n\\section*{Acknowledgements}\nThe authors thank the anonymous referee for a helpful review of the manuscript. This paper includes data collected by the Kepler mission. Funding for the Kepler mission is provided by the NASA Science Mission directorate. The data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NNX13AC07G and by other grants and contracts. We acknowledge with thanks the variable star observations from the AAVSO International Database contributed by observers worldwide and used in this research.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}