diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzewrz" "b/data_all_eng_slimpj/shuffled/split2/finalzzewrz" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzewrz" @@ -0,0 +1,5 @@ +{"text":"\\section{}\n\n\\section{Introduction}\nQuantum magnets in one dimension are a basic class of many-body systems \nin condensed matter and statistical physics \n(see e.g., Refs.~\\onlinecite{Giamarchi,Affleck}). \nThey have offered various kinds of topics \nin both experimental and theoretical studies for a long time. \nIn particular, the spin-$\\frac{1}{2}$ XXZ chain is a simple though\nrealistic system in this field. The Hamiltonian is defined by \n\\begin{equation}\n{\\cal H}^{\\rm XXZ}=J\\sum_{j}(S_{j}^{x}S_{j+1}^{x}+S_{j}^{y}S_{j+1}^{y}\n+\\Delta_{z}S_{j}^{z}S_{j+1}^{z}),\n\\label{eq:XXZ}\n\\end{equation}\nwhere $S_{j}^{\\alpha}$ is $\\alpha$-component of a \nspin-$\\frac{1}{2}$ operator on $j$-th site, \n$J>0$ is the exchange coupling constant, \nand $\\Delta_{z}$ is the anisotropy parameter. \nThis model\nis exactly solved by integrability methods,~\\cite{Korepin,Takahashi} \nand the ground-state phase diagram has been completed. \nThree phases appear depending on $\\Delta_{z}$; \nthe antiferromagnetic (AF) phase with a N\\'eel order \n$\\langle S_{j}^{z}\\rangle=-\\langle S_{j+1}^{z}\\rangle$ \n($\\Delta_z>1$), the critical Tomonaga-Luttinger liquid (TLL) phase \n($-1<\\Delta_z\\leq 1$), and the fully polarized phase with \n$\\langle S_{j}^{z}\\rangle=1\/2$ ($\\Delta_z\\leq -1$). \nIn and around the TLL phase, the low-energy and long-distance properties \ncan be understood via effective field theory techniques such \nas bosonization and conformal field \ntheory (CFT).~\\cite{Giamarchi,Affleck,Tsvelik,Gogolin,Francesco} \nThese theoretical results nicely explain experiments of several quasi \none-dimensional (1D) magnets. \nThe deep knowledge of this model is also useful for analyzing \nplentiful related magnetic systems, such as \nspin-$\\frac{1}{2}$ chains with some perturbations \n(e.g. external fields,~\\cite{Alcaraz95} \nadditional magnetic anisotropies,~\\cite{Oshikawa97,Affleck99,Essler98,Kuzmenko09}\ndimerization~\\cite{Haldane82,Papenbrock03,Orignac04}), \ncoupled spin chains,~\\cite{Shelton96,Kim2000} \nspatially anisotropic 2D or 3D spin \nsystems,~\\cite{Starykh04,Balents07,Starykh07} etc.\n\n\n\nA recent direction \nof studying spin chains is to \nestablish solid correspondences between the model~(\\ref{eq:XXZ}) \nand its effective theory. For example, Lukyanov and \nhis collaborators~\\cite{Lukyanov97,Lukyanov99,Lukyanov03} \nhave analytically predicted \ncoefficients of bosonized spin operators in the TLL phase. \nHikihara and Furusaki~\\cite{Hikihara98,Hikihara04} \nhave also determined them numerically in the same chains \nwith and without a uniform Zeeman term. \nUsing these results, one can now calculate amplitudes of \nspin correlation functions as well as their critical exponents. \nFurthermore, effects of perturbations on an XXZ chain can also be \ncalculated with high accuracy. \nIt therefore becomes possible to quantitatively compare \ntheoretical and experimental results in quasi 1D magnets. \nThe purpose of the present study is to attach a new relationship \nbetween the spin-$\\frac{1}{2}$ XXZ chain and its bosonized effective theory. \nNamely, we numerically evaluate coefficients of \nbosonized dimer operators in the TLL phase of the XXZ chain. \nDimer operators $(-1)^jS_j^\\alpha S_{j+1}^\\alpha$, \nas well as spin operators, are\nfundamental degrees of freedom in spin-$\\frac{1}{2}$ AF chains. \nIn fact, the leading terms of both bosonized spin and dimer \noperators have the same scaling dimension $1\/2$ \nat the $SU(2)$-symmetric AF point $\\Delta_z=1$ (see Sec.~\\ref{sec:dimer}). \n\n\nIn Refs.~\\onlinecite{Hikihara98,Hikihara04}, Hikihara and Furusaki have \nused density-matrix renormalization-group (DMRG) method \nin an efficient manner \nin order to accurately evaluate coefficients of spin operators \nof an XXZ chain in a magnetic field. \nInstead of such a direct powerful method, we \nutilize the relationship between a dimerized XXZ chain and its effective \nsine-Gordon theory~\\cite{Essler98,Essler04} to \ndetermine the coefficients of dimer operators \n(defined in Sec.~\\ref{sec:dimer}), i.e., excitation gaps in dimerized spin \nchains are evaluated by numerical diagonalization method and are \ncompared with the gap formula of the effective sine-Gordon theory. \nIn other words, we derive the information on uniform \nspin-$\\frac{1}{2}$ XXZ chains from dimerized (deformed) chains. \nMoreover, we also determine the coefficients of both spin and dimer \noperators for the spin-$\\frac{1}{2}$ Heisenberg (i.e., XXX) AF chain with \nan additional next-nearest-neighbor (NNN) coupling $J_2=0.2411J$ \nin the similar strategy. As seen in Sec.~\\ref{subsec:SU2_dimer}, \nevaluated coefficients are more reliable for the $J$-$J_2$ model,\nsince the marginal terms vanish in its effective theory. \n\n\n\n\nThe plan of this paper is as follows. \nIn Sec.~\\ref{sec:dimer}, we shortly summarize the bosonization of \nXXZ spin chains. Both the XXZ chain with dimerization \nand the chain in a staggered magnetic field are mapped to a sine-Gordon model. \nWe also consider the AF Heisenberg chain with NNN coupling $J_2=0.2411J$. \nIn Sec.~\\ref{sec:delta}, we explain how to obtain the coefficients of \ndimer and spin operators by using numerical diagonalization method. \nThe evaluated coefficients are listed \nin Tables~\\ref{tb:dimer_coeff} and \\ref{tb:a1} \nand Fig.~\\ref{fig:dimer_coeff}. These are the main results of this paper. \nFor comparison, the same dimer coefficients are also calculated \nby using the formula of the ground-state energy of the sine-Gordon model. \nWe find that the coefficients fixed by the gap formula are more reliable. \nWe apply these coefficients to several systems and physical quantities \nrelated to an XXZ chain (dimerized spin chains under a magnetic field, \nspin ladders with a four-spin exchange and \noptical response of dimerized 1D Mott insulators) in Sec.~\\ref{sec:apply}. \nFinally our results are summarized in Sec.~\\ref{sec:con}.\n\n\n\n\n\n\n\\section{Dimerized chain and sine-Gordon model}\n\\label{sec:dimer}\nIn this section, we explain the relationship between a dimerized XXZ \nchain and the corresponding sine-Gordon theory \nin the easy-plane region $-1<\\Delta_z\\leq 1$.\nXXZ chains in a staggered field and the AF Heisenberg chain with \nNNN coupling $J_2=0.2411J$ are also discussed. \nThe coefficients of dimer operators are \ndefined in Eq.~(\\ref{eq:dimer}). \n\n\n\n\n\\subsection{Bosonization of spin-$\\frac{1}{2}$ XXZ chain}\n\\label{subsec:XXZ}\nWe first review the effective theory \nfor undimerized spin chain~(\\ref{eq:XXZ}). \nAccording to the standard strategy, \nXXZ Hamiltonian (\\ref{eq:XXZ}) is bosonized as\n\\begin{align}\n{\\cal H}_{\\rm eff}^{\\rm XXZ}=\\int{\\rm d}x\\Big\\{\\frac{v}{2}\n[K^{-1}(\\partial_{x}\\phi)^{2}+K(\\partial_{x}\\theta)^{2}] \\nonumber\\\\\n-v\\frac{\\lambda}{2\\pi}\\cos\\big(\\sqrt{16\\pi}\\phi\\big)+\\cdots\\Big\\},\n\\label{eq:XXZ_eff_H}\n\\end{align}\nin the TLL phase. \nHere, $\\phi(x)$ and $\\theta(x)$ are dual scalar fields, \nwhich satisfy the commutation relation, \n\\begin{equation}\n[\\phi(x),\\theta(x')]=-{\\rm i}\\vartheta_{\\rm step}(x-x'),\n\\end{equation}\nwith $x=j a$ ($a$ is the lattice spacing). \nAs we see in Eq.~(\\ref{eq:normalization}), \n$\\cos(\\sqrt{16\\pi}\\phi)$ is irrelevant in $-1<\\Delta_z<1$, and becomes \nmarginal at the $SU(2)$-symmetric AF Heisenberg point $\\Delta_z=1$. \nThe coupling constant $\\lambda$ has been determined \nexactly.~\\cite{Lukyanov98,Lukyanov03} \nTwo quantities $K$ and $v$ denote the TLL parameter and spinon velocity, \nrespectively, which can be exactly evaluated from \nBethe ansatz:~\\cite{Giamarchi,Cabra98}\n\\begin{subequations}\n\\label{eq:K_v}\n\\begin{align}\nK=&\\frac{\\pi}{2(\\pi-\\cos^{-1}\\Delta_{z})}=\\frac{1}{4\\pi R^{2}}\n=\\frac{1}{2\\eta},\\\\\nv=&Ja\\frac{\\pi\\sqrt{1-\\Delta_{z}^{2}}}{2\\cos^{-1}\\Delta_{z}}\n=Ja\\frac{\\sin(\\pi\\eta)}{2(1-\\eta)}.\n\\end{align}\n\\end{subequations}\nHere we have introduced new parameters $\\eta$ and $R$. \nThe former is the critical exponent of two-point spin correlation functions\nand used in the discussion below. \nThe latter is called the compactification radius.\nIt fixes the periodicity \nof fields $\\phi$ and $\\theta$ as $\\phi\/\\sqrt{K}\\sim\\phi\/\\sqrt{K}+2\\pi R$ \nand $\\sqrt{K}\\theta\\sim \\sqrt{K}\\theta+1\/R$. \nUsing the scalar fields $\\phi$ and $\\theta$, \nwe can obtain the bosonized representation of spin operators: \n\\begin{subequations}\n\\label{eq:spin_boson}\n\\begin{align}\nS_{j}^{z}\\approx&\\frac{a}{\\sqrt{\\pi}}\\partial_{x}\\phi\n+(-1)^{j}a_{1}\\cos(\\sqrt{4\\pi}\\phi)+\\cdots,\\\\\nS_{j}^{+}\\approx&{\\rm e}^{{\\rm i}\\sqrt{\\pi}\\theta}\n\\left[b_{0}(-1)^{j}+b_{1}\\cos(\\sqrt{4\\pi}\\phi)+\\cdots\\right],\n\\end{align}\n\\end{subequations}\nwhere $a_n$ and $b_n$ are non-universal constants, and some of them \nwith small $n$ have been determined accurately in \nRefs.~\\onlinecite{Lukyanov97,Lukyanov99,Lukyanov03,Hikihara98,Hikihara04}. \nIn this formalism, vertex operators are normalized \nas~\\cite{Lukyanov97,Lukyanov99,Lukyanov03}\n\\begin{equation}\n\\langle{\\rm e}^{{\\rm i}q\\phi(x)}{\\rm e}^{-{\\rm i}q\\phi(x')}\\rangle\n=\\left(\\frac{a}{|x-x'|}\\right)^{\\frac{Kq^{2}}{2\\pi}}\n{\\rm at}\\;|x-x'|\\gg a.\n\\label{eq:normalization}\n\\end{equation}\nThis means that the operator ${\\rm e}^{{\\rm i}q\\phi(x)}$ has scaling \ndimension $Kq^{2}\/(4\\pi)$. \n\n\nIn addition to the spin operators, the bosonized forms of \nthe dimer operators are known to \nbe~\\cite{Giamarchi,Affleck,Tsvelik,Gogolin} \n\\begin{subequations}\n\\label{eq:dimer}\n\\begin{align}\n(-1)^{j}(S_{j}^{x}S_{j+1}^{x}+S_{j}^{y}S_{j+1}^{y})\n\\approx&d_{xy}\\sin(\\sqrt{4\\pi}\\phi)+\\cdots,\\\\\n(-1)^{j}S_{j}^{z}S_{j+1}^{z}\n\\approx&d_{z}\\sin(\\sqrt{4\\pi}\\phi)+\\cdots.\n\\end{align}\n\\end{subequations}\nIn contrast to the spin operators, the coefficients $d_{xy}$ and $d_z$ \nhave never been evaluated so far. To determine them is \nthe subject of this paper. It seems to be possible to calculate $d_{xy,z}$ \nby utilizing Eq.~(\\ref{eq:spin_boson}) and operator-product-expansion \n(OPE) technique,~\\cite{Gogolin,Tsvelik,Francesco} \nbut it requires the correct values of all the \nfactors $a_n$ and $b_n$.~\\cite{Hikihara04} \nTherefore, we should interpret that the \ndimer coefficients $d_{xy,z}$ are independent of spin coefficients \n$a_n$ and $b_n$. \n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Bosonization of dimerized spin chain}\n\\label{subsec:dimer}\nNext, let us consider a bond-alternating XXZ chain whose Hamiltonian is \ngiven as\n\\begin{align}\n{\\cal H}^{{\\rm XXZ}\\mathchar`-\\delta}=&J\\sum_{j}\\left[\n(1+(-1)^{j}\\delta_{xy})(S_{j}^{x}S_{j+1}^{x}\n+S_{j}^{y}S_{j+1}^{y})\\right.\\nonumber\\\\\n&\\left.+(\\Delta_{z}+(-1)^{j}\\delta_{z})\nS_{j}^{z}S_{j+1}^{z}\\right].\n\\label{eq:dimer_XXZ}\n\\end{align}\nIn the weak dimerization regime of $|\\delta_{xy,z}|\\ll 1$, \nthe bosonization is applicable and the dimerization terms can be treated \nperturbatively. From the formula~(\\ref{eq:dimer}), the effective Hamiltonian \nof Eq.~(\\ref{eq:dimer_XXZ}) is \n\\begin{align}\n{\\cal H}_{\\rm eff}^{{\\rm XXZ}\\mathchar`-\\delta}&=\\int{\\rm d}x\\Big\\{\n\\frac{v}{2}[K^{-1}(\\partial_{x}\\phi)^{2}+K(\\partial_{x}\\theta)^{2}]\n\\nonumber\\\\\n&+\\frac{J}{a}(\\delta_{xy}d_{xy}+\\delta_{z}d_{z})\n\\sin(\\sqrt{4\\pi}\\phi)+\\cdots \\Big\\}.\n\\label{eq:XXZ_dimer_eff}\n\\end{align}\nHere, we have neglected all of the irrelevant terms \nincluding $\\cos(\\sqrt{16\\pi}\\phi)$. \nThis is nothing but an integrable sine-Gordon model (see e.g., \nRefs.~\\onlinecite{Essler98,Essler04} and references therein). \nThe $\\sin(\\sqrt{4\\pi}\\phi)$ term has a scaling dimension $K$, \nand is relevant when $K<2$, i.e., $-0.7071<\\Delta_z\\leq 1$. \nIn this case, an excitation gap opens and a dimerization \n$\\langle S_{j}^{\\alpha}S_{j+1}^{\\alpha}\n-S_{j+1}^{\\alpha}S_{j+2}^{\\alpha}\\rangle\\neq0$ occurs.\nThe excitation spectrum of the sine-Gordon model \nhas been known,~\\cite{Essler98,Essler04} and three types of \nelementary particles appear; a soliton, \nthe corresponding antisoliton, and bound states \nof the soliton and the antisoliton (called breathers). \nThe soliton and antisoliton have the same mass gap $E_S$. \nThere exist $[4\\eta-1]$ breathers, in which $[A]$ stands for the integer \npart of $A$. The mass of soliton and \n$n$-th breather $E_{B_n}$ are related as follows.\n\\begin{equation}\nE_{B_n}=2 E_S \\sin\\left(\\frac{n\\pi}{2(4\\eta-1)}\\right), \n\\quad n=1,\\cdots,[4\\eta-1]. \\label{eq:mass_relation}\n\\end{equation}\nThe breather mass in units of the soliton mass \nis shown in Fig.~\\ref{fig:spectrum_SG} \nas a function of $\\Delta_z$. Note that there is no breather in the \nferromagnetic side $\\Delta_z<0$, and the lightest breather with \nmass $E_{B_1}$ is always heavier than the soliton \nin the present easy-plane regime. \nFollowing Refs.~\\onlinecite{Zamolodchikov95,Lukyanov97}, the soliton mass \nis also analytically represented as \n\\begin{align}\n\\frac{E_S}{J}=&\\frac{v}{Ja}\\frac{2}{\\sqrt{\\pi}}\n\\frac{\\Gamma\\left(\\frac{1}{8\\eta-2}\\right)}{\\Gamma\\left(\\frac{2}{4-1\/\\eta}\n\\right)}\n\\nonumber\\\\\n&\\times\\left[\\frac{Ja}{v}\\frac{\\pi (\\delta_{xy}d_{xy}+\\delta_{z}d_{z})}{2}\n\\frac{\\Gamma\\left(\\frac{4-1\/\\eta}{4}\\right)}\n{\\Gamma\\left(\\frac{1}{4\\eta}\\right)} \\right]^{\\frac{2}{4-1\/\\eta}}.\n\\label{eq:dimergap}\n\\end{align}\nIn addition, the difference between the ground-state energy \n${\\cal E}_{\\rm free}$ of \nthe free-boson theory~(\\ref{eq:XXZ_eff_H}) with $\\lambda=0$ \nper site and that of the sine-Gordon theory~(\\ref{eq:XXZ_dimer_eff}), \n${\\cal E}_{\\rm SG}$, has been predicted as~\\cite{Zamolodchikov95,Lukyanov97}\n\\begin{align}\n\\frac{\\Delta{\\cal E}_{\\rm GS}}{J}=\\frac{{\\cal E}_{\\rm free}-{\\cal E}_{\\rm SG}}{J}\n=\\frac{1}{4}\\frac{v}{Ja}\n\\left(\\frac{Ja}{v}\\frac{E_{S}}{J}\\right)^2\\tan\n\\left(\\frac{\\pi}{2}\\frac{1}{4\\eta-1}\\right).\n\\label{eq:E_GS}\n\\end{align}\nHowever, we should note that the above formula is invalid for the \nferromagnetic side $\\Delta_z\\leq 0$ ($\\eta\\leq 1\/2$) \nsince it diverges at the XY point $\\Delta_z=0$ ($\\eta=1\/2$). \n\n\n\nA similar sine-Gordon model also emerges\nin spin-$\\frac{1}{2}$ XXZ chains in a staggered field, \n\\begin{equation}\n{\\cal H}^{\\rm stag}={\\cal H}^{\\rm XXZ}+\\sum_{j}(-1)^{j}h_{\\rm s}S_{j}^{z}.\n\\label{eq:XXZ_stag}\n\\end{equation}\nThe staggered field $h_{\\rm s}$ induces a relevant perturbation \n$\\cos(\\sqrt{4\\pi}\\phi)$. \nTherefore, the resultant effective Hamiltonian is \n\\begin{equation}\n{\\cal H}_{\\rm eff}^{\\rm stag}={\\cal H}_{\\rm eff}^{\\rm XXZ}+\n\\int{\\rm d}x \\frac{h_{\\rm s}}{a}a_{1}\n\\cos(\\sqrt{4\\pi}\\phi).\n\\label{eq:XXZ_stag_eff}\n\\end{equation}\nIf we redefine the scalar field $\\phi$ as $\\phi+\\sqrt{\\pi}\/4$, \nthe form of Eq.~(\\ref{eq:XXZ_stag_eff}) becomes equivalent to \nthat of Eq.~(\\ref{eq:XXZ_dimer_eff}). \nThus, the soliton gap of the model~(\\ref{eq:XXZ_stag_eff}) is \nequal to Eq.~(\\ref{eq:dimergap}) with the replacement \nof $\\delta_{xy}d_{xy}+\\delta_{z}d_{z}\\to h_{\\rm s}a_{1}\/J$. \nNamely the soliton gap of the model~(\\ref{eq:XXZ_stag_eff}) is given by\n\\begin{align}\n\\frac{E_S}{J}=&\\frac{v}{Ja}\\frac{2}{\\sqrt{\\pi}}\n\\frac{\\Gamma\\left(\\frac{1}{8\\eta-2}\\right)}{\\Gamma\\left(\\frac{2}{4-1\/\\eta}\n\\right)}\n\\nonumber\\\\\n&\\times\\left[\\frac{Ja}{v}\\frac{\\pi (h_{\\rm s}a_{1})}{2J}\n\\frac{\\Gamma\\left(\\frac{4-1\/\\eta}{4}\\right)}\n{\\Gamma\\left(\\frac{1}{4\\eta}\\right)} \\right]^{\\frac{2}{4-1\/\\eta}}.\n\\label{eq:staggap}\n\\end{align}\nThis type of staggered-field induced gaps has been observed in some \nquasi 1D magnets with an alternating gyromagnetic tensor or \nDzyaloshinskii-Moriya interaction such as \nCu benzoate.~\\cite{Oshikawa97,Affleck99,Essler98,Kuzmenko09,Dender97}\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.35\\textwidth]{sG_spec_v03.eps}\n\\end{center}\n\\caption{(Color online) Ratio of the $n$-th breather mass \n$E_{B_n}$ to the soliton mass $E_S$ as a function of \nthe XXZ anisotropy $\\Delta_z$ in the sine-Gordon \nmodel~(\\ref{eq:XXZ_dimer_eff}) or (\\ref{eq:XXZ_stag_eff}).}\n\\label{fig:spectrum_SG}\n\\end{figure}\n\n\nMasses of the soliton, antisoliton and breathers are related to \nthe excitation gaps of \nthe original lattice systems, Eqs.~(\\ref{eq:dimer_XXZ}) and \n(\\ref{eq:XXZ_stag}). The soliton and antisoliton \ncorrespond to the lowest excitations which change the $z$ component of \ntotal spin $S^z_{\\rm tot}=\\sum_jS_j^z$ by $\\pm 1$. \nOn the other hand, the lightest breather is regarded as the lowest \nexcitation with $\\Delta S^z_{\\rm tot}=0$. \nAt the $SU(2)$-symmetric AF point $\\Delta_z=1$, there are \nthree breathers. \nThe soliton, antisoliton and lightest breather are degenerate and \nform the spin-1 triplet excitations (so-called magnons). \nThe second lightest breather is interpreted as the singlet excitation \nwith $\\Delta S_{\\rm tot}=0$. \nIn the ferromagnetic regime $\\Delta_z<0$, where any breather disappears, \nthe lowest soliton-antisoliton scattering state \nwould correspond to the excitation gap in the sector of \n$\\Delta S_{\\rm tot}^z=0$. \n\n\n\\subsection{${\\bol J}$-${\\bol J}_{\\bol 2}$ antiferromagnetic spin chain}\n\\label{subsec:J-J2}\nIn the previous two subsections, we have completely \nneglected effects of irrelevant perturbations in \nthe low-energy effective theory. However, \nas already noted, the $\\lambda$ term becomes nearly \nmarginal when the anisotropy $\\Delta_z$ approaches unity. \nIn this case, the $\\lambda$ term is expected to affect \nseveral physical quantities. \nActually, such effects have been studied in both \nthe models~(\\ref{eq:dimer_XXZ}) [Ref.~\\onlinecite{Orignac04}] \nand (\\ref{eq:XXZ_stag}) [Refs.~\\onlinecite{Oshikawa97,Affleck99}]. \n\n\n\nIt is known~\\cite{Haldane82} that a small AF NNN coupling $J_2$ \ndecreases the value of $\\lambda$ in the $SU(2)$-symmetric AF \nHeisenberg chain. Okamoto and Nomura~\\cite{Okamoto92} have shown that \nthe marginal interaction vanishes, i.e., $\\lambda\\to 0$ \nin the following model: \n\\begin{equation}\n{\\cal H}^{\\rm nnn}=\\sum_{j}(J\\bol{S}_{j}\\cdot\\bol{S}_{j+1}\n+J_{2}\\bol{S}_{j}\\cdot\\bol{S}_{j+2}),\n\\label{eq:J-J2chain}\n\\end{equation}\nwith $J_2=0.2411J$. On the $J_2\/J$ axis, this model is \nlocated at the Kosterlitz-Thouless transition point between \nthe TLL and a spontaneously dimerized phase. \nFrom this fact, if we replace ${\\cal H}^{\\rm XXX}$ with \n${\\cal H}^{\\rm nnn}$ in the $SU(2)$-symmetric models~(\\ref{eq:dimer_XXZ}) \nand (\\ref{eq:XXZ_stag}), namely, if we consider the following models:\n\\begin{subequations}\n\\label{eq:modified}\n\\begin{align}\n\\tilde{\\cal H}^{{\\rm XXX}\\mathchar`-\\delta}\n=&{\\cal H}^{\\rm nnn}+\\sum_{j}(-1)^{j}\\delta J\\bol{S}_{j}\\cdot\\bol{S}_{j+1},\n\\label{eq:modified_dimer}\\\\\n\\tilde{\\cal H}^{\\rm stag}=&{\\cal H}^{\\rm nnn}+\\sum_{j}(-1)^{j}h_{\\rm s}S_{j}^{z},\n\\label{eq:modified_stag}\n\\end{align}\n\\end{subequations}\nthen their effective theories are much \ncloser to a pure sine-Gordon model. \nIn other words, the predictions from the sine-Gordon model, \nsuch as Eqs.~(\\ref{eq:dimergap}) and (\\ref{eq:staggap}), \nbecome more reliable. \n\n\n\n\n\n\n\n\n\n\n\n\\section{Coefficients of Dimer and Spin Operators}\n\\label{sec:delta}\nFrom the discussions in Sec.~\\ref{sec:dimer}, one can readily find a way \nof extracting the values of $d_{xy,z}$ and $a_1$ in Eqs.~(\\ref{eq:dimer}) \nand (\\ref{eq:spin_boson}) as follows. \nWe first calculate some low-energy levels in \n$S_{\\rm tot}^z=\\pm1$ and $S_{\\rm tot}^z=0$ sectors of the models \n(\\ref{eq:dimer_XXZ}), (\\ref{eq:XXZ_stag}) and (\\ref{eq:modified}) \nby means of numerical diagonalization method. \nSince all the Hamiltonians~(\\ref{eq:dimer_XXZ}), (\\ref{eq:XXZ_stag}) and \n(\\ref{eq:modified}) commute with $S^{z}_{\\rm tot}=\\sum_{j}S_{j}^{z}$, \nthe numerical diagonalization can be performed in the Hilbert subspace \nwith each fixed $S^{z}_{\\rm tot}$. \nIn order to extrapolate gaps to the thermodynamic limit \nwith reasonable accuracy, we use appropriate finite-size scaling \nmethods~\\cite{Cardy84,Cardy86,Cardy86b,Shanks55} \nfor spin chains under periodic boundary condition (total number of \nsites $L=8$, 10, $\\cdots$, 28, 30). \nSecondly, the coefficients $d_{xy,z}$ and $a_1$ of \nthe spin-$\\frac{1}{2}$ XXZ chain and the $J$-$J_2$ chain are determined \nvia the comparison between the sine-Gordon gap formula~(\\ref{eq:dimergap}) \nand numerically evaluated spin gaps \nfor various values of $\\delta_{xy,z}$ and $h_{\\rm s}$. \nIn this procedure, (as already mentioned) the energy difference \nbetween the lowest (i.e., ground-state) and the second lowest levels \nof the $S_{\\rm tot}^z=0$ sector (gap with $\\Delta S_{\\rm tot}^z=0$) and \nthat between the ground-state level and the lowest level of \nthe $S_{\\rm tot}^z=\\pm1$ sector (gap with $\\Delta S_{\\rm tot}^z=\\pm1$) \nare respectively interpreted as the breather (or \nsoliton-antisoliton scattering state) and soliton masses \nin the sine-Gordon scheme. \n\n\n\n\n\\subsection{TLL phase and Numerical diagonalization}\n\\label{subsec:TLLandED}\nIn this subsection, we focus on the TLL phase of uniform \nspin-$\\frac{1}{2}$ XXZ chains~(\\ref{eq:XXZ}) and \ntest the reliability of our numerical diagonalization. \nThe low-energy properties are described by Eq.~(\\ref{eq:XXZ_eff_H}), \nwhich is a free boson theory (i.e., CFT with central charge $c=1$) \nwith some irrelevant perturbations. \nGenerally, the finite-size scaling formula for \nthe excitation spectrum in any CFT has been proved~\\cite{Cardy84,Cardy86} \nto be\n\\begin{align}\n\\Delta E_{\\cal O}\\equiv E_{\\cal O}-E_{0}=\\frac{2\\pi v}{La} [{\\cal O}]+\\cdots.\n\\label{eq:finite_size}\n\\end{align}\nHere $E_0$ and $E_{\\cal O}$ are respectively the ground-state energy and \nthe energy of an excited state generating from a primary field ${\\cal O}$ \nin the given CFT. Remaining quantities $[{\\cal O}]$, $v$, and $La$ are the \nscaling dimension of the operator ${\\cal O}$, the excitation velocity \nand the system length, respectively. \nIn the case of the spin chain~(\\ref{eq:XXZ}), \nthe bosonization formula~(\\ref{eq:spin_boson}) indicates that \n$E_{e^{\\pm i\\sqrt{\\pi}\\theta}}$ and $E_{e^{\\pm i\\sqrt{4\\pi}\\phi}}$ \ncorrespond to the excitation energies in the $S_{\\rm tot}^z=\\pm 1$ \nand $S_{\\rm tot}^z=0$ sectors, respectively. The irrelevant perturbations \ncan also contribute to the finite-size \ncorrection to excitation energies. From the $U(1)$ and translational \nsymmetries of the XXZ chain~(\\ref{eq:XXZ}), one can show that the finite-size \ngap $\\Delta E_{\\Delta S_{\\rm tot}^z=\\pm 1}$ has no significant \nmodification from the perturbations, while the correction to \n$\\Delta E_{\\Delta S_{\\rm tot}^z= 0}$ is proportional \nto $L^{1-[{\\rm e}^{{\\rm i}2\\sqrt{4\\pi}\\phi}]}$. \nTherefore, the following finite-size scaling formulas are predicted:\n\\begin{subequations}\n\\label{eq:gap_correction}\n\\begin{align}\n\\Delta E_{\\Delta S_{\\rm tot}^z=\\pm 1}\\approx\n\\frac{2\\pi v}{La}\\frac{1}{4K}+\\cdots,\\\\\n\\Delta E_{\\Delta S_{\\rm tot}^z=0}\\approx\n\\frac{2\\pi v}{La}K+c_0 L^{1-4K}+\\cdots,\n\\end{align}\n\\end{subequations}\nwith $c_0$ being a non-universal constant. \nHere we have used $[{\\rm e}^{{\\rm i}n\\sqrt{\\pi}\\theta}]=n^2\/(4K)$ \nand $[{\\rm e}^{{\\rm i}n\\sqrt{4\\pi}\\phi}]=n^2K$. \nAt the $SU(2)$-symmetric AF point $\\Delta_z=1$, \n$\\Delta E_{\\Delta S_{\\rm tot}^z=\\pm 1}=\\Delta E_{\\Delta S_{\\rm tot}^z=0}\n\\equiv \\Delta E_{\\rm su2}$ holds and the marginal $\\lambda$ term modifies the \nscaling form of the spin gap. \nThe marginal term is known to yield a logarithmic correction \nas follows:~\\cite{Cardy86b} \n\\begin{align}\n\\label{eq:gap_log}\n\\Delta E_{\\rm su2}\\approx\\frac{2\\pi v}{La}\\left(\\frac{1}{2}+\n\\frac{c_1}{\\ln L}+\\frac{c_2}{(\\ln L)^2}+\\cdots\\right).\n\\end{align}\nHere $c_{1,2}$ are non-universal constants. \n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.35\\textwidth]{TLL_v04.eps}\n\\end{center}\n\\caption{(Color online) (a) Numerically evaluated gaps with \n$\\Delta S^{z}_{\\rm tot}=\\pm1$ (circles) \nand $\\Delta S^{z}_{\\rm tot}=0$ (triangles) \nfor XXZ chains with $\\Delta_z=0.6$ and finite length $L$.\nThe solid curve $8.019\\times10^{-4}+2.977\/L$ (dashed curve \n$1.312\\times10^{-3}+5.982\/L-4.764\/L^{1.8376}$) is \ndetermined by fitting the circles (triangles). (b) Gaps of finite-size \nHeisenberg chains with $\\Delta_z=1$. The solid curve is \n$\\Delta E_{\\rm su2}\/J=2.173\\times 10^{-4}\n+4.965\/L-2.203\/(L \\ln L)+1.200\/(L (\\ln L)^2)$.}\n\\label{fig:Del0_6TLL}\n\\end{figure}\n\n\nAs an example, numerically evaluated gaps with \n$\\Delta S^{z}_{\\rm tot}=\\pm1$ and $\\Delta S^{z}_{\\rm tot}=0$ \nin the case of $\\Delta_z=0.6$ are \nrespectively represented as circles and triangles in \nFig.~\\ref{fig:Del0_6TLL}(a). \nCircles are nicely fitted by the solid curve \n$\\Delta E_{\\Delta S_{\\rm tot}^z=\\pm 1}\/J=8.019\\times10^{-4}+2.977\/L$. \nThis result is consistent with the fact that\nan easy-plane anisotropic XXZ model is gapless in the thermodynamic limit \nand that the exact coefficient of the $1\/L$ term is $2\\pi v\/(4JK)=3$ at \n$\\Delta_z=0.6$. Similarly, triangles can be fitted by \n$\\Delta E_{\\Delta S_{\\rm tot}^z=0}\/J=\n1.312\\times10^{-3}+5.982\/L-4.764\/L^{1.8376}$ where $1.8376=1-4K$. \nThe factor 5.982 of the $1\/L$ term is very close \nto $2\\pi v K\/(Ja)=6.040$. The spin gap at $SU(2)$-symmetric point \nis also represented in Fig.~\\ref{fig:Del0_6TLL}(b). \nFollowing the formula~(\\ref{eq:gap_log}), we can correctly determine \nthe fitting curve $\\Delta E_{\\rm su2}\/J=2.173\\times 10^{-4}\n+4.965\/L-2.203\/(L \\ln L)+1.200\/(L (\\ln L)^2)$, \nin which the factor of the second term is nearly equal to \n$\\pi v\/(Ja)=4.935$. \nThese results support the reliability of our numerical diagonalization.\nWe note that a more precise finite-size scaling analysis for \nAF Heisenberg model has been performed in Ref.~\\onlinecite{Nomura93}.\n\n\n\n\n\n\n\n\\subsection{Dimer coefficients of XY model}\n\\label{subsec:XY_dimer}\nNext, let us move onto the evaluation of excitation gaps in \ndimerized XXZ chains. In this case, since the system is not critical, \nthe above finite-size scaling based on CFT cannot be applied.\nInstead, we utilize Aitken-Shanks method~\\cite{Shanks55} to extrapolate \nour numerical data to the values in the thermodynamic limit.\n\n\nIn this subsection, we consider a special dimerized XY chain \nwith $\\Delta_{z}=\\delta_{z}=0$. It is mapped to a \nsolvable free fermion system through Jordan-Wigner \ntransformation. Therefore, our numerically determined coefficients in \nEq.~(\\ref{eq:dimer}) can be compared with the exact value. \nThe lowest energy gap with $\\Delta S_{\\rm tot}^z=\\pm 1$, \nwhich corresponds to the soliton mass $E_S$, is exactly evaluated as \n\\begin{equation}\n\\Delta E_{\\Delta S_{\\rm tot}^z=\\pm 1}\/J=\\delta_{xy}. \\label{eq:XY_soliton}\n\\end{equation}\nComparing Eq.~(\\ref{eq:XY_soliton}) with Eq.~(\\ref{eq:dimergap}), \nwe obtain the exact coefficient \n\\begin{equation}\nd_{xy}=1\/\\pi=0.3183\n\\end{equation}\nat the XY case $\\Delta_{z}=\\delta_{z}=0$. The exact solution also tells us \nthat the excitation gap with $\\Delta S_{\\rm tot}^z=0$ is \n\\begin{align}\n\\Delta E_{\\Delta S_{\\rm tot}^z=0}\/J=2\\delta_{xy}. \\label{eq:XY_breather}\n\\end{align}\nThis is consistent with the sine-Gordon prediction that any breather \ndisappears and the relation $E_{B_1}=2E_S$ holds just \nat the XY point $\\Delta_z=0$. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.35\\textwidth]{gap_XY_v04.eps}\n\\end{center}\n\\caption{(Color online) Numerically evaluated gaps \n$\\Delta E_{\\Delta S_{\\rm tot}^z=\\pm 1}$ (circles) and \n$\\Delta E_{\\Delta S_{\\rm tot}^z=0}$ (triangles) in dimerized XY models with \n$\\Delta_{z}=\\delta_{z}=0$. Solid and dashed lines are \nthe exact results determined via Jordan-Wigner transformation. \nThese lines respectively correspond to the soliton and breather masses \nin the framework of sine-Gordon theory.}\n\\label{fig:gap_XY}\n\\end{figure}\n\nFigure~\\ref{fig:gap_XY} shows the comparison between the energy gap \ncalculated by numerical diagonalization with Aitken-Shanks process \nand Eq.~(\\ref{eq:XY_soliton}) [or Eq.~(\\ref{eq:XY_breather})].\nExcept for $\\Delta E_{\\Delta S_{\\rm tot}^{z}=0}$ \nin the weak dimerized regime $\\delta_{xy}\\alt 0.1$, \nnumerically calculated gaps coincide well with the exact value. \nWe have found that when $\\delta_{xy,z}$ becomes smaller, \nthe precision of Aitken-Shanks method is decreased due to a large \nsize dependence of gaps. \n\n\n\n\n\\subsection{Dimer coefficients of XXZ model}\n\\label{subsec:XXZ_dimer}\n\n\n\n\\begin{table*}\n\\caption{\\label{tb:dimer_coeff} Dimer coefficients ($d_{xy}$ and $d_z$), \nTLL parameter $K$, compactification radius $R$, spinon velocity $v$ \nof spin-$\\frac{1}{2}$ XXZ chain. Dimerization-induced gaps are \nalso listed in the final column. The final line is the result for \nthe $J$-$J_2$ chain~(\\ref{eq:J-J2chain}). The same data of $d_{xy,z}$ are also \nshown in Fig.~\\ref{fig:dimer_coeff}.}\n\\begin{ruledtabular}\n\\begin{tabular}{lllllll}\n\\multicolumn{1}{c}{$\\Delta_z$} \n& \\multicolumn{1}{c}{$d_{xy}$} & \n\\multicolumn{1}{c}{$d_z$} & \\multicolumn{1}{c}{$K$} & \\multicolumn{1}{c}{$R$} & \n\\multicolumn{1}{c}{$v\/(Ja)$} & \\multicolumn{1}{c}{soliton gap $E_S\/J$} \\\\\n\\hline\n$1$ & 0.228 (0.204) & 0.110 (0.097) & 0.5 & 0.3989($=1\/\\sqrt{2\\pi}$) & 1.571($=\\pi\/2$) & $3.535(\\delta_{xy}d_{xy}+\\delta_{z}d_{z})^{0.6667}$\\\\\n$0.9$ & 0.278 (0.261) & 0.141 (0.131) & 0.5838 & 0.3692 & 1.518 & $3.268(\\delta_{xy}d_{xy}+\\delta_{z}d_{z})^{0.7061}$\\\\\n$0.8$ & 0.297 (0.284) & 0.154 (0.146) & 0.6288 & 0.3557 & 1.465 & $3.147(\\delta_{xy}d_{xy}+\\delta_{z}d_{z})^{0.7293}$\\\\\n$0.7$ & 0.309 (0.299) & 0.165 (0.159) & 0.6695 & 0.3448 & 1.410 & $3.057(\\delta_{xy}d_{xy}+\\delta_{z}d_{z})^{0.7516}$\\\\\n$0.6$ & 0.318 (0.310) & 0.174 (0.169) & 0.7094 & 0.3349 & 1.355 & $2.986(\\delta_{xy}d_{xy}+\\delta_{z}d_{z})^{0.7748}$\\\\\n$0.5$ & 0.324 (0.318) & 0.182 (0.177) & 0.75 & 0.3257 & 1.299 & $2.934(\\delta_{xy}d_{xy}+\\delta_{z}d_{z})^{0.8}$\\\\\n$0.4$ & 0.327 (0.323) & 0.188 (0.185) & 0.7924 & 0.3169 & 1.242 & $2.902(\\delta_{xy}d_{xy}+\\delta_{z}d_{z})^{0.8281}$\\\\\n$0.3$ & 0.328 (0.325) & 0.193 (0.191) & 0.8375 & 0.3082 & 1.184 & $2.893(\\delta_{xy}d_{xy}+\\delta_{z}d_{z})^{0.8602}$\\\\\n$0.2$ & 0.328 (0.325) & 0.197 (0.196) & 0.8864 & 0.2996 & 1.124 & $2.918(\\delta_{xy}d_{xy}+\\delta_{z}d_{z})^{0.8980}$\\\\\n$0.1$ & 0.324 (0.323) & 0.200 (0.200) & 0.9401 & 0.2910 & 1.063 & $2.991(\\delta_{xy}d_{xy}+\\delta_{z}d_{z})^{0.9434}$\\\\\n$0$ & 0.318 (0.318) & 0.202 (0.203) & 1 & 0.2821($=1\/\\sqrt{4\\pi}$) & 1 & $3.141(\\delta_{xy}d_{xy}+\\delta_{z}d_{z})$\\\\\n$-0.1$ & 0.309 (0.311) & 0.202 (0.204) & 1.068 & 0.2730 & 0.9353 & $3.431(\\delta_{xy}d_{xy}+\\delta_{z}d_{z})^{1.073}$\\\\\n$-0.2$ & 0.297 (0.302) & 0.200 (0.204) & 1.147 & 0.2634 & 0.8685 & $4.008(\\delta_{xy}d_{xy}+\\delta_{z}d_{z})^{1.172}$\\\\\n$-0.3$ & 0.278 (0.289) & 0.194 (0.203) & 1.241 & 0.2533 & 0.7990 & $5.308(\\delta_{xy}d_{xy}+\\delta_{z}d_{z})^{1.317}$\\\\\n$-0.4$ & 0.252 (0.273) & 0.184 (0.199) & 1.355 & 0.2423 & 0.7263 & $9.214(\\delta_{xy}d_{xy}+\\delta_{z}d_{z})^{1.550}$\\\\\n$-0.5$ & 0.213 (0.248) & 0.163 (0.191) & 1.5 & 0.2303 & 0.6495 & $33.25(\\delta_{xy}d_{xy}+\\delta_{z}d_{z})^{2}$\\\\\n$J$-$J_{2}$ model & 0.364 (0.361) & 0.188 (0.182) & 0.5 & 0.3989 & 1.174 & $3.208(\\delta_{xy}d_{xy}+\\delta_{z}d_{z})^{0.6667}$\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table*}\n\n\n\nIn the easy-plane region $-1<\\Delta_{z}<1$, \nany generic analytical way of determining the coefficients \nin Eq.~(\\ref{eq:dimer}) has never been known except for \nthe above special point $\\Delta_z=\\delta_z=0$. \nTo obtain $d_{xy}$ (respectively $d_{z}$), we numerically \ncalculate excitation gaps \nat the points $\\delta_{xy}$ $(\\delta_{z})=0.05$, \n0.1, $\\cdots$, 0.3 with fixing $\\delta_{z}(\\delta_{xy})=0$. \nAlthough both $\\Delta E_{\\Delta S_{\\rm tot}^{z}=0}$ and \n$\\Delta E_{\\Delta S_{\\rm tot}^{z}=\\pm 1}$ are applicable to determine \n$d_{xy,z}$ in principle, \nwe use only the latter gap since it more smoothly converges \nto its thermodynamic-limit value via Aitken-Shanks process, \ncompared to the former. \nIn fact, Eq.~(\\ref{eq:gap_correction}) suggests that \n$\\Delta E_{\\Delta S_{\\rm tot}^{z}=0}$ is subject to effects of \nirrelevant perturbations and therefore contains complicated \nfinite-size corrections. \nCoefficients $d_{xy}$ ($d_{z}$) can be determined for each \n$\\delta_{xy}$ ($\\delta_{z}$) from Eq.~(\\ref{eq:dimergap}). \nSince the field theory result~(\\ref{eq:dimergap}) is generally \nmore reliable as the perturbation $\\delta_{xy,z}$ is smaller, \nwe should compare Eq.~(\\ref{eq:dimergap}) with excitation gaps \ndetermined at sufficiently small values of $\\delta_{xy,z}$. \nHowever, the extrapolation to thermodynamic limit by Aitken-Shanks method\nis less precise in such a small dimerization region \nmainly due to large finite-size effects.~\\cite{Papenbrock03,Orignac04}\nTherefore, we adopt coefficients $d_{xy,z}$ extracted from the gaps at \nrelatively large dimerization $\\delta_{xy(z)}=0.1$ and $0.3$, and \nthey are listed in Table~\\ref{tb:dimer_coeff}: the values outside \n[inside] parentheses are the data for \n$\\delta_{xy(z)}=0.3$ [0.1]. The anisotropy dependence of \nthe same data $d_{xy,z}$ is depicted in Fig.~\\ref{fig:dimer_coeff}. \nThe data in Table~\\ref{tb:dimer_coeff} and Fig.~\\ref{fig:dimer_coeff} \nare the main result of this paper. The difference \nbetween $d_{xy(z)}$ outside and inside the parentheses \nin Table~\\ref{tb:dimer_coeff} could be interpreted as \nthe \"strength\" of irrelevant perturbations neglected in the effective \nsine-Gordon theory or the \"error\" of our numerical strategy. \nThe neglected operators must \nbring a renormalization of coefficients $d_{xy,z}$, and the \"error\" \nwould become larger as the system approaches the Heisenberg point \nsince (as already mentioned) the $\\lambda$ term becomes marginal \nat the point.\n\n\n\n\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{dimercoeff_v10.eps}\n\\end{center}\n\\caption{(Color online) XXZ-anisotropy ($\\Delta_z$) dependence of \ndimer coefficients $d_{xy}$ and $d_z$. Filled [Open] circles represent \n$d_{xy}$ determined from dimerization gap at \n$(\\delta_{xy},\\delta_z)=(0.3,0)$ [$=(0.1,0)$]. Similarly, \nfilled [open] triangles show $d_z$ determined from \ndimerization gap at $(\\delta_{xy},\\delta_z)=(0,0.3)$ [$=(0,0.1)$].} \n\\label{fig:dimer_coeff}\n\\end{figure}\n\n\n\n\n\nWe here discuss the validity of the numerically determined \n$d_{xy,z}$ in Table~\\ref{tb:dimer_coeff} and Fig.~\\ref{fig:dimer_coeff}. \nTable~\\ref{tb:dimer_coeff} shows that in the wide range \n$-0.3\\alt \\Delta_z\\alt0.9$, the difference (error) between $d_{xy,z}$ \noutside and inside the parentheses is less than 8 $\\%$. As expected, \none finds that the error gradually increases when the anisotropy \n$\\Delta_z$ approaches unity. Similarly, the error is large in \nthe deeply ferromagnetic regime $\\Delta_z\\alt-0.3$. \nThis is naturally understood from the fact that as $\\Delta_z$ is \nnegatively increased, the dimerization term $\\sin(\\sqrt{4\\pi}\\phi)$ \nbecomes less relevant and effects of other irrelevant terms is \nrelatively strong. Indeed, for $\\Delta_z<-0.7071$ ($K>2$), \nthe dimerization does not yield any spin gap and our method of \ndetermining $d_{xy,z}$ cannot be used. \nFurthermore, it is worth noting that the spin gap is \nconvex downward as a function of dimerization $\\delta_{xy,z}$ \nin the ferromagnetic side $\\Delta_z<0$, and \nthe accuracy of the fitting therefore depreciates. \n\n\nIn addition to coefficients $d_{xy,z}$, \nlet us examine dimerization gaps and \nthe quality of fitting by Eq.~(\\ref{eq:dimergap}). \nExcitation gaps for $\\Delta_{z}=0.6$ are shown \nin Fig.~\\ref{fig:gap_Del0_6} as an example.\nRemarkably, both soliton-gap curves~(\\ref{eq:dimergap}) with the values \n$d_{xy,z}$ outside and inside the parentheses in Table~\\ref{tb:dimer_coeff} \nfit the numerical data $\\Delta E_{\\Delta S_{\\rm tot}^{z}=\\pm 1}$ \nin the broad region $0\\leq\\delta_{xy(z)}\\leq 0.3$ with reasonable accuracy. \nThe former solid curve is slightly better that the latter. \nThe breather gaps $\\Delta E_{\\Delta S_{\\rm tot}^{z}=0}$ and \ncorresponding fitting curves \nare also shown in Fig.~\\ref{fig:gap_Del0_6}. \nThis breather curve is determined by combining the solid \ncurve~(\\ref{eq:dimergap}) and the soliton-breather \nrelation~(\\ref{eq:mass_relation}). \nIt slightly deviates from numerical data, especially, \nin a relatively large dimerization regime $0.15\\alt \\delta_{xy(z)}$. \nAs mentioned above, this deviation would be attributed to \nirrelevant perturbations. \nThe breather-soliton mass ratio $E_{B_{1}}\/E_{S}$ \n[see Eq.~(\\ref{eq:mass_relation})] in the sine-Gordon \nmodel~(\\ref{eq:XXZ_dimer_eff}) and the numerically evaluated \n$\\Delta E_{\\Delta S_{\\rm tot}^{z}=0}\/\\Delta E_{\\Delta S_{\\rm tot}^{z}=\\pm1}$ \nare shown in Fig.~\\ref{fig:compare_ED_SG}. \nThese two values are in good agreement with each other \nin the wide parameter region \n$-0.5<\\Delta_{z}< 1$, although their difference becomes slightly \nlarger in the region $0.5\\alt\\Delta_{z}\\alt 1$, which includes \nthe point $\\Delta_z=0.6$ in Fig.~\\ref{fig:gap_Del0_6}. \nGaps $\\Delta E_{\\Delta S_{\\rm tot}^{z}=\\pm 1}$ for dimerized XXZ \nchains with several values of both $\\delta_{xy}$ and $\\delta_z$ are plotted \nin Fig.~\\ref{fig:gap_Del0_6all}. It shows that the numerical data are \nquantitatively fitted by the {\\it single} gap formula~(\\ref{eq:dimergap}). \nAll of the results in Figs.~\\ref{fig:gap_Del0_6}-\\ref{fig:gap_Del0_6all} \nindicates that a simple sine-Gordon model~(\\ref{eq:XXZ_dimer_eff}) can \ndescribe the low-energy physics of the dimerized spin \nchain~(\\ref{eq:dimer_XXZ}) \nwith reasonable accuracy in the wide easy-plane regime. \nThis also supports the validity of our \nnumerical approach for fixing the coefficients $d_{xy,z}$. \n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.35\\textwidth]{gap_Del0_6_v04.eps}\n\\end{center}\n\\caption{(Color online) Excitation gaps \n$\\Delta E_{\\Delta S_{\\rm tot}^{z}=\\pm 1}$ (circles) and \n$\\Delta E_{\\Delta S_{\\rm tot}^{z}=0}$ (triangles) \nof the dimerized XXZ model~(\\ref{eq:dimer_XXZ}) with $\\Delta_z=0.6$. \nSolid and dashed-dotted curves are fitted by the gap \nformula~(\\ref{eq:dimergap}) with coefficients outside and \ninside parentheses in Table~\\ref{tb:dimer_coeff}, respectively. \nThe dashed curve represents the lightest breather mass which is fixed \nby combining the solid soliton curve and Eq.~(\\ref{eq:mass_relation}).} \n\\label{fig:gap_Del0_6}\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.35\\textwidth]{bs_ratio_v04c.eps}\n\\end{center}\n\\caption{(Color online) Ratio between two numerically evaluated gaps \n$\\Delta E_{\\Delta S_{\\rm tot}^{z}=0}\/\\Delta E_{\\Delta S_{\\rm tot}^{z}=\\pm1}$\n(circles) in the dimerized chain~(\\ref{eq:dimer_XXZ}) \nwith $\\delta_{xy}=0.3$ and $\\delta_z=0$. Solid curve is the soliton-breather \nmass ratio $E_{B_1}\/E_S$ in the effective sine-Gordon \ntheory~(\\ref{eq:XXZ_dimer_eff}). Note that in the ferromagnetic side \n$\\Delta_z<0$, there is no breather and $E_{B_1}$ is replaced with \nthe mass gap of soliton-antisoliton scattering states $2E_S$, \nnamely, $E_{B_1}\/E_S\\to 2E_S\/E_S=2$.}\n\\label{fig:compare_ED_SG}\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.35\\textwidth]{gap_all_v04.eps}\n\\end{center}\n\\caption{(Color online) Numerically evaluated gaps \n$\\Delta E_{\\Delta S_{\\rm tot}^{z}=\\pm1}$ of \ndimerized XXZ chains with several values of \nboth parameters $\\delta_{xy}$ and $\\delta_z$ at $\\Delta_z=0.6$ and $-0.2$. \nSolid curves are Eq.~(\\ref{eq:dimergap}) with $d_{xy}$ and $d_z$ \nin Table~\\ref{tb:dimer_coeff}. In the ferromagnetic case $\\Delta_z=-0.2$, \nthe analytical curve successfully fits the numerical data for a wide \nweakly-dimerized regime $\\delta_{xy,z}\\ll 1$, \nwhile the deviation occurs for the strongly-dimerized one.}\n\\label{fig:gap_Del0_6all}\n\\end{figure}\n\n\n\n\n\\subsection{Dimer coefficients of SU(2)-symmetric models}\n\\label{subsec:SU2_dimer}\nAt the $SU(2)$-symmetric AF point, \nthe $\\lambda$ term in the effective Hamiltonian~(\\ref{eq:XXZ_eff_H}) \nbecomes marginal and induces logarithmic corrections to several \nphysical quantities. Such a logarithmic fashion often makes the accuracy \nof numerical methods decrease. \nInstead of numerical approaches, using the asymptotic form \nof the spin correlation function~\\cite{Affleck98} \nand OPE technique,~\\cite{Gogolin,Tsvelik} \nOrignac~\\cite{Orignac04} has predicted \n\\begin{equation}\n\\label{eq:dimer_SU2}\nd_{xy}=2d_z = \\frac{2}{\\pi^{2}}\\left(\\frac{\\pi}{2}\\right)^{1\/4}=0.2269\n\\end{equation}\nat the $SU(2)$-symmetric point. Substituting Eq.~(\\ref{eq:dimer_SU2}) \ninto Eq.~(\\ref{eq:dimergap}), \nthe spin gap in a $SU(2)$-symmetric AF chain with dimerization \n$\\delta_{xy}=\\delta_z\\equiv\\delta$ (${\\cal H}^{{\\rm XXX}\\mathchar`-\\delta}$) \nis determined as\n\\begin{align}\n\\label{eq:gap_SU2_dimer}\n\\Delta E_{\\rm su2}\/J = 1.723 \\delta^{2\/3}.\n\\end{align}\nThe marginal term however produces a correction to this result. \nIt has been shown in Ref.~\\onlinecite{Orignac04} that the \nspin gap in the model ${\\cal H}^{{\\rm XXX}\\mathchar`-\\delta}$ \nis more nicely fitted with\n\\begin{equation}\n\\label{eq:gap_SU2_dimer_corr}\n\\Delta E_{\\rm su2}\/J = \\frac{1.723\\delta^{2\/3}}\n{\\left(1+0.147\\ln\\Big|\\frac{0.1616}{\\delta}\\Big|\\right)^{1\/2}},\n\\end{equation}\nfrom the renormalization-group argument. \nAs can be seen from Eq. (\\ref{eq:gap_SU2_dimer_corr}), \nthe logarithmic correction is not significantly large \nfor the spin gap. We may therefore apply the way based on the \nsine-Gordon model in Sec.~\\ref{subsec:XXZ_dimer} \neven for the present AF Heisenberg model. The resultant data are listed in \nthe first line of Table~\\ref{tb:dimer_coeff}. \nEvaluated coefficients $d_{xy}=0.228$ (0.204) and $d_{z}=0.110$ (0.097) \nare fairly close to the results of Eq.~(\\ref{eq:dimer_SU2}). \nThis suggests that the effect of the marginal operator on \nthe spin gap is really small. \nWe should also note that $d_{xy}= 2d_z$ is approximately \nrealized, which is required from the $SU(2)$ symmetry. \nThe numerically calculated spin gap $\\Delta E_{\\rm su2}$, \nEq.~(\\ref{eq:gap_SU2_dimer_corr}), and the curve of the gap \nformula~(\\ref{eq:dimergap}) \nare shown in Fig.~\\ref{fig:gap_Heisen_dimer}(a). It is found that \neven the curve without any logarithmic correction \ncan fit the numerical data within semi-quantitative level. \nAt least, parameters $d_{xy,z}$ at the $SU(2)$-symmetric point can \nbe regarded as effective coupling constants when we naively approximate \na dimerized Heisenberg chain as a simple sine-Gordon model. \n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.35\\textwidth]{gap_Heisen_dimer_v04.eps}\n\\end{center}\n\\caption{(Color online) Spin gaps (circles) of (a) the Heisenberg model \nwith dimerization $\\delta_{xy}=\\delta_{z}=\\delta$ and \n(b) the dimerized $J$-$J_{2}$ model~(\\ref{eq:modified_dimer}).\nBoth solid curves in panels (a) and (b) are determined from \nthe gap formula~(\\ref{eq:dimergap}). The dashed curve in panel (a) \nrepresents Eq.~(\\ref{eq:gap_SU2_dimer_corr}).} \n\\label{fig:gap_Heisen_dimer}\n\\end{figure}\n\nAs discussed in Sec.~\\ref{subsec:J-J2}, logarithmic corrections vanish \nin the $J$-$J_{2}$ model~(\\ref{eq:J-J2chain}) due to the absence of \nthe marginal operator. As expected, Fig.~\\ref{fig:gap_Heisen_dimer}(b) \nshows that the spin gap $\\Delta E_{\\rm su2}$ is accurately fitted by \nthe sine-Gordon gap formula~(\\ref{eq:dimergap}) in the wide range \n$0\\leq\\delta\\leq0.3$. Therefore, the coefficients $d_{xy,z}$ of the $J$-$J_2$ \nmodel (the final line of Table~\\ref{tb:dimer_coeff}) are highly reliable. \nRemarkably, the difference between the values \noutside and inside the parentheses is much smaller than that \nof the Heisenberg model (the first and last line of \nTable~\\ref{tb:dimer_coeff}). \nHere, to determine $d_{xy,z}$ of the $J$-$J_{2}$ model, \nwe have used its spinon velocity $v=1.174Ja$, \nwhich has been evaluated in Ref.~\\onlinecite{Okamoto97}. \n\n\n\n\n\n\n\n\n\n\\subsection{Coefficients of spin operator}\n\\label{subsec:spin_op}\nIn this subsection, we discuss the spin-operator coefficient $a_{1}$ \nin Eq.~(\\ref{eq:spin_boson}). \nAlthough $a_1$ for the easy-plane XXZ model has been evaluated \nanalytically~\\cite{Lukyanov97,Lukyanov99,Lukyanov03} and \nnumerically,~\\cite{Hikihara98,Hikihara04} those for \nthe $SU(2)$-symmetric Heisenberg chain and the $J$-$J_2$ model \nhave never been studied. The existent data also help us to check \nthe validity of our method. \nFrom the bosonization formula~(\\ref{eq:spin_boson}), the $z$-component \nspin correlation function has the following asymptotic form: \n\\begin{align}\n\\langle S_{j}^{z}S_{j'}^{z}\\rangle=\n-\\frac{1}{4\\pi^{2}\\eta|j-j'|^{2}}+\n\\frac{A_{1}^{z}(-1)^{j-j'}}{|j-j'|^{1\/\\eta}}+\\cdots,\n\\end{align}\nin the easy-plane TLL phase. \nThe amplitude $A_1^z$ is related to $a_1$ as\n\\begin{align}\nA_{1}^{z}=a_{1}^{2}\/2.\n\\label{eq:amp_coeff}\n\\end{align}\nLukyanov and his collaborators \\cite{Lukyanov97,Lukyanov99} have predicted \n\\begin{align}\n&A_{1}^{z}=\\frac{2}{\\pi^{2}}\\left[\\frac{\\Gamma(\\frac{\\eta}{2-2\\eta})}\n{2\\sqrt{\\pi}\\Gamma(\\frac{1}{2-2\\eta})}\\right]^{1\/\\eta}\\nonumber\\\\\n&\\times\\exp\\left[\\int_{0}^{\\infty}\\frac{{\\rm d}t}{t}\n\\left(\\frac{\\sinh[(2\\eta-1)t]}{\\sinh(\\eta t)\\cosh[(1-\\eta)t]}-\\frac{2\\eta-1}{\\eta}\n{\\rm e}^{-2t}\\right)\\right].\n\\label{eq:a1_anal}\n\\end{align}\nThe same amplitude has been calculated by using DMRG in \nRefs.~\\onlinecite{Hikihara98,Hikihara04}. \n\n\\begin{table*}\n\\caption{\\label{tb:a1} Spin-operator coefficients $a_1$ \nof spin-$\\frac{1}{2}$ XXZ chain and the $J$-$J_2$ chain. \nValues in column (A), (B), and (C) correspond to the analytical prediction \nfrom Refs.~\\onlinecite{Lukyanov97,Lukyanov99,Lukyanov03}, the result by \nDMRG in Refs.~\\onlinecite{Hikihara98,Hikihara04}, and ours, respectively.}\n\\begin{ruledtabular}\n\\begin{tabular}{lllllll}\n\\multicolumn{1}{c}{$\\Delta_z$} & \n\\multicolumn{1}{c}{$a_1$ (A)\n} & \n\\multicolumn{1}{c}{$a_1$ (B)\n} & \n\\multicolumn{1}{c}{$a_1$ (C) \n} \n& \\multicolumn{1}{c}{$\\eta$} &\n\\multicolumn{1}{c}{$v\/(Ja)$} & \\multicolumn{1}{c}{soliton gap $E_S\/J$} \\\\\n\\hline\n1 & & & 0.4724 (0.4325) & 1 & 1.571 & $3.535(a_{1}h_{\\rm s}\/J)^{0.6667}$\\\\\n0.9 & 0.7049 & 0.64 & 0.5327 (0.4830) & 0.8564 & 1.518 & $3.268(a_{1}h_{\\rm s}\/J)^{0.7061}$\\\\\n0.8 & 0.6069 & 0.587 & 0.5226 (0.4808) & 0.7952 & 1.465 & $3.147(a_{1}h_{\\rm s}\/J)^{0.7293}$\\\\\n0.7 & 0.5464 & 0.54 & 0.5019 (0.4693) & 0.7468 & 1.410 & $3.057(a_{1}h_{\\rm s}\/J)^{0.7516}$\\\\\n0.6 & 0.5008 & 0.499 & 0.4771 (0.4530) & 0.7048 & 1.355 & $2.986(a_{1}h_{\\rm s}\/J)^{0.7748}$\\\\\n0.5 & 0.4629 & 0.4626 & 0.4505 (0.4338) & 0.6667 & 1.299 & $2.934(a_{1}h_{\\rm s}\/J)^{0.8}$ \\\\\n0.4 & 0.4297 & 0.4297 & 0.4235 (0.4127) & 0.6310 & 1.242 & $2.902(a_{1}h_{\\rm s}\/J)^{0.8281}$\\\\\n0.3 & 0.3994 & 0.3995 & 0.3966 (0.3903) & 0.5970 & 1.184 & $2.893(a_{1}h_{\\rm s}\/J)^{0.8602}$\\\\\n0.2 & 0.3712 & 0.3713 & 0.3701 (0.3670) & 0.5641 & 1.124 & $2.918(a_{1}h_{\\rm s}\/J)^{0.8980}$\\\\\n0.1 & 0.3443 & 0.3443 & 0.3440 (0.3430) & 0.5319 & 1.063 & $2.991(a_{1}h_{\\rm s}\/J)^{0.9434}$\\\\\n0 & 0.3183 & 0.3183 & 0.3183 (0.3183) & 0.5 & 1 & $3.141(a_{1}h_{\\rm s}\/J)$ \\\\\n$J$-$J_2$ model & & & 0.4693 (0.4668) & 1 & 1.174 & $3.208(a_{1}h_{\\rm s}\/J)^{0.6667}$\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table*}\n\nIn order to determine $a_{1}$, we use XXZ models \nin a staggered field~(\\ref{eq:XXZ_stag}). \nFollowing the similar way to Sec.~\\ref{subsec:XXZ_dimer}, \nwe can extract the coefficient $a_1$ by fitting numerically evaluated \ngaps of the model~(\\ref{eq:XXZ_stag}) \nthrough the sine-Gordon gap formula~(\\ref{eq:staggap}). \nWe numerically estimate the gaps at $h_{\\rm s}\/J=0.01$, 0.02, $\\cdots$, 0.09, \n0.1, 0.2, and 0.3 via Aitken-Shanks method. \nThe results are listed in column (C) of Table~\\ref{tb:a1}. \nSimilarly to the case of dimerization, we adopt spin gaps at \nrelatively large staggered fields $h_s\/J=0.1$ and $0.3$ to determine \nthe coefficients $a_1$. The value outside (inside) the parentheses \nin Table~\\ref{tb:a1} corresponds to $a_1$ fixed at $h_{\\rm s}\/J=0.1$ (0.3). \nNote that the XY model in a staggered field is solvable through \nJordan-Wigner transformation, and as a result \nthe coefficient $a_1$ is exactly evaluated as\n\\begin{equation}\n\\label{eq:XY_stagg}\na_1 = 1\/\\pi=0.3183.\n\\end{equation}\nThe table clearly shows that the values \nat $h_{\\rm s}\/J=0.1$ are closer to those of the previous prediction in \nRefs.~\\onlinecite{Lukyanov97,Lukyanov99,Lukyanov03,Hikihara98,Hikihara04}.\nWe emphasize that our results gradually deviate from the analytical \nprediction from Eq.~(\\ref{eq:a1_anal}) as the system approaches \nthe $SU(2)$-symmetric point. The same property also appears \nin the DMRG results in Refs.~\\onlinecite{Hikihara98,Hikihara04}. \nActually, $A_1^z$ in Eq.~(\\ref{eq:a1_anal}) diverges when $\\Delta_z\\to1$. \nHowever, the bosonization formula~(\\ref{eq:spin_boson}) for spin \noperators must be still used even around $\\Delta_z=1$. \nThus we should realize that the relation~(\\ref{eq:amp_coeff}) is \nbroken and $a_1$ remains to be finite at the $SU(2)$-symmetric point. \nFigure~\\ref{fig:gap_Del0_9_stag} represents the numerically evaluated \ngaps $\\Delta E_{\\Delta S_{\\rm tot}^{z}=\\pm1}$, and three fitting curves \nfixed by $a_1$ (A) and $a_1$ (C) outside and inside the parentheses in \nTable~\\ref{tb:a1}. Our coefficient $a_1$ successfully fits the numerical \ndata semi-quantitatively in the wide regime $0.01\\alt h_{\\rm s}\/J\\alt 0.3$, \nwhile the curve of $a_1$ (A) is valid only in an extremely weak \nstaggered-field regime $00$ \ninduces the Zeeman splitting, and the gap of the magnon excitation \nwith $S^z=1$ ($-1$) decreases (increases) as \n$\\Delta E_{\\Delta S_{\\rm tot}^z=\\pm1}\\mp H$. \nWhen $H$ becomes larger than the value of the zero-field spin gap, \nthe $S^z=1$ magnon condensation takes place and \na field-induced TLL phase emerges with an incommensurate Fermi wave number \n$k_F=\\pi-2\\pi \\langle S_j^z\\rangle$. \nTherefore, the curve of the spin gap \nas a function of dimerization $\\delta$ \nis directly interpreted as the ground-state phase boundary of \nthe model~(\\ref{eq:XXZ_dimer_field}), if the vertical axis (spin gap) is \nreplaced with the strength of the magnetic field $H$. \nIt is shown in Fig.~\\ref{fig:dimer_field}. \n\nThe critical point between the dimerized and TLL phases \ncan be determined from experiments with varying $H$. \nComparing the experimentally obtained critical field $H_c$ and \nthe phase diagram of Fig.~\\ref{fig:dimer_field} in \nquasi 1D dimerized spin-$\\frac{1}{2}$ compounds,\none can evaluate the strength of the dimerization $\\delta$. \n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{dimer_magfield_v03.eps}\n\\end{center}\n\\caption{(Color online) Ground-state phase diagram of \nthe dimerized spin chains under a magnetic field $H$, \nEq.~(\\ref{eq:XXZ_dimer_field}). Each curve represents the phase boundary \nbetween the dimer and field-induced TLL phases.}\n\\label{fig:dimer_field}\n\\end{figure}\n\n\n\n\n\\subsection{Two-leg spin ladder with a four-spin interaction}\nWe next consider an $SU(2)$-symmetric two-leg spin-$\\frac{1}{2}$ AF \nladder with a four-spin exchange, whose Hamiltonian is given by \n\\begin{align}\n{\\cal H}^{\\rm lad} =& \\sum_{j}\\sum_{r=1,2}\nJ\\bol{S}_{r,j}\\cdot\\bol{S}_{r,j+1}\n+\\sum_j J_\\perp \\bol{S}_{1,j}\\cdot\\bol{S}_{2,j}\\nonumber\\\\\n&+\\sum_j\nJ_4 (\\bol{S}_{1,j}\\cdot\\bol{S}_{1,j+1})\n(\\bol{S}_{2,j}\\cdot\\bol{S}_{2,j+1}). \n\\label{eq:ladder}\n\\end{align}\nThe symbol $r$ denotes the chain index. Three quantities $J>0$, $J_\\perp$ \nand $J_4$ respectively stand for the intrachain-, interchain- and \nfour-spin coupling constants. There are at least two kinds of \nphysical origin of the four-spin term $J_4$. The first is that \noptical phonon modes with a spin-Peierls type coupling can cause a negative \n$J_{4}$.~\\cite{Nersesyan97} \nThe second is that the higher-order expansion of hopping terms \nin half-filled electron ladders \nwith a strong on-site Coulomb repulsion.~\\cite{Takahashi77,MacDonald88} \nIn fact, the cyclic exchange term defined on each plaquette in the ladder \ncontains a positive $J_4$ term, which is known to have scaling dimension 1 \nand be most relevant in all the four-spin couplings of \nthe cyclic term in the weak rung-coupling regime $J\\gg |J_\\perp|,|J_4|$. \n\n\nThe model~(\\ref{eq:ladder}) has been analyzed \nby some groups.~\\cite{Nersesyan97,Kolezhuk98,Nomura09} \nThere appear four kinds of competing phases: the rung-singlet, Haldane, \ncolumnar-dimer, and staggered dimer phases.~\\cite{Starykh04,Starykh07} \nIn particular, the ground-state phase diagram in the region of \n$J_\\perp>0$ and $J_4>0$ has been numerically \ncompleted in Ref.~\\onlinecite{Nomura09}. \n\n\nHere, we show that the data in Tables~\\ref{tb:dimer_coeff} and \\ref{tb:a1} \nallow us to construct the phase diagram of the model~(\\ref{eq:ladder}) \nin the weak rung-coupling regime with reasonable accuracy. \nFrom the bosonization, the low-energy effective Hamiltonian of \nEq.~(\\ref{eq:ladder}) reads\n\\begin{align}\n{\\cal H}^{\\rm lad}_{\\rm eff} =& \\int dx \\sum_{q=\\pm}\n\\frac{v}{2}[K^{-1}(\\partial_x\\phi_q)^2+K(\\partial_x\\theta_q)^2]\\nonumber\\\\\n& +\\frac{1}{a}(J_\\perp\\frac{\\bar a^2}{2}-J_4\\frac{(3d)^2}{2})\n\\cos(\\sqrt{8\\pi}\\phi_+)\\nonumber\\\\\n&+\\frac{1}{a}(J_\\perp\\frac{\\bar a^2}{2}+J_4\\frac{(3d)^2}{2})\n\\cos(\\sqrt{8\\pi}\\phi_-)\\nonumber\\\\\n&+\\frac{1}{a}J_\\perp\\bar a^2 \\cos(\\sqrt{2\\pi}\\theta_-)+\\cdots. \n\\label{eq:ladder_eff}\n\\end{align}\nHere we have defined boson fields \n$\\phi_\\pm=(\\phi_1\\pm\\phi_2)\/\\sqrt{2}$ and \n$\\theta_\\pm=(\\theta_1\\pm\\theta_2)\/\\sqrt{2}$, where $\\phi_r$ and $\\theta_r$ \nare dual fields of the $r$-th chain (see Sec.~\\ref{subsec:XXZ}). \nIn Eq. (\\ref{eq:ladder_eff}), we have extracted only the most relevant part \nof the rung couplings. The $SU(2)$ symmetry requires the relations \n$v=\\pi Ja\/2$, $K=1\/2$, $a_1=b_0\\equiv\\bar a$ and $d_{xy}=2d_z\\equiv 2d$. \nDue to this symmetry, three vertex terms in Eq.~(\\ref{eq:ladder_eff}) \nhave the same scaling dimension 1. \nThe $(\\phi_+,\\theta_+)$ sector is equivalent to a sine-Gordon model. \nA Gaussian-type transition is expected at \n$J_\\perp \\bar a^2- J_4 (3d)^2=0$ if other irrelevant perturbations are \nnegligible. On the other hand, the $(\\phi_-,\\theta_-)$ sector is \na self-dual sine-Gordon model,~\\cite{Lecheminant02} \nwhich is known to yield an Ising-type transition due to the competition \nbetween $\\cos(\\sqrt{8\\pi}\\phi_-)$ and $\\cos(\\sqrt{2\\pi}\\theta_-)$. \nThe transition occurs as the strength of \ntwo coupling constants becomes equal, namely, \n$|J_\\perp\\bar a^2+ J_4 (3d)^2|\/2=|J_\\perp\\bar a^2|$. \nSince we have already obtained the values of $\\bar a$ and $d$ \n(see Tables~\\ref{tb:dimer_coeff} and \\ref{tb:a1}), \nwe can draw the phase transition curves in \nthe $J_\\perp$-$J_4$ space in the weak rung-coupling regime, \nwhich are shown in Fig.~\\ref{fig:ladder}. \nThe two transition curves are represented as \n\\begin{subequations}\n\\label{eq:curves}\n\\begin{align}\n\\label{eq:curve1}\nJ_4 =& \\Big(\\frac{\\bar a}{3d}\\Big)^2 J_\\perp \\approx 2.05 J_\\perp,\\\\ \nJ_4 =& -3 \\Big(\\frac{\\bar a}{3d}\\Big)^2 J_\\perp \\approx -6.15 J_\\perp.\n\\label{eq:curve2}\n\\end{align}\n\\end{subequations}\nEach phase is characterized by the locked boson fields and their position: \nIn the columnar [staggered] dimer phase, \n$\\phi_+$ and $\\phi_-$ are respectively pinned at $\\sqrt{\\pi\/8}$ and 0 \n[$0$ and $\\sqrt{\\pi\/8}$] and \n$(-1)^j\\langle{\\bol S}_{1,j}\\cdot{\\bol S}_{1,j+1}\n+{\\bol S}_{2,j}\\cdot{\\bol S}_{2,j+1}\\rangle\\propto \n\\langle\\sin(\\sqrt{2\\pi}\\phi_+)\\cos(\\sqrt{2\\pi}\\phi_-)\\rangle \\neq0$ \n[$(-1)^j\\langle{\\bol S}_{1,j}\\cdot{\\bol S}_{1,j+1}\n-{\\bol S}_{2,j}\\cdot{\\bol S}_{2,j+1}\\rangle\\propto \n\\langle\\cos(\\sqrt{2\\pi}\\phi_+)\\sin(\\sqrt{2\\pi}\\phi_-)\\rangle \\neq0$]. \nIn the rung-singlet (Haldane) phase, $\\theta_-$ is pinned instead of \n$\\phi_-$ and $\\langle\\phi_+\\rangle=\\sqrt{\\pi\/8}$ ($0$), \nwhich corresponds to a non-zero ``even\"-(``odd\"-)type nonlocal string \norder parameter.~\\cite{Shelton96,Kim2000,Nakamura03} \n\n\n\n\nIt has been shown in Ref.~\\onlinecite{Shelton96} that \nEq.~(\\ref{eq:ladder_eff}) can be fermionized. \nThe resulting Hamiltonian consists of\nthree copies of massive Majorana fermions and another one \n(For detail, see e.g. Refs.~\\onlinecite{Shelton96,Tsvelik,Gogolin}). \nThe mass of the Majorana triplet $M_t$ and that of the remaining one \n$M_s$ are given by\n\\begin{subequations}\n\\label{eq:gap_ladder}\n\\begin{align}\nM_t\\propto& J_\\perp\\bar a^2- J_4 (3d)^2, \\\\\nM_s\\propto& 3 J_\\perp\\bar a^2+ J_4 (3d)^2. \n\\end{align}\n\\end{subequations}\nThe transition curves in Fig.~\\ref{fig:ladder} are identified \nwith $M_t=0$ and $M_s=0$. \nAt $M_s=0$, the low-energy physics is governed by the gapless \nsinglet fermion which is equivalent to a critical Ising chain \nin a transverse field. The transition at $M_s=0$ therefore belongs to \nthe Ising universality class with central charge $c=1\/2$. \nOn the other hand, three copies of massless Majorana fermions, \nwhich appear at $M_t=0$, are equivalent to an $SU(2)_2$ \nWess-Zumino-Witten (WZW) theory~\\cite{Tsvelik,Gogolin,Francesco} \nwith central charge $c=3\/2$. Thus, the transition at $M_t=0$ is expected \nto be a $c=3\/2$ (first-order) type if the marginal current-current \ninteraction~\\cite{Shelton96,Starykh04,Starykh07}\nomitted in Eq.~(\\ref{eq:ladder_eff}) is \nirrelevant (relevant). In Ref.~\\onlinecite{Nomura09}, \nthe transition has been proved to be described by \na $SU(2)_2$ WZW theory at least in the region of $J\\gg J_\\perp,J_4>0$. \nThis suggests that the marginal term is irrelevant there. \nThe Majorana fermion with the mass $M_t$ corresponds to \na spin-triplet excitation (magnon), and \nanother fermion with mass $M_s$ is a spin-singlet excitation, \nwhich is believed to be continuously connected to two-magnon bound \nstate observed in the strong rung-coupling regime. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{LadderPhaseDiagram_v04.eps}\n\\end{center}\n\\caption{(Color online) Ground-state phase diagram of the spin ladder~(\\ref{eq:ladder}) \nin the weak rung-coupling regime. There are two transition curves, \n$J_4\\approx 2.05 J_\\perp$ and $J_4\\approx -6.15 J_\\perp$. \nThe former is $c=3\/2$ or first-order type, \nwhile the latter is in the Ising universality class with \n$c=1\/2$ (see the text).}\n\\label{fig:ladder}\n\\end{figure}\n\n\nFinally, we note that \nin the extremely weak rung-coupling limit, the coupling constants of \nvertex operators in Eq.~(\\ref{eq:ladder_eff}) would be less valid \nsince coefficients $\\bar a$ and $d$ are determined from gaps \ninduced by relatively large staggered field ($h_s\/J=0.1$ or $0.3$) \nand dimerization ($\\delta_{xy,z}=0.1$ or $0.3$), respectively. \nThe true transition curves might somewhat deviate from our \nprediction~(\\ref{eq:curves}). \nOur result is expected to be more reliable \nin a moderate rung-coupling regime. In fact, \na numerical study in Ref.~\\onlinecite{Nomura09} has shown that \nthe phase boundary is located at $J_4\/J_\\perp\\sim 2$ around $J_\\perp\/J=0.25$ \n(see Fig.~6 in Ref.~\\onlinecite{Nomura09}), being consistent \nwith Eq.~(\\ref{eq:curve1}). \nWe stress that our coefficients $\\bar a$ and $d$ provides an easy way of \nestimating the phase boundary although it is a rough approximation compared \nwith other sophisticated strategies such as DMRG and renormalization-group \ncalculations. If we replace the intrachain term in Eq.~(\\ref{eq:ladder}) \nwith two $J$-$J_2$ chains~(\\ref{eq:J-J2chain}), the intrachain \nmarginal interaction omitted in Eq.~(\\ref{eq:ladder_eff}) disappears. \nIn this case, the prediction from the effective \ntheory~(\\ref{eq:ladder_eff}) becomes more reliable even in the weak \nrung-coupling limit $J_\\perp\/J,J_4\/J\\to0$. \nFrom the data of the $J$-$J_2$ model in \nTables~\\ref{tb:dimer_coeff} and \\ref{tb:a1}, \ntwo transition curves in the modified ladder are \n\\begin{subequations}\n\\label{eq:curves_v2}\n\\begin{align}\nJ_4 \\approx& 0.69 J_\\perp,\\\\ \nJ_4 \\approx& -2.08 J_\\perp.\n\\end{align}\n\\end{subequations}\n\n\n\n\n\n\n\n\n\n\\subsection{Optical response of dimerized spin chains}\nOptical responses in Mott insulators including multiferroic compounds \nhave been investigated intensively.\nQuite recently, the authors in Ref.~\\onlinecite{Katsura09} have \ntheoretically studied the optical conductivity in a 1D \nionic-Hubbard type Mott insulator with Peierls instability, \nwhose strong coupling limit is equal to a \nspin-$\\frac{1}{2}$ dimerized Heisenberg chain, ${\\cal H}^{{\\rm XXX}-\\delta}$. \nThe results in Ref.~\\onlinecite{Katsura09} would be relevant to, \nfor example, organic Mott insulators such as TTF-BA.~\\cite{Kagawa10} \nIn this system, the uniform electric polarization $P$ along the 1D chain \nis shown to be proportional to the dimer operator:\n\\begin{equation}\nP = g a\\sum_j(-1)^j {\\bol S}_{j}\\cdot{\\bol S}_{j+1},\n\\end{equation}\nwhere $g$ is the coupling constant between the polarization and dimer \noperators. Therefore, $P$ can be bosonized as \n\\begin{equation}\nP\\approx 3d g \\int dx \\sin(\\sqrt{4\\pi}\\phi(x))+\\cdots,\n\\label{eq:P_bosonized}\n\\end{equation}\nwith $d_{xy}=2d$ and $d_{z}=d$. \nFrom Eq.~(\\ref{eq:P_bosonized}), one can calculate $P$ and \nrelated observables by means of the bosonization \nfor the dimerized spin chain. It has been shown that \nthe spin-singlet excitation, i.e., the breather with mass $E_{B_2}$, \nis observed as the lowest-frequency sharp peak \nin the optical conductivity measurements. Since the mass $E_{B_2}$ is evaluated \nfrom the sine-Gordon theory as \n\\begin{equation}\nE_{B_2}\/J=\\sqrt{3}E_S\/J=2.924 \\delta^{2\/3},\n\\end{equation}\nwe can extract the value of $\\delta$ from the peak position \nof the optical conductivity. The exact expectation value of \nvertex operators in the sine-Gordon model has been predicted in \nRef.~\\onlinecite{Lukyanov97}. According to it, \nthe polarization density is calculated to be \n\\begin{equation}\n\\langle P\\rangle\/L=({\\cal A}\/3)^{3\/2}(E_S a\/v)^{1\/2}3dg,\n\\end{equation}\nwith ${\\cal A}\\approx3.041$ and $L$ being the chain length. \nThis provides an experimental way of estimating the coupling constant \n$g$, which is usually difficult to determine \nin other multiferroic compounds. \n\n\n\n\n\\section{Conclusions}\n\\label{sec:con}\nWe have numerically evaluated coefficients of \nbosonized dimer and spin operators in spin-$\\frac{1}{2}$ XXZ \nmodel~(\\ref{eq:XXZ}) and $J$-$J_2$ \nmodel~(\\ref{eq:J-J2chain}), \nby using the correspondence between the excitation gap of deformed models \nwith dimerization (or with staggered Zeeman term) and the gap formula \nfor the sine-Gordon theory. \nThis is a new strategy\nrelying on a solid relationship between the lattice models \nand their low-energy effective theories. \nOur numerical approach is relatively easy compared with \nanother method based on DMRG, developed in \nRefs.~\\onlinecite{Hikihara98,Hikihara04}, although the accuracy \nis expected to be better in the latter method.\nThe obtained coefficients are summarized in \nTables~\\ref{tb:dimer_coeff} and \\ref{tb:a1} and Fig.~\\ref{fig:dimer_coeff}. \nIn addition to these coefficients, \nwe have pointed out a dangerous nature of applying \nthe correlation amplitude~(\\ref{eq:a1_anal}) \nas coefficients of bosonized spin operators near the \n$SU(2)$-symmetric point $\\Delta_z=1$ in Sec.~\\ref{subsec:spin_op}. \nFurthermore, we have also used the formula for ground-state \nenergy of sine-Gordon model to calculate the same dimer coefficients\nin Sec.~\\ref{subsec:gs}. We conclude that the excitation-gap \nformula~(\\ref{eq:dimergap}) is more suitable than the ground-state \nenergy formula~(\\ref{eq:E_GS}) for determining coefficients \nof bosonized operators. \n\n\nPhysical quantities associated with dimer and spin operators \ncan be evaluated accurately by utilizing the dimer and spin coefficients. \nAs examples, we have determined ground-state phase diagrams of \ndimerized spin chains in a uniform field and \na two-leg spin ladder with a four-spin interaction \nin Sec.~\\ref{sec:apply}. In addition, we have \nshown how to estimate the electromagnetic coupling constant and \nthe strength of the dimerization from the optical observables \nin a ferroelectric dimerized spin chain. \nThese applications clearly indicate high potential of the data in \nTables~\\ref{tb:dimer_coeff} and \\ref{tb:a1}. \n\n\nAn interesting future direction is to apply a similar method to \nother 1D systems including fermion and boson models. \nOur method in this paper can be applied to lattice systems which \nhave a well-established low-energy effective theory, in principle. \n\n\n\n\n\\begin{acknowledgements}\nThe authors thank Kiyomi Okamoto, Masaki Oshikawa, and T\\^oru Sakai \nfor useful comments.\nS.T. and M.S. were supported by Grants-in-Aid for JSPS Fellows \n(Grant No.\\ 09J08714) and for Scientific Research from MEXT \n(Grant No.\\ 21740295 and No. 22014016), respectively. \nThe program package, TITPACK version 2.0, developed by Hidetoshi Nishimori, \nwas used in the numerical diagonalization in Sec.~\\ref{sec:delta}. \n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \\label{sec:Intro}\nThe muon anomalous magnetic moment is one of the most precisely measured observables in particle physics. It can also be predicted by theory with a high accuracy, and is therefore an ideal quantity for testing the standard model and for finding possible deviations from it caused by new physics \\cite{CzMa, DaMa}. The great interest in muon anomaly is motivated by the present discrepancy of about 3 to 4 $\\sigma$ between theory and experiment. For recent reviews, see Refs. \\cite{MiRa, JeNy, Miller, PDG} (see also the bibliography in e.g, \\cite{Carloni:2015}). New generation measurements of muon $g-2$ planned at Fermilab \\cite{Fermi} and JPARC \\cite{JPARC} are expected to produce results with experimental errors at the level of $16 \\times 10^{-11}$, a factor of 4 smaller compared to the Brookhaven measurement \\cite{BNL06}. This therefore requires a precision at the same level also for the theoretical result.\n\nThe largest theoretical uncertainties are related to the hadronic contribution to $a_\\mu$, which comes mainly from energies at which the confined quarks are strongly interacting and the QCD perturbative treatment breaks down. The evaluation of the nonperturbative effects is usually done by means of dispersion\nrelations in conjunction with experimental data. Low energy effective theories and lattice QCD are also used. Efforts are currently made to increase the precision of these calculations, regarding both the hadronic vacuum polarization and the hadronic light-by-light scattering (a compilation of recent studies is presented in \\cite{Benayounetal}). \n\nThe hadronic vacuum polarization (VP), which is numerically the most significant term, contributes with about $43 \\times 10^{-11}$ units to the theoretical error. It is dominated by the two-pion contribution, which brings more than 70\\% of the leading-order hadronic contribution. \n\nThe two-pion contribution to the VP is expressed in terms of the modulus squared of the pion electromagnetic form factor. It has been measured in $e^+e^-$-annihilation experiments by CMD2 \\cite{CMD2,CMD204,CMD206}, \nSND \\cite{SND}, \\emph{BABAR} \\cite{BaBar,BaBar1}, KLOE \\cite{KLOE1, KLOE2, KLOE3} and BESIII \\cite{BESIII}, and from the hadronic decays of the $\\tau$ lepton by CLEO \\cite{Anderson:1999ui}, ALEPH \\cite{Schael:2005am, Davier:2013sfa}, \n OPAL \\cite{Ackerstaff:1998yj} and Belle \\cite{Fujikawa:2008ma}. Due to experimental difficulties in the identification of low-energy pions, the data below 0.6 GeV have very large uncertainties, except for \\emph{BABAR} and KLOE.\nThe recent data published by BESIII \\cite{BESIII} are restricted to energies above 0.6 GeV. Two new detectors, CMD-3 and SND, now operating at the VEPP-2000 $e^+e^-$ collider in Novosibirsk, are expected to bring accurate data of greatest interest for the $a_\\mu$ evaluation \\cite{Achasov:2014xsa, CMD3, SND2}. Preliminary results reported in \\cite{Fedotovich:2015kna, EidelmanCD} indicate as goal an accuracy comparable to that of \\emph{BABAR} and KLOE experiments. \n\nThe lack of data of sufficient precision at low energies, combined with the fact that the integration kernel exhibits a drastic increase in this region, leads to a relatively large uncertainty of the corresponding contribution to the muon anomaly \\cite{Davier:2009, Davier:2011, Teubner}. \nThe contribution to $a_\\mu$ from below 0.63 GeV, obtained using a fit of the pion form factor in the region near threshold and the direct integration of a compilation of data on $e^+e^-\\to\\pi^+\\pi^-$ cross section between 0.30 and 0.63 GeV, is quoted in \\cite{Davier:2009} with an error of $13.1 \\times 10^{-11}$, while the direct integration in this range of the \\emph{BABAR} data alone leads to an error of $14.7 \\times 10^{-11} $. \n\nThe present large uncertainty of the two-pion contribution to $a_\\mu$ from energies below 0.63 GeV motivated us recently \\cite{Ananthanarayan:2013zua} to investigate it theoretically in a framework based on the analyticity and unitarity properties of the pion form factor. The main idea was to use, instead of the poorly known modulus, the phase of the form factor, which is equal by the Fermi-Watson theorem \\cite{Fermi:2008zz, Watson} to the $\\pi\\pi$ scattering $P$-wave phase shift, which has been calculated with high precision from Chiral Perturbation Theory (ChPT) and Roy equations \\cite{ACGL, Caprini:2011ky, GarciaMartin:2011cn}. Above the inelastic threshold, where the Fermi-Watson theorem is no longer valid and the\nphase of the form factor is not known, we have used an integral condition on the form-factor modulus, derived using measurements of the \\emph{BABAR} experiment \\cite{BaBar,BaBar1} up to\n3 GeV and the asymptotic behavior of the form factor predicted by perturbative QCD \\cite{Farrar:1979aw,Lepage:1979zb, Melic:1998qr} above that energy.\n\n The knowledge of the phase on a part of the unitarity cut and of the modulus on the other part of the cut is not sufficient for uniquely predicting the form factor. However, as shown first in \\cite{IC}, from this information one can derive rigorous upper and lower bounds on the modulus below the inelastic threshold, in particular in the low energy region. To increase the strength of the bounds, we have used as input also several values of the modulus from\nthe region $0.65-0.71$ GeV, measured with higher precision by the $e^+e^-$ experiments CMD2 \\cite{CMD2}, SND \\cite{SND} \\emph{BABAR} \\cite{BaBar,BaBar1} and KLOE 13 \\cite{KLOE3}. \nThe method amounts to a parametrization-free analytic extrapolation\nfrom higher energies to the low energy region of interest for the improved calculation of the muon anomaly.\nIt led to a two-pion contribution to $a_\\mu$ from the region below 0.63 GeV which agreed with other recent determinations and had\na smaller uncertainty \\cite{Ananthanarayan:2013zua}. \n\nIn the present paper we present an update of the work \\cite{Ananthanarayan:2013zua}, improving certain details of the analysis. The main improvement is a proper treatment by Monte Carlo simulations of the statistical errors of the data used as input, which will allow us to attach an uncertainty to the result at a precise confidence level (C.L.). \nAlso, better tools \\cite{MiSc} for combining different predictions accounting for their possible correlations are used.\nIn addition to the $e^+e^-$ data from the region $0.65-0.71$ GeV used as input in \\cite{Ananthanarayan:2013zua}, we also consider the KLOE independent measurements\nreported in \\cite{KLOE2} and the very recent data of BESIII \\cite{BESIII}. We include also the data obtained in the same energy region from\n$\\tau$-lepton decays by the \nCLEO\\cite{Anderson:1999ui}, ALEPH \\cite{Schael:2005am, Davier:2013sfa}, OPAL\\cite{Ackerstaff:1998yj} and Belle \\cite{Fujikawa:2008ma} collaborations. \n\nThe outline of this paper is as follows: \nin Sec. \\ref{sec:aim} we formulate our aim and review the conditions used as input. In Sec. \\ref{sec:input} we give a detailed description of the \nexperimental information used as input and in Sec. \\ref{sec:errors} we describe the Monte Carlo simulation used for implementing the statistical uncertainties of the input data and the prescription of combining the predictions from different experiments. \n Section \\ref{sec:results} contains our results and\nSec. \\ref{sec:conclusion} a summary and our conclusions. The paper has two Appendices: in \nAppendix \\ref{sec:A} we present the solution of the functional extremal problem formulated in Sec. \\ref{sec:aim}, which is the mathematical basis of our approach. In\nAppendix \\ref{sec:B} we discuss the extraction of the pion form factor from the $e^+e^-$ and $\\tau$-decay experiments, giving a short overview of various corrections applied. \n\n\\vspace{0.3cm}\n\\section{Formalism\\label{sec:aim}}\nWe consider the leading order (LO) two-pion contribution to $a_\\mu$, which does not contain the\nvacuum polarization effects but includes one-photon final-state radiation (FSR). We are interested in finding the two-pion contribution to $a_\\mu$ from the interval of energies ranging from $\\sqrt{t_\\text{low}}$ to $\\sqrt{t_\\text{up}}$,\nwhich is expressed in terms of the pion electromagnetic form factor $F(t)$ as\n\\begin{widetext} \n\\begin{equation} \\label{eq:amu}\na_\\mu^{\\pi\\pi(\\gamma), {\\text{LO}}} [\\sqrt{t_\\text{low}}, \\sqrt{t_\\text{up}}] = \\frac{\\alpha^2 m_\\mu^2}{12 \\pi^2}\\int_{t_\\text{low}}^{t_\\text{up}} \\frac{dt}{ t} \\, K(t)\\, \\beta^3_\\pi(t) \\, \n|F(t)|^2 |F_\\omega(t)|^2 \\left(1+\\frac{\\alpha}{\\pi}\\,\\eta_\\pi(t)\\right).\\end{equation}\\end{widetext} \nIn this relation, $\\beta_\\pi(t)=(1-4 m_\\pi\/t)^{1\/2}$ is the two-pion phase space relevant for $e^+ e^-\\to \\pi^+\\pi^-$ annihilation ($m_\\pi$ being the charged pion mass), and\n\\begin{equation} \\label{eq:K}\nK(t) = \\int_0^1 du(1-u)u^2(t-u+m_\\mu^2u^2 )^{-1}\n\\end{equation}\nis the QED kernel function. This function is known to exhibit a drastic increase at low $t$ \\cite{CzMa}.\n\nThe integrand in (\\ref{eq:amu}) contains the pion electromagnetic form factor $F(t)$ in the isospin limit, defined by\n\\begin{equation}\\label{eq:def} \\langle \\pi^+(p')\\vert J_\\mu^{\\rm elm} \n\\vert \\pi^+(p)\\rangle= (p+p')_\\mu F(t), ~ t=(p-p')^2,\n\\end{equation}\n which is a real analytic function in the $t$ complex plane cut along the real semiaxis $t\\ge 4 m_\\pi^2$.\nThe remaining factors in (\\ref{eq:amu}) denote corrections not included in the form factor: $F_\\omega(t)$ accounts for the isospin violation due to $\\rho-\\omega$ mixing and is parametrized as\n\\cite{Leutwyler:2002hm, Hanhart:2012wi}:\n\\begin{align}\\label{eq:rhoomega}\n F_\\omega(t)= 1+\\epsilon\\frac{t}{(m_\\omega-i\\Gamma_\\omega\/2)^2 -t}\\,,\n\\end{align}\nwhere $\\epsilon=1.9\\times 10^{-3}$. Finally, $\\eta_\\pi(t)$ is the FSR correction, calculated in scalar QED \\cite{FSR1, FSR2}.\n\nWe are interested in the contribution to (\\ref{eq:amu}) of the energies below 0.63 GeV. For convenience we shall use in what follows the simplified notation \n\\begin{equation}\\label{eq:not}\na_\\mu\\equiv a_\\mu^{\\pi\\pi(\\gamma),\\, {\\text{LO}}} [2 m_\\pi, 0.63 \\,\\text{GeV}].\n\\end{equation}\n We now formulate the conditions on the form factor $F(t)$ adopted as input for constraining the above quantity. Following Ref. \\cite{Ananthanarayan:2013zua}, we write these conditions as:\n\\vskip0.2cm\n 1. Fermi-Watson theorem \\cite{Fermi:2008zz, Watson}, which implies:\n\\begin{equation}\\label{eq:watson}\n{\\rm Arg} [F(t+i\\epsilon)]=\\delta_1^1(t), \\quad\\quad 4 m_\\pi^2 \\le t \\le t_{\\text{in}},\n\\end{equation}\nwhere $\\delta_1^1(t)$ is the phase-shift of the $P$-wave of $\\pi\\pi$ elastic scattering and \n$t_{\\text{in}}$ is the first inelastic threshold.\n\n 2. Normalization at $t=0$ and the value of the charge radius $\\langle r^2_\\pi \\rangle$, expressed by:\n\\begin{equation}\\label{eq:taylor}\n\tF(0) = 1, \\quad \\quad \\left[\\frac{dF(t)}{dt}\\right]_{t=0} =\\displaystyle\\frac{1}{6} \\langle r^2_\\pi \\rangle.\n\\end{equation}\n\n 3. An integral condition on the modulus squared above the inelastic threshold, written in the form\n\\begin{equation}\\label{eq:L2}\n \\displaystyle\\frac{1}{\\pi} \\int_{t_{\\text{in}}}^{\\infty} dt \\rho(t) |F(t)|^2 \\leq I,\n \\end{equation}\nwhere $\\rho(t)$ is a suitable positive-definite weight, for which the integral converges and an accurate evaluation of $I$ is possible. \n\n 4. The value at one spacelike energy, known from experiment: \n\\begin{equation}\\label{eq:val}\nF(t_s)= F_s \\pm \\epsilon_s, \\qquad t_s<0.\n\\end{equation}\n\n 5. The modulus at one energy in the elastic region of the timelike axis, known from experiment: \n\\begin{equation}\\label{eq:mod}\n|F(t_t)|= F_t \\pm \\epsilon_t, \\qquad 4 m_\\pi^2< t_t n-1$, this is an indication that the individual errors are underestimated. If the ratio $\\chi^2(0)\/(n-1)$ is not very far from 1, the procedure suggested in \\cite{MiSc, PDG} is to rescale the variance $\\sigma^2 (\\bar a)$ calculated with (\\ref{eq:sigma}) by the factor $\\chi^2(0)\/(n-1)$. In our work, this kind of procedure was applied first for combining the results obtained with different measurements by each experiment. Then the predictions of various experiments were combined leading to a global average. \n\n\\begin{figure}[thb]\\vspace{0.5cm}\n\\begin{center}\n \\includegraphics[width = 8cm]{histNotime.eps}\n\\caption{Distribution of $a_\\mu$ values obtained from the Monte Carlo sample of pseudodata, without input modulus. The vertical lines delimitate the region of 68.3\\% C.L. \\label{fig:fig1}}\n\\end{center}\n\\vspace{0.3cm}\n\\end{figure}\n\n\n\\section{Results \\label{sec:results}}\n\n\nIt is instructive to first give the value obtained without using as input the measurements of the timelike modulus.\nIn Fig. \\ref{fig:fig1} we show the distribution of the $a_\\mu$ sample, obtained using as input the Bern phase and the first spacelike point from (\\ref{eq:val}). It may be readily seen that the distribution is not fully symmetrical, as it should be for a Gaussian distribution. From this distribution, by applying the 68.3\\% criterion we obtained for $a_\\mu$ the value $(130.865^{+4.124}_{-5.460}) \\times 10^{-10}$, and Madrid phase gives a similar result, $(131.933 ^{+3.438 }_{-5.922}) \\times 10^{-10}$. Since these results are not statistically independent, the most conservative procedure is to take the simple average of the central values and of the uncertainties. This gives\n\\begin{equation}\\label{eq:nomod}\n a_\\mu^{\\pi\\pi, \\, \\text{LO}}\\,[2m_\\pi ,\\, 0.63 \\, \\text{GeV}]=(131.399 ^{+3.780 }_{-5.691}) \\times 10^{-10}.\n\\end{equation}\nThe large error shows that the constraining power of the phase and the spacelike data is rather low.\n\n\n\n\n\\begin{figure}[thb]\\vspace{0.5cm}\n\\begin{center}\n \\includegraphics[width = 8cm]{histBabar.eps}\n\\caption{Distribution of a typical $a_\\mu$ sample obtained from the Monte Carlo simulation of pseudodata, with an input modulus measured by \\emph{BABAR} in the region $0.65-0.71$ GeV. Details of the input are given in the text. The vertical lines delimitate the region of 68.3\\% C.L. \\label{fig:fig2}}\n\\end{center}\n\\vspace{0.3cm}\n\\end{figure}\n\n\nBy including as input the modulus measured at one energy from the region $0.65 - 0.71$ GeV, the determination (\\ref{eq:nomod}) is considerably improved. In Fig. \\ref{fig:fig2} we show for illustration the distribution of the $a_\\mu$ sample, obtained using as input the Bern phase, the highest energy \\emph{BABAR} point shown in Fig. \\ref{fig:epemtau} and the first spacelike point from (\\ref{eq:val}). The distribution is much narrower than that shown in Fig. \\ref{fig:fig1} and more symmetrical, allowing the extraction of a smaller standard deviation by means of the 68.3\\% C.L. criterion. \n\nSimilar distributions of $a_\\mu$ have been obtained for all the values of the input modulus measured in the region $0.65-0.71$ GeV, shown in Fig. \\ref{fig:epemtau}. The procedure was applied for each of the input phases, Bern and Madrid. The calculation was performed using as input each of the two spacelike values (\\ref{eq:val}) and the best prediction was retained. \n\nIn Figs. \\ref{fig:fig2e} and \\ref{fig:fig2t} we show the 68.3\\% C.L. intervals of $a_\\mu$ obtained from the Monte Carlo simulation described in the previous section, for all the timelike points used as input from the $e^+e^-$ and $\\tau$ experiments. The results have been obtained using as input the Bern phase \\cite{ACGL, Caprini:2011ky}. The Madrid phase leads to similar results. The bounds obtained with various values of the timelike modulus reflect the quality of data shown in\nFigs. \\ref{fig:epemtau}, ranging between the most accurate, \\emph{BABAR}, and those with the largest errors, OPAL. \n\n\n\\begin{figure}[htb]\\vspace{0.4cm}\n\\begin{center}\n \\includegraphics[width = 9cm]{eeBern.eps}\n\\caption{Allowed intervals at 68.3\\% C.L. for the quantity\n$a_\\mu \\equiv a_\\mu^{\\pi\\pi(\\gamma),\\, {\\text{LO}}} [2 m_\\pi, \\,0.63\\, \\text{GeV}]\\times 10^{10}$, as a function of the energy in the region $(0.65 - 0.71)\\, \\text{GeV}$ where the timelike modulus used as input was measured in $e^+e^-$ experiments. \\label{fig:fig2e}}\\end{center}\n\\vspace{0.cm}\n\\end{figure}\n\n\\begin{figure}[htb]\\vspace{0.4cm}\n\\begin{center}\n \\includegraphics[width = 9cm]{tauBern.eps}\n\\caption{Allowed intervals at 68.3\\% C.L. for the quantity $a_\\mu \\equiv a_\\mu^{\\pi\\pi(\\gamma),\\, {\\text{LO}}} [2 m_\\pi, \\,0.63 \\,\\text{GeV}]\\times 10^{10}$, as a function of the energy in the region $0.65 - 0.71\\, \\text{GeV}$ where the timelike modulus used as input was measured in $\\tau$ experiments. \\label{fig:fig2t}}\\end{center}\n\\vspace{0.cm}\n\\end{figure}\n\nWe have then applied the averaging procedure described in the previous section, for combining the predictions available from different measurements of each experiment. The average was obtained using the robust prescription (\\ref{eq:av}). For estimating the error, we have computed $\\chi^2(f)$ defined in (\\ref{eq:chisq}) and compared it with the number of degrees of freedom, $n-1$, where $n$ is the number of points in each panel of Figs. \\ref{fig:fig2e} and \\ref{fig:fig2t}. It turned out that in all cases the ratio $\\chi^2(0)\/(n-1)$ was less than 1 and increased for a positive correlation, reaching unity for $f$ in general in the range $0.40-0.70$.\n \nSome pathologies were encountered however in a few cases. One type of pathology is illustrated in Fig. \\ref{fig:fig3a}, where we show the dependence on $f$ of the ratio $\\chi^2(f)\/(n-1)$ and of the standard deviation for the input from CMD2 and Madrid phase. In this case, the ratio becomes 1 only for values of $f$ close to 1, where the variance $\\sigma^2(f)$ calculated according to (\\ref{eq:sigma}) starts to decrease\\footnote{One can show that in all cases when the individual errors are different, $\\sigma$ exhibits a decrease above a certain $f$ and vanishes for $f=1$. In the particular case of equal errors, the variance grows linearly with $f$, as discussed in \\cite{MiSc}.}. This happens because the individual values are much closer than expected from the ascribed errors. As discussed in \\cite{MiSc}, in such cases the averaging cannot reduce the overall error, as the blind application of the prescription would indicate. Therefore, for this case we adopt the modified prescription of taking the \nmaximum variance for $f$ in the range (0, 1). The value of $\\chi^2$ corresponding to this $f$ is smaller than 1, which is due to the fact that the individual values to be averaged are very close. We encountered a similar situation with the data from ALEPH and both phases. In these cases, the combined error is not much less than the individual errors entering the combination.\n\nA different type of pathology was encountered with KLOE 11 data: in this case, for both phases, the individual values are rather different and their errors are rather small. As a consequence, $\\chi^2(f)\/(n-1)$ becomes 1 for $f$ close to 0. However, the corresponding variance (\\ref{eq:sigma}) turns out to be much smaller than estimated from the spread of the individual values. Since these values are based on measurements of the modulus at different energies by the same experiment, the differences among them indicate a problem with the data and an error reduction by their combination is not reliable. Therefore, as a conservative error, we adopted in this case too the maximum variance for $f$ in the range (0, 1), whose magnitude is comparable with those of the individual errors. We illustrate this case in Fig. \\ref{fig:fig3b}, where we show the dependence on $f$ of the ratio $\\chi^2(f)\/(n-1)$ and of the standard deviation for the input from KLOE 11 and Madrid phase.\n\nExcept these special cases, the standard deviation was calculated using (\\ref{eq:sigma}), with the covariance matrix (\\ref{eq:cov}) corresponding to the fraction $f$ determined from the equation $\\chi^2(f)=n-1$. A typical situation is shown in Fig. \\ref{fig:fig3c}, where we show the dependence on $f$ of the ratio $\\chi^2(f)\/(n-1)$ and of the standard deviation for the input from BESIII and Madrid phase.\n\nIn Table \\ref{table:amu}, we present the results of the average procedure for all the $e^+e^-$ and $\\tau$ experiments.\nFor completeness, we give the results obtained separately with the Bern and the Madrid phase. \n\n\\begin{figure}[htb]\\vspace{0.5cm}\n\\begin{center}\n \\includegraphics[width = 6.7cm]{cmd2madrid.eps}\n\\caption{Dependence on $f$ of the ratio $\\chi^2(f)\/(n-1)$ and the standard deviation, $\\sigma\\equiv\\sqrt{\\sigma^2(f)}$ for the timelike data measured by CMD2 and Madrid phase. The equality $\\chi^2(f)\/(n-1)=1$ holds for large values of $f$. \\label{fig:fig3a}}\\end{center}\n\\vspace{0.2cm}\n\\end{figure}\n\n\\begin{figure}[htb]\\vspace{0.5cm}\n\\begin{center}\n \\includegraphics[width = 6.7cm]{kloemadrid.eps}\n\\caption{Dependence on $f$ of the ratio $\\chi^2(f)\/(n-1)$ and the standard deviation, $\\sigma\\equiv\\sqrt{\\sigma^2(f)}$ for the timelike data measured by KLOE 11 and Madrid phase. The equality $\\chi^2(f)\/(n-1)=1$ holds for small values of $f$. \\label{fig:fig3b}}\\end{center}\n\\vspace{0.2cm}\n\\end{figure}\n\n\n\\begin{figure}[htb]\\vspace{0.5cm}\n\\begin{center}\n \\includegraphics[width = 6.7cm]{besmadrid.eps}\n\\caption{Dependence on $f$ of the ratio $\\chi^2(f)\/(n-1)$ and the standard deviation, $\\sigma\\equiv\\sqrt{\\sigma^2(f)}$ for the timelike data measured by BESIII and Madrid phase. The error is obtained with $f$ determined from the equation $\\chi^2(f)\/(n-1)=1$. \\label{fig:fig3c}}\\end{center}\n\\vspace{0.2cm}\n\\end{figure}\n\n\n \\begin{table}\\vspace{0.3cm}\n \\begin{tabular}{l c c }\\hline \\hline\n & ~~Bern phase & ~~~ Madrid phase\\\\\\hline\n CMD2 06 & ~ $ 131.804 \\pm 1.563$ & ~~ $ 131.396 \\pm 1.585 $ \\\\\n SND 06 & ~ $ 133.535 \\pm 1.371 $ & ~~ $ 133.102 \\pm 1.306 $ \\\\\n \\emph{BABAR} 09 & ~ $ 134.338 \\pm 0.939 $ & ~~ $134.086 \\pm 0.862 $ \\\\\n KLOE 11 & ~ $ 132.560 \\pm 1.220 $ & ~~ $ 132.017 \\pm 1.035 $ \\\\\n KLOE 13 & ~ $ 132.864 \\pm 1.413 $ & ~~ $ 132.343 \\pm 1.224 $ \\\\\n BESIII 15 & ~ $ 131.958 \\pm 1.725 $ & ~~ $ 132.753 \\pm 1.719 $ \\\\\\hline\n CLEO 00 & ~ $134.478 \\pm 1.389 $ & ~~ $ 133.897 \\pm 1.183 $ \\\\\n ALEPH 05 & ~ $ 133.114 \\pm 1.703 $ & ~~ $ 132.298 \\pm 1.783 $ \\\\\n Belle 05 & ~ $ 134.588 \\pm 1.227 $ & ~~ $ 134.280 \\pm 1.136 $ \\\\\n OPAL 12 & ~ $ 131.176\\pm 2.803 $ & ~~ $ 129.910 \\pm 2.970 $ \\\\\n\\hline\\hline\n \\end{tabular}\\caption{Central values and 68.3\\% C.L. standard deviations for the quantity $a_\\mu^{\\pi\\pi(\\gamma),\\, {\\text{LO}}} [2 m_\\pi, \\,0.63 \\,\\text{GeV}] \\times 10^{10}$, \n obtained by averaging the results shown in Figs. \\ref{fig:fig2e} and \\ref{fig:fig2t} for each experiment. \\label{table:amu}}\n \\end{table}\n\n The last step is to combine the individual values obtained with measurements by different experiments. The correlation between these values is difficult to assess \\emph{a priori}. There is of course a consistent common information going as input into all these determinations. However, the most important input, which has the crucial role in error reduction, is the modulus of the form factor in the region $0.65-0.71$ GeV measured by different experiments, which makes the difference between the values given in Table \\ref{table:amu}. Some correlation might exist also between these measurements, but there is no consensus in the views on their treatment \\cite{Davier:2011, Teubner}. We therefore applied the same averaging procedure \\cite{MiSc} suitable for cases when the correlations are not known.\n\nThe data from $e^+e^-$-annihilation and $\\tau$-decay experiments are consistent in the region $0.65-0.71$ GeV, so the results from all the 10 experiments can be combined into a single average. \nThe ratio $\\chi^2(0)\/(N-1)$, where $N=10$, turned out to be smaller than 1, which indicates a positive correlation between the predictions of various experiments. By applying the prescription given in \\cite{MiSc}, the correlation was found to be $f=0.4$ for Bern phase and $f=0.3$ for Madrid phase, leading to the values of $a_\\mu$ equal to $(133.425 \\pm 0.793) \\times 10^{-10}$ and $(133.092 \\pm 0.653)\\times 10^{-10}$, respectively\\footnote{Assuming the values not correlated, one would obtain considerably smaller errors, 0.437 and 0.402, respectively.}. Taking the simple average of the central values and errors obtained with the two phases, which are not statistically independent, we obtain\nthe conservative final estimate\\footnote{The separate combination of the results obtained with data from $e^+e^-$ and $\\tau$-decay experiments leads to the values $(133.018 \\pm 0.766) \\times 10^{-10}$ and $( 133.785 \\pm 0.993)\\times 10^{-10}$, respectively.}\n \\begin{equation}\\label{eq:average}\na_\\mu^{\\pi\\pi(\\gamma),\\, {\\text{LO}}} [2 m_\\pi, \\,0.63 \\,\\text{GeV}] =(133.258 \\pm 0.723 )\\times 10^{-10}.\n \\end{equation}\nThis result is consistent with our previous result reported in Ref. \\cite{Ananthanarayan:2013zua} and has a slightly smaller error. We emphasize that in \\cite{Ananthanarayan:2013zua} the prediction based on the present formalism was combined also with the direct integration of the cross section measured by \\emph{BABAR} at energies below 0.63 GeV, while in this work we do not use data from low energies. \n \n\n\\section{Discussions and Conclusion \\label{sec:conclusion}}\n\n\nIn this work, we have studied the two-pion contribution from energies below 0.63 GeV to the muon $g-2$,\n by exploiting analyticity and unitarity of the pion pion electromagnetic form factor. The motivation of the work is the relatively large error (of about $13.1 \\times 10^{-11}$, see Ref. \\cite{Davier:2009}) of this contribution obtained by direct data integration, explained by experimental difficulties in identifying pion pairs at low energies and the behavior of the QED kernel $K(t)$ in the integral (\\ref{eq:amu}) expressing $a_\\mu$ in terms of the pion form factor modulus. \n\nThe main idea of our approach was to use, instead of the modulus, the\nphase of the pion form factor in the elastic region, equal by Fermi-Watson theorem to the phase shift of the P-wave $\\pi\\pi$ amplitude, known with precision from the solution of Roy equations \\cite{ACGL, Caprini:2011ky, GarciaMartin:2011cn}. We have also used a conservative integral constraint on the modulus above the inelastic threshold, derived from \\emph{BABAR} data \\cite{BaBar} and perturbative QCD \\cite{Farrar:1979aw, Lepage:1979zb, Melic:1998qr}, and two precise measurements of the form factor at spacelike values of the momentum transfer \\cite{Horn, Huber}.\n\nA significant contribution to the final precision is brought by the inclusion in the input of several measurements of the modulus of the form factor at higher energies, from the $e^+e^-$-annihilation experiments CMD2, SND, \\emph{BABAR}, KLOE 11, KLOE 13 and BESIII, and the $\\tau$-decay experiments CLEO, ALEPH, OPAL and Belle. In practice we considered the energy region $0.65-0.71$ GeV, where the modulus is measured with a better accuracy, and which is close enough to the low-energy region of interest such as to have a significant constraining power. From this input, using techniques of functional optimization theory, \nwe derived rigorous constraints on the contribution to $a_\\mu$ of the energies below 0.63 GeV, where the experimental data are poor. We emphasize that the formalism exploits in an optimal way the input information and requires no parametrization of the pion form factor. Furthermore, the results do not depend on the unknown phase of the pion form factor above the inelastic threshold. \n\nThe present analysis supersedes our previous work \\cite{Ananthanarayan:2013zua}, where the same mathematical formalism was applied with data from only 4 $e^+e^-$ experiments (CMD2, SND, \\emph{BABAR} and KLOE 13). We included now as input data in the region $0.65-0.71$ GeV from 2 additional $e^+e^-$ experiments, KLOE 2011 and BESIII 2016, and the measurements in the same region reported by 4 $\\tau$-decays experiments. The analysis has been also improved by a proper treatment with statistical tools of the uncertainties and the correlations between the input data. For each timelike input from the region $0.65-0.71$ GeV of a given experiment, we have evaluated a range for $a_\\mu$ at a 68.3\\% confidence level. The results obtained with a definite input from the region $0.65-0.71$ GeV have been then combined using a statistical prescription suitable for cases when the correlations among the individual measurements are not precisely known \\cite{PDG, MiSc}. The combination of the values obtained with data \nfrom \nvarious \nexperiments was done using the same prescription, which defines a robust central value and leads to a conservative error. By this procedure we have increased the reliability of our determination. \n\nThe final outcome of our\nanalysis is expressed in Eq. (\\ref{eq:average}). Our result is consistent with and more precise than the previous result reported in \\cite{Ananthanarayan:2013zua}. It has an uncertainty smaller by about $6 \\times 10^{-11}$ than the direct integration of the cross section below 0.63 GeV \\cite{Davier:2009}.\n\nOur work demonstrates that very general methods of unitarity and analyticity\ncan be combined with high precision data from one sector to obtain stringent\nconstraints in another sector. Using in addition suitable\nstatistical methods to account for the uncertainties and the correlations of the input data, we obtained a significant\nimprovement of the low-energy two-pion contribution to $a_\\mu$. While the central\nvalue continues to remain stable, which in itself is a remarkable result,\nthe fact that it has been possible to lower the uncertainty in this region\nby nearly a factor of two makes the results in this paper to be of\nsignificance. Until the accurate data expected from the CMD-3 and SND experiments at the VEPP-2000 $e^+ e^\u2212$ collider in Novosibirsk become available, our result represents the most precise and robust determination of the low-energy hadronic contribution to muon $g-2$.\n\n\n\n\\subsection*{ACKNOWLEDGEMENTS} The authors would like to dedicate this work to the loving memory\nof their friend and collaborator I. Sentitemsu Imsong who sadly passed away\non October 30, 2015 while this work was in progress. \nWe thank S. Eidelman, F. Jegerlehner, B. Malaescu, A. Nyffeler and M. Zhang\nfor useful discussions in various stages of the work, and D. Boito for valuable information on the updated OPAL data. I.C. acknowledges support from CNCS-UEFISCDI, Contract \nIdei-PCE 121\/2011 and from ANCS, Contract PN 16 42 01 01\/2016. The work of D.D. was supported by Deutsche Forschungsgemeinschaft Research Unit \nFOR 1873 \"Quark Flavour Physics and Effective Theories\". \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{\\textbf{Scientific Background}}\nGenome-wide association studies (GWAS) aim to find statistical associations between genetic variants and traits of interest using data from a large number of individuals \\cite{mccarthy2008genome, manolio2013bringing}.\nWhen multiple correlated traits are studied simultaneously, joint, multi-trait approaches can be more advantageous than studying the traits individually, due to increased power from taking into account cross-trait covariances and reduced multiple-testing burden by performing a single test for association to a set of traits \\cite{allison1998multiple, ferreira2009multivariate}\n\nThe most commonly used multi-trait GWAS approaches are based on a multivariate analysis of variance (MANOVA) or canonical correlation analysis (CCA) \\cite{ferreira2009multivariate}. However, these are applicable only to cases where the number of traits is small. When analysing the effects of genetic variants on molecular traits (gene or protein expression levels, metabolite concentrations) or imaging features, we have to deal with a large number, often an order of magnitude or more greater than the sample size, of correlated traits simultaneously. For such cases, the standard procedure is still to conduct univariate linear regression or ANOVA tests for each genetic variant against each trait separately. While efficient algorithms exist to undertake this task, \\cite{shabalin2012matrix,qi2014,ongen2015fast}\nthe massive multiple-testing problem results in a significant loss of statistical power.\n\n An alternative approach to multi-trait GWAS has been to reverse the functional relation between genotypes and traits, and fit a multivariate regression model that predicts genotypes from multiple traits simultaneously, instead of the usual approach to regress traits on genotypes. The first study to do this explicitly used logistic regression and showed a significant increase in power compared to univariate methods \\cite{o2012multiphen}. Although the method as presented in \\cite{o2012multiphen} is still only valid when the number of traits is small, extending multivariate regression methods to high-dimensional settings is straightforward. Thus a recent study used L2-regularized linear regression of single nucleotide polymorphisms (SNPs) on gene expression traits to identify trans-acting expression quantitative trait loci (trans-eQTLs), and showed that this approach aggregates evidence from many small trans-effects while being unaffected by strong expression correlations \\cite{banerjee2020reverse}. \n\n\n Despite these advances, several limitations and open questions remain unanswered in high-dimensional GWAS. First, linear models effectively search for the linear combination of traits that is most strongly associated to the genetic variant, but there is no \\textit{a priori} biological reason why only linear combinations should be considered. Second, while regularization allows to deal with high-dimensional traits, it does not address the problem of variable selection. For instance, in the case of gene expression, we expect that trans-eQTLs are potentially associated with \\emph{many}, but not \\emph{all} genes. Indeed, in \\cite{banerjee2020reverse} a secondary set of univariate tests is carried out to select genes associated to trans-eQTLs identified by the initial multi-variate regression. Finally, a systematic biological validation and comparison of the available methods is lacking. \n Here we address these questions by considering a wider range of machine learning methods (in particular, random forests regression (RFR) and support vector regression (SVR)) for reverse genotype prediction from gene expression traits. Hypothesizing that true trans-eQTL associations are mediated by transcription regulatory networks, we use simulated data from the DREAM5 Systems Genetics Challenge,\n and real data from 1,012 segregants of a cross between two budding yeast strains \\cite{albert2018genetics} together with the YEASTRACT database of known transcriptional interactions \\cite{monteiro2020yeastract}, to validate and compare these methods against\n L2-regularized linear regression (ridge regression (RR)).\n\n\n\\section{\\textbf{Materials and Methods}}\n\n\\subsection{\\textit{\\textbf{Reverse genotype prediction}}}\nFor genotype prediction using machine learning models, the expression values were treated as explanatory variables whereas the genotype value of a variant was treated as a response variable. The prediction performance was measured by computing the root mean squared error (RMSE) between the predicted and the actual genotype value of variants. \n\n\\subsection{\\textit{\\textbf{Trans-eQTL target prediction}}}\nTrans-eQTL target prediction was done using weights assigned to the features by the machine learning methods. We computed the area under the receiver operating characteristic (AUROC) curve to measure prediction performance. The weights (feature importance in case of RFR, and coefficients for RR and SVR) were compared against the true targets in the ground truth for each variant. \n\n\\subsection{\\textit{\\textbf{Datasets}}}\nThe simulated data for our experiments was obtained from \nDREAM5 Systems Genetic Challenge A (\\url{https:\/\/www.synapse.org\/\\#!Synapse:syn2820440\/wiki\/}), generated by the Sys\\-Gen\\-SIM software \\cite{pinna2011simulating}. The DREAM data consists of simulated genotype and transcriptome data of synthetic gene regulatory networks. The dataset consists of 15 sub-datasets, where 5 different networks are provided and for each network 100, 300 and 999 samples are simulated. Every sub-dataset contains 1000 genes. Due to space restrictions, only results for Network 1 and data with 999 samples will be shown.\n\nIn the DREAM data, each genetic variant is associated to a unique causal gene that mediates its effect. We therefore defined ground-truth trans-eQTL targets for each variant as the causal gene's direct targets in the ground-truth network.\n25\\% of the variants acted in \\textit{cis}, meaning they affected expression of their causal gene directly. The remaining 75\\% of the variants acted in \\textit{trans}. Since the identities of the \\textit{cis} and \\textit{trans} eQTLs are unknown, we computed the P-values of genotype-gene expression associations between matching variant-gene pairs using Pearson correlation and selected all genes with P-values less than 1\/750 to identify cis-acting eQTLs.\n\n\n\nThe yeast data used in this paper was obtained from \\cite{albert2018genetics}. The expression data contains expression values for 5,720 genes in 1,012 segregants. The genotype data consists of binary genotype values for 42,052 genetic markers in the same 1,012 segregants.\n\nBatch and optical density (OD) effects, as given by the covariates provided in \\cite{albert2018genetics}, were removed from the expression data using categorical regression, as implemented in the \\textit{statsmodels} python package. The expression data was then normalized to have zero mean and unit standard deviation with respect to each sample.\n\nTo match variants to genes, we considered the list of genome-wide significant eQTLs provided by \\cite{albert2018genetics} whose confidence interval (of variable size) overlapped with an interval covering a gene plus 1,000 bp upstream and 200 bp downstream of the gene position. We defined its matching variant as the most strongly associated variant from the list. This resulted in a list of 2,884 genes and for each of these genes.\n\nNetworks of known transciptional regulatory interactions in yeast (S. cerevisiae) were obtained from the YEASTRACT (Yeast Search for Transcriptional Regulators And Consensus Tracking) \\cite{monteiro2020yeastract}. Regulation matrices were obtained from \\url{http:\/\/www.yeastract.com\/formregmatrix.php}.\nWe re\\-trieved the ground-truth matrix containing all reported interactions of the type \\emph{DNA binding and expression evidence}. Self regulation was removed from the ground-truths. The data from the ensembl database (release 83, December 2015) \\cite{yates2020ensembl} was used to map gene names to their identifiers. After overlaying the ground-truth with the set of genes with matching cis-eQTL, a ground-truth network of 80 transcription factors (TFs) with matching cis-eQTL and 3,394 target genes was obtained.\n\nThe expression dataset was then filtered to contain only the genes present in the ground truth network, and ground-truth trans-eQTL sets for 80 genetic variants were defined as direct targets of the corresponding TFs in the ground-truth network.\n\n\n\\subsection{\\textit{\\textbf{Experimental settings}}}\nFor both datasets we used 90\\% of the samples as training samples, whereas the rest of 10\\% samples were \nheld-out as test samples.\n\nRidge Regression (RR), Random Forest Regression (RFR), and Support Vector Regression (SVR) were implemented using Python library \\textit{scikit-learn}. The regularization strength (alpha) was set to 10,000 for RR (the higher value for alpha is chosen following \\cite{banerjee2020reverse}) and the rest of the parameters were set to their defaults. The default parameters were used for RFR and SVR. SVR was used with a linear kernel. For trans-eQTL predictions, univariate linear correlation was also used to compare with the regression methods mentioned above. \n\n\\section{\\textbf{Results}}\n\\subsection{\\textit{\\textbf{Random forest regression predicts genotypes best}}}\nWe predicted the genotypes for variants using the expression data from genes as predictors, using Random Forest Regression (RFR), Support Vector Regression (SVR) and Ridge Regression (RR) methods. RMSE was then measured for each predicted variant on 100 held-out individuals.\n\nRFR achieves the best prediction performance by achieving lowest RMSE for both simulated and yeast data (Fig. \\ref{fig:1}). \n\nNo difference was observed in the DREAM data between the RMSE distributions for all variants vs the 75\\% of trans-acting only variants (see Materials and Methods). In the yeast data, we observed that removing genes on the same chromosome as the variant lowered RMSE values, suggesting that part of the prediction performance can be explained by local cis-effects on gene expression (Fig. \\ref{fig:1}).\n\n\\begin{figure}\n \\begin{minipage}[t]{.50\\linewidth}\n \\textbf{A}\\\\\n \\includegraphics[width=\\linewidth]{fig1A.png}\n \\end{minipage}\n \\hfill\n \\begin{minipage}[t]{.50\\linewidth}\n \\textbf{B}\\\\\n \\includegraphics[width=\\linewidth]{fig6A.png}\n \\end{minipage}\n \\caption{{\\textbf{A.} The RMSE distribution for all variants (blue) and for all trans-acting-only variants (red) in DREAM5 simulated data. \\textbf{B.} The RMSE distribution for all variants using all genes (blue) and excluding genes on the same chromosome as the variant (red) as predictors in yeast data.}}\n \\label{fig:1}\n\\end{figure}\n\n\n\\subsection{\\textit{\\textbf{Feature importances are predictive of true trans-eQTL associations}}}\n\n\\begin{figure}\n\\vspace{1mm}\n \n \\begin{minipage}[t]{.50\\linewidth}\n \\textbf{A}\\\\\n \\includegraphics[width=\\linewidth]{fig3A.png}\n \\end{minipage}\n \\hfill\n \\begin{minipage}[t]{.50\\linewidth}\n \\textbf{B}\\\\\n \\includegraphics[width=\\linewidth]{fig8A.png}\n \\end{minipage}\n \n \\caption{\\textbf{A.} Bar plots show the proportion of variants with trans-eQTL target prediction AUROC $> 0.7$ (blue) and $> 0.8$ (red) in DREAM5 simulated data. \\textbf{B.} Bar plots show the number of variants with trans-eQTL target prediction AUROC $\\geq 0.6$ (blue) and $\\geq 0.7$ (red) in yeast data. Genes on the same chromosome were excluded as predictors for each SNP.}\n \\label{fig:2}\n \n\\vspace{-5mm}\n\\end{figure}\n\nTo evaluate the ability of reverse genotype prediction methods to identify true trans-eQTL targets of a given variant, we defined true trans associations as direct target genes of a variant's causal gene in the ground-truth network and used feature importances\/coefficients in the genotype prediction model to predict how likely a gene is a trans-eQTL of a given variant (see \\textit{Material and Methods}). \nPerformance was measured using the area under the receiver operating curve (AUROC).\n\nIn DREAM5 simulated data, for all methods, more than $\\sim 55\\%$, resp. $ \\sim 65\\%$ of variants with at least one trans-eQTL target in the ground-truth network had AUROC$>0.8$, resp. $0.7$, with univariate linear correlation and ridge regression performing slightly better than RFR and SVR (Fig. \\ref{fig:2}A). For yeast data, feature importances were only modestly predictive, with 20-30\\%, resp. 10-15\\%, of TF cis-eQTLs obtaining AUROCs $>0.6$, resp. $>0.7$ (Fig. \\ref{fig:2}B).\n\nFinally we tested whether genotype prediction accuracy can be used as a proxy for trans-eQTL prediction accuracy, that is, in the absence of ground-truth networks, can we use genotype prediction accuracy to filter variants whose model feature weights are indicative of true trans-eQTL targets? However, we did not observe any correlation between the genotype prediction accuracies and trans-eQTL target prediction accuracies for any of the methods. In Fig. \\ref{fig:3}, we only show the scatter plots for random forest resgression (RFR). Similar trends were observed for the remaining methods as well but these are not shown here due to to space limitations.\n\n\\begin{figure}\n\\vspace{10mm}\n \\begin{minipage}[t]{0.50\\linewidth}\n \\textbf{A}\\\\\n \\includegraphics[width=\\linewidth]{fig5.png}\n \\end{minipage}\n \\hfill\n \\begin{minipage}[t]{0.50\\linewidth}\n \\textbf{B}\\\\\n \\includegraphics[width=\\linewidth]{fig10.png}\n \\end{minipage}\n\n \\caption{Scatter plots show trans-eQTL prediction accuracy (AUROC) vs genotype prediction performance (RMSE) for Random Forest Regression. \\textbf{(A)}. For DREAM5 simulated data. \\textbf{(B)}. For yeast data where genes on the same chromosome were excluded as predictors for each SNP.}\n \\label{fig:3}\n\\vspace{-5mm}\n\\end{figure}\n\n\\subsection{\\textit{\\textbf{RMSE plotted against the whole genome produces a map of transcriptional hotspots}}}\nTranscriptional hotspots are regions of the genome associated with widespread changes in gene expression \\cite{albert2018genetics}. We learned prediction models for all 2,884 SNPs in the yeast genome that were associated with local changes in gene expression and plotted the RMSE for each predicted SNP against its genome position.\nRFR showed a wide variation in RMSE values for SNPs, across the whole genome, allowing to delineate genomic ranges with high and low regulatory activity. Whereas RR and SVR showed much less variation, and did not allow to separate high and low activity regions on most chromosomes (Fig. \\ref{fig:4}). The regions detected by RFR overlapped only partially with traditional hotspot maps based on univariate correlations, again suggesting that non-linear methods like random forest may detect biological signals missed by traditional methods.\n\n\\begin{figure}[h!]\n\\vspace{5mm}\n \\centering\n \\includegraphics[width=1.0\\linewidth]{fig_11.png}\n \\caption{Expression hotspot maps showing the RMSE values vs genome position for 2884 SNPs in the yeast genome, for random forest (RFR, top), ridge regression (Ridge, middle), and support vector regression (SVR, bottom). Genes on the same chromosome were excluded as predictors for each SNP.}\n \\label{fig:4}\n\\end{figure}\n\n\\section{\\textbf{Conclusion}}\n\nWe analyzed the use of machine learning regression methods for genotype prediction in high-dimensional multi-trait GWAS. Reverse genotype prediction from multiple trait combinations is based on the hypotheses that variants whose genotypes can be predicted with higher accuracy are more likely to have an effect on a large number of traits, and that feature importances in the trained models indicate the strength of association between variants and individual traits. However, existing studies have not presented conclusive evidence for these hypotheses, because they only performed downstream analysis for the highest scoring variants.\n\nOur results support the hypotheses only partially. We observed that RMSE values vary across genetic variants and could be used as a measure for transcriptional activity, and that model feature importances were predictive of how likely a given gene is a true trans-eQTL target of a given variant. However, RMSE values and AUROCs for target prediction showed no correlation between them. This is further illustrated by the fact that RFR achieved best prediction performance overall among the tested methods, yet performed somewhat weaker on the target prediction task. However it should be noted that computing feature importances for correlated features in machine learning models is a non-trivial task \\cite{cammarota2021variable}, and this may explain the suboptimal performance of RFR on this task.\n\nIn summary, feature importance weights in machine learning models that predict genotypes from high-dimensional sets of traits identify biologically relevant variant-trait associations, but comparing the relative importance of variants through these models in a GWAS-like manner using a single test statistic remains an open challenge. \n\\bibliographystyle{ieeetr}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nSpontaneous translational symmetry breaking, i.e., crystallization, is one of the most interesting problems in condensed matter. But as stressed by Robert Laughlin in his book \\cite{Laughlin:2005fu}, the liquid-solid transition remains one of the outstanding unsolved problems in theoretical physics. To date, there is no quantum mechanical model in which crystallization can be observed first hand.\n\nIn contradistinction, translational symmetry breaking occurs quite often in the self-consistent models of the condensed matter. For repulsive interactions, Ref.~\\cite{Bach:1994kk} showed that the energy levels in the unrestricted Hartree-Fock approximation are always fully populated. This result automatically implies symmetry breaking whenever the last occupied level of the translational invariant Hartree-Fock Hamiltonian is only partially populated. Ref.~\\cite{Prodan:2001gv} proved the existence of symmetry breaking in the fermionic Hartree approximation for short, attractive interactions. The precise conditions and the mechanism of symmetry breaking in the Kohn-Sham equations was discussed in Ref.~\\cite{Prodan:2005qg}. The Wigner crystallization of the electron liquid was studied in the Refs.~\\cite{Likos:1997ud} through a combination Monte-Carlo and Density Functional Theory calculations. The symmetry breaking in the mean field approximation of the bossonic Hubbard model was recently discussed in Ref.~\\cite{Oelkers:2007lr}. The modern theory of freezing \\cite{Ramakrishnan:1997lh,Baus:1990wq} is based on the assumption that the linear density response equation of the liquid displays a crystalized self-consistent solution as soon as the system crosses the liquid-solid phase boundary. And the list can continue.\n\nThe crystallization in fermionic self-consistent models, such as Hartree, Hartree-Fock and Kohn-Sham, is triggered by a strong coupling between the bare electrons and holes at the opposite sites of the Fermi surface. In well defined conditions, these models were shown to display robust fixed points that break the translational symmetry of the many-body Hamiltonian \\cite{Prodan:2001gv,Prodan:2005qg}. Almost like a rule, for metallic systems, the crystallization was observed to be accompanied by a gap opening in the mean-field spectrum. \n\nIn this work, we deal with the translational symmetry breaking in self-interacting 1 dimensional (1D) quantum liquids. According to a result by Mermin \\cite{Mermin:1968bv}, true crystallization cannot occur in 1 and 2D for specific short range interacting systems. This result, however, does not entirely exclude crystallization in 1D, as shown by explicit 1D models \\cite{Haldane:1980cr}, or mathematically rigorous statements \\cite{Giuliani:2006dq,Giuliani:2007rr,Giuliani:2009lq}. In any case, the issue is extremely puzzling because the self-consistent models have a tendency to display robust crystallization precisely in lower dimensions.\n\nHere we propose a many-body Hamiltonian, which is a band insulator with a particular two-body interaction that takes into account only the electron-hole coupling mentioned above. While the model is fairly simple, it still displays interesting features such as a second order singularity of the ground state energy as function of interaction strength, an insulator-metal phase transition and also crystallization in its mean-field approximation. Based on these features, we can explicitly see how how the crystallization seen in the mean-fied approximation relates to the exact many-body solution.\n\n\n\\section{The many-body model} \n\nWe start with several considerations that will allow us to place the many-body model introduced here relative to the Luttinger liquid \\cite{Haldane:1981wd}. The Luttinger liquid concept is now generally accepted to apply to all conducting spinless fermion systems in one dimension \\cite{Haldane:1981wd}, whenever such conducting behavior can be established. The Luttinger liquid state is, however, known to be unstable against an insulating pinned charge-density-wave state \\cite{Haldane:1981wd} and in fact the full phase diagram of the 1D Fermi systems is not presently known. It will be interesting to establish a solvable many-body model that is relevant to a region of the phase diagram not covered by the Luttinger liquid concept. We will try to argue that this is the case for the present model. \n\n\nLet us start from the 1D spinless Fermi gas on a circle of length $L$,\nself-interacting via a two-body potential $v$. The dynamics of the gas is generated \nby the following general Hamiltonian:\n\\begin{equation}\\label{Hamiltonian}\n H_{\\mbox{\\tiny{Gen}}}=\\frac{1}{2}\\sum_{k} (k^2-k_{F}^2)a^{\\dagger}_{k}a_{k}\n +\\frac{1}{L}\\sum_{k}v_k\\hat{\\rho}_{k}\\hat{\\rho}_{-k} \\ (\\equiv T+V),\n\\end{equation}\nwhere $a^{\\dagger}_{k}$ creats a fermion in the state\n$\\phi_{k}(x)=e^{ikx}\/\\sqrt{L}$, $k=2n\\pi\/L$, $n=0,\\pm 1,\\ldots$, and\n$\\hat{\\rho}_{k}=\\sum_{k^\\prime}a^{\\dagger}_{k^\\prime+k}a_{k^\\prime}$. \nThe Fermi wave-vector, $k_F$, is assumed equal to one of the $k_n$ vectors. \nWe follow Ref.~\\cite{Heidenreieh:1980oq} and introduce the Luttinger variables as:\n\\begin{eqnarray}\n a_{1q}\\equiv a_{q+k_{F}},\\ q\\in [-k_{F},\\infty], \\ \n a_{2q}\\equiv a_{q-k_{F}},\\ q\\in [-\\infty,k_{F}).\n\\end{eqnarray}\nNote that no fictitious states have been introduced. With these new\nvariables, the kinetic energy becomes\n\\begin{equation}\n T=\\sum_{q\\geq-k_F} q \\left(k_F+q\/2\\right)a^{\\dagger}_{1q}a_{1q}\n -\\sum_{qk_{F},\n\\end{array} \\right.\n\\end{equation}\nin which case $H_0$ describes an insulator with two non-dispersive bands, viewed in an extended Brillouin zone, separated by a gap $2\\Delta$. The proposed full many-body Hamiltonian is\n\\begin{equation}\nH= \\sum_{-2k_{F}\\leq\n k<2k_{F}}(\\varepsilon_{k}-\\varepsilon_{F})a^{\\dagger}_{k}a_{k}+\\frac{v_{0}}{L}[(\\hat{\\rho}^{(+)}_{2k_F})^2-(\\hat{\\rho}^{(-)}_{2k_F})^2].\n\\end{equation}\nTo summarize, the above Hamiltonian assumes an energy cut-off at $2k_F$, or a two band approximation, non-dispersive bands separated by a gap $2\\Delta$, and a singular two-body interaction that takes into account only the electron-hole couplings that were argued to be relevant for the translational symmetry breaking.\n\n\\section{Diagonalizing the Hamiltonian}\n\nIt is convenient to render $k$ from $\\pm k_F$ and express $a_{k}$ in terms of creation and\ndestruction operators with respect to the ground state $\\Psi_0$ of $H_0$, for which all the states with\n$k\\in[-k_{F},k_{F})$ are occupied:\n\\begin{equation}\n a_{q-k_{F}}=\\left\\{\n \\begin{array}{l}\n b_{q},\\ -k_{F}\\leq q<0 \\medskip \\\\\n c^{\\dagger}_{q},\\ 0\\leq qj$ or $|M-m|>j'$, a statement that clarifies the allowed values of $m$.\nEq.~(\\ref{Hreduced}) allows one to calculated the whole energy spectrum of $H$ and solving it is no more complicated than diaganolizing a one-particle tight-binding model in 1D. The many-body eigenvalues fall into distinct manifolds that can be labeled by $j$, $j'$ and $M$. To \ncalculate thermodynamic functions, we also need to calculate the degeneracy of each manifold and for that we need to compute how many highest weight vectors $|j j\\rangle \\otimes |j^\\prime j^\\prime\\rangle$ are there for each $j$ and $j'$. This will not be done here.\n\n\\section{Thermodynamic limit}\n\nThings greatly simplify in the thermodynamic limit $L$$\\rightarrow$$\\infty$, when \n$j$, $j^\\prime$ and $M$ take macroscopic values (proportional to $L$). For \nlarge $L$, we normalize $m$ by $j$ and work with $x\\equiv m\/j$ ($|x|\\leq 1$) as our variable, which now becomes continuous. For fixed $j$, $j'$ and $M$ and with the representation $\\Psi=\\sum_m c_m \\phi_m$, where $c$ becomes a function of $x$, the action of the Hamiltonian per unit length becomes:\n\\begin{equation}\\label{Hami}\n \\frac{1}{L}Hc(x)=\\frac{v_0}{L^2}\\partial_x a(x) \\partial_x\n c(x)+v(x)c(x).\n\\end{equation}\nFor each manifold $\\{j,j^\\prime,M\\}$, the potentials $a(x)$ and\n$v(x)$ can be easily derived from Eq. (\\ref{descrete}),\n\\begin{eqnarray}\n a(x)&=&\\sqrt{(1-x^2)[(j^\\prime\/j)^2-(x-M\/j)^2]} \\nonumber \\\\\n v(x)&=&v_0\\left(j\/L\\right)^2[1+(j^\\prime\/j)^2-(x-M\/j)^2-x^2] \\nonumber \\\\\n &+&2\\Delta (j\/L)[2x-M\/j]+2v_0\\left(j\/L\\right)^2a(x).\n\\end{eqnarray}\nEq.~(\\ref{Hami}) is defined on the interval where the factor under the \nsquare root in $a(x)$ is positive and\nzero boundary must be imposed at the ends of this interval.\n\n\nFrom Eq.~(\\ref{Hami}), one can see that, when $L$$\\rightarrow$$\\infty$, the energy per unit length is given by $v(x)$. The ground state is contained in the manifold\n$j=j^\\prime=j_{\\max}=\\frac{1}{4}N_0$, and $M=0$. This can be established analytically but we have also verified the statement numerically. For this manifold, $v(x)$ is\nequal to:\n\\begin{equation}\n v_{\\min}(x)=n_0^2v_0(1-x^2)\/4+n_0\\Delta x,\n\\end{equation}\nwith $n_0\\equiv \\lim\\limits_{L\\rightarrow \\infty}N_0\/L=k_F\/\\pi$. Given that $x$ is \nconstrained to $|x|\\leq 1$, the minimum of this potential, which defines the ground state energy\nper unit length, corresponds to \n$x_0=\\max\\{\\frac{2\\Delta}{n_0v_0},-1\\}$\nand the energy per unit length is:\n\\begin{equation}\\label{GR}\n\t\\lim\\limits_{L\\rightarrow\\infty}\\frac{E_0}{L}=\n\\left\\{\\begin{array}{l}\n\t -n_0\\Delta, \\ 2\\Delta>n_0|v_0| \\medskip \\\\\n\tn_0 \\Delta[\\frac{n_0v_0}{4\\Delta}+\\frac{\\Delta}{n_0v_0}], \\ 2\\Deltan_0|v_0| \\medskip \\\\\n\t2\\Delta^2\/v_0, \\ 2\\Delta < n_0|v_0|.\n\t\\end{array}\\right.\n\\end{equation}\nFrom here, we conclude that for $v_0$ larger than the critical value $v_c=2\\Delta\/n_0$, we have \nmacroscopic occupation of states with $|k|>k_F$.\n\n\\section{Mean field analysis} \n\nWe define the mean field problem by the following \nsubstitution:\n\\begin{equation}\n\tV\\rightarrow \\frac{v_0}{L}[\\langle\\hat{\\rho}^{(+)}_{2k_F}\\rangle \n\t\\hat{\\rho}^{(+)}_{2k_F}-\\langle\\hat{\\rho}^{(-)}_{2k_F}\\rangle \\hat{\\rho}^{(-)}_{2k_F}],\n\\end{equation}\nwhere $\\langle\\hat{\\rho}^{(\\pm)}_{2k_F}\\rangle$ denotes the expectation value of \n$\\hat{\\rho}^{(\\pm)}_{2k_F}$ on the ground state of the mean field Hamiltonian, which \nhas to be calculated self-consistently. If $\\alpha$$\\equiv$$\\langle\\hat{\\rho}_{2k_F}\\rangle$, then $\\langle \\hat{\\rho}^{(+)}_{2k_F}\\rangle$=$\\mbox{Re}[\\alpha]$\nand $\\langle\\hat{\\rho}^{(-)}_{2k_F}\\rangle$=$i\\mbox{Im}[\\alpha]$. We assume in the \nfollowing that $\\alpha$ is a real positive number. The mean field Hamiltonian becomes:\n\\begin{equation}\n\tH_{\\mbox{\\tiny{MF}}}=\\sum\\limits_q \\left\\{ \\Delta (b_q^\\dagger \n\tb_q-c_qc_q^\\dagger)+\\frac{\\alpha v_0}{L}(b_q^\\dagger \n\tc_q^\\dagger+c_qb_q)\\right\\}.\n\\end{equation}\nWe can diagonalize the quadratic mean field Hamiltonian by using the following \nBogoliubov substitution:\n\\begin{equation}\n\tb_q=\\cos \\theta \\tilde{b}_q+\\sin \\theta \\tilde{c}_q^\\dagger, \\ \n\tc_q=-\\sin \\theta \\tilde{b}_q^\\dagger+\\cos \\theta \\tilde{c}_q,\n\\end{equation}\nwith\n\\begin{equation}\\label{alpha}\n\t\\tan 2 \\theta=-\\frac{\\alpha v_0}{\\Delta L}, \\ \\ \\theta\\in [0,\\pi\/2].\n\\end{equation}\nThe mean field Hamiltonian becomes\n\\begin{equation}\n\tH_{\\mbox{\\tiny{MF}}}=\\frac{\\Delta}{\\cos \n\t2\\theta}\\sum\\limits_q(\\tilde{b}_q^\\dagger\\tilde{b}_q+\n\t\\tilde{c}_q^\\dagger\\tilde{c}_q-1),\n\\end{equation}\nand\n\\begin{equation}\n\t\\hat{\\rho}^+_{2k_F}=-\\frac{\\sin\n 2\\theta}{2}\\sum\\limits_q(\\tilde{b}_q^\\dagger\\tilde{b}_q-\n \\tilde{c}_q\\tilde{c}_q^\\dagger) \n\t+\\frac{\\cos2\\theta}{2}\\sum\\limits_q\n\t(\\tilde{b}_q^\\dagger\\tilde{c}_q^\\dagger+\n \\tilde{c}_q\\tilde{b}_q).\\nonumber\n\\end{equation}\nIt remains to determine $\\alpha\\equiv \n\\langle\\hat{\\rho}_{2k_F}\\rangle$, self-consitently. Of course, there is always the trivial solution $\\alpha$=0 but we will show that, if $v_0$ is negative enough, the mean field approximation has nontrivial self-consistent solutions.\n\nIndeed, let us assume $1\\geq\\cos 2\\theta>0$, in which case the ground state of $H_{\\mbox{\\tiny{MF}}}$ has all the $\\tilde{b}_q$ and \n$\\tilde{c}_q$ states empty and the expectation value of $\\hat{\\rho}^+_{2k_F}$ \nbecomes\n\\begin{equation}\n\t\\alpha=\\nicefrac{1}{2} N_0\\sin 2\\theta,\n\\end{equation}\nwhich together with Eq.~(\\ref{alpha}) leads to\n\\begin{equation}\n\t\\cos 2\\theta=-\\frac{2\\Delta}{n_0v_0} \\ \\mbox{and} \\ \\alpha= \\frac{N_0}{2} \\sqrt{1-\\left ( \\frac{2\\Delta}{n_0v_0} \\right )^2}.\n\\end{equation}\nThis self-consistent solution is in line with the starting assumptions that $\\alpha>0$ and $\\cos 2\\theta>0$, therefore it is a valid solution. This solution exists as long as \n$2\\Deltan_0|v_0| \\medskip \\\\\n n_0^2 v_0\/2, \\ 2\\Delta 2$ the Yamabe and Ricci flows do not agree, since the Yamabe flow preserves the conformal class of the metric but the Ricci flow does not in general.\\\\\nA Yamabe soliton \\cite{barbosa} correspond to self-similar solution of the Yamabe flow, is defined on a Riemannian or pseudo-Riemannian manifold $(M, g)$ as:\n\\begin{equation}\\label{1.2}\n \\frac{1}{2}\\pounds_V g = (r-\\lambda)g,\n\\end{equation}\nwhere $\\pounds_V g$ denotes the Lie derivative of the metric $g$ along the vector field $V$, $r$ is the scalar curvature and $\\lambda$ is a constant. Moreover a Yamabe soliton is said to be expanding, steady, shrinking depending on $\\lambda$ being positive, zero, negative respectively. If $\\lambda$ is a smooth function then \\eqref{1.2} is called almost Yamabe soliton \\cite{barbosa}. \\\\\nMany authors such as \\cite{roy2}, \\cite{roy3}, \\cite{dong}, \\cite{ghosh} have studied Yamabe solitons on some contact manifolds.\\\\\nIn 2015, N. Basu and A. Bhattacharyya \\cite{nbab} established the notion of conformal Ricci soliton \\cite{soumendu}, \\cite{roy} as:\n\\begin{equation}\\label{1.3}\n\\pounds_V g + 2S = \\Big[2\\lambda - \\Big(p + \\frac{2}{n}\\Big)\\Big]g,\n\\end{equation}\nwhere $S$ is the Ricci tensor, $p$ is a scalar non-dynamical field(time dependent scalar field), $\\lambda$ is constant, $n$ is the dimension of the manifold. \\\\\nUsing \\eqref{1.2} and \\eqref{1.3}, we introduce the notion of conformal Yamabe soliton as:\\\\\\\\\n\\textbf{Definition 1.1:} A Riemannian or pseudo-Riemannian manifold $(M,g)$ of dimension $n$ is said to admit conformal Yamabe soliton if\n\\begin{equation}\\label{1.4}\n \\pounds_V g+\\Big[2\\lambda -2r - \\Big(p + \\frac{2}{n}\\Big)\\Big]g=0,\n \\end{equation}\n where $\\pounds_V g$ denotes the Lie derivative of the metric $g$ along the vector field $V$, $r$ is the scalar curvature and $\\lambda$ is a constant, $p$ is a scalar non-dynamical field(time dependent scalar field), $n$ is the dimension of the manifold. The conformal Yamabe soliton is said to be expanding, steady, shrinking depending on $\\lambda$ being positive, zero, negative respectively. If the vector field $V$ is of gradient type i.e $V = grad(f)$, for $f$ is a smooth function on $M$, then the equation \\eqref{1.4} is called conformal gradient Yamabe soliton. \\\\\\\\\n The notion of $*$-Ricci tensor on almost Hermitian manifolds and $*$-Ricci tensor of real hypersurfaces in non-flat complex space were introduced by Tachibana \\cite{tachi} and Hamada \\cite{hama} respectively where the $*$-Ricci tensor is defined by:\n\\begin{equation}\\label{1.5}\n S^*(X,Y)=\\frac{1}{2}(\\Tr \\{\\varphi\\circ R(X,\\varphi Y)\\}),\n\\end{equation}\nfor all vector fields $X,Y$ on $M^n$, $\\varphi$ is a (1,1)-tensor field and $\\Tr$ denotes Trace.\\\\\nIf $S^*(X,Y)=\\lambda g(X,Y)+\\nu \\eta(X)\\eta(Y)$ for all vector fields $X,Y$ and $\\lambda$, $\\nu$ are smooth functions, then the manifold is called $*$-$\\eta$-Einstein manifold.\\\\\nFurther if $\\nu=0$ i.e $S^*(X,Y)=\\lambda g(X,Y)$ for all vector fields $X,Y$ then the manifold becomes $*$-Einstein.\\\\\nIn 2014, Kaimakamis and Panagiotidou \\cite{kaipan} introduced the notion of $*$-Ricci soliton which can be defined as:\n\\begin{equation}\\label{1.6}\n \\pounds_V g + 2S^* + 2\\lambda g=0,\n\\end{equation}\nfor all vector fields $X,Y$ on $M^n$ and $\\lambda$ being a constatnt.\\\\\nUsing \\eqref{1.2} and \\eqref{1.6}, we develop the notion of $*$-Yamabe soliton as:\\\\\\\\\n\\textbf{Definition 1.2:} A Riemannian or pseudo-Riemannian manifold $(M,g)$ of dimension $n$ is said to admit $*$-Yamabe soliton if\n\\begin{equation}\\label{1.7}\n \\frac{1}{2}\\pounds_V g = (r^*-\\lambda)g,\n\\end{equation}\nwhere $\\pounds_V g$ denotes the Lie derivative of the metric $g$ along the vector field $V$, $r^* = \\Tr(S^*)$ is the $*$- scalar curvature and $\\lambda$ is a constant. The $*$-Yamabe soliton is said to be expanding, steady, shrinking depending on $\\lambda$ being positive, zero, negative respectively. If the vector field $V$ is of gradient type i.e $V = grad(f)$, for $f$ is a smooth function on $M$, then the equation \\eqref{1.7} is called $*$-gradient Yamabe soliton.\\\\\\\\\nThe outline of the article goes as follows:\\\\\nIn section 2, after a brief introduction, we have discussed some needful results which will be used in the later sections. Section 3 deals with some applications of torse forming potential vector field on conformal Yamabe soliton. In this section we have contrived conformal Yamabe soliton own up to Riemannian connection, semi-symmetric metric connection and projective semi-symmetric connection with torse forming vector field to accessorize the nature of this soliton on Riemannian manifold and we have proved Theorem 3.1, Theorem 3.3 and Theorem 3.5 concerning those mentioned connections. Section 4 is devoted to utilize of torse forming potential vector field on $*$-Yamabe soliton with respect to Riemannian connection and we have evolved a theorem to develop the essence of this soliton. In section 5, we have constructed an example to illustrate the existence of the conformal Yamabe soliton on 3-dimensional Riemannian manifold.\\\\\\\\\n\\section{\\textbf{Preliminaries}}\nA nowhere vanishing vector field $\\tau$ on a Riemannian or pseudo-Riemannian manifold $(M,g)$ is called torse-forming \\cite{yano} if\n\\begin{equation}\\label{2.1}\n \\nabla_X \\tau = \\phi X+\\alpha(X)\\tau,\n\\end{equation}\nwhere $\\nabla$ is the Levi-Civita connection of $g$, $\\phi$ is a smooth function and $\\alpha$ is a 1-form.\nMoreover The vector field $\\tau$ is called concircular \\cite{chen}, \\cite{kyano} if the 1-form $\\alpha$ vanishes identically in the equation \\eqref{2.1}. The vector field $\\tau$ is called concurrent \\cite{Schouten}, \\cite{kyano1} if in \\eqref{2.1} the 1-form $\\alpha$ vanishes identically and the function $\\phi = 1$. The vector field $\\tau$ is called recurrent if in \\eqref{2.1} the function $\\phi = 0$. Finally if in \\eqref{2.1} $\\phi = \\alpha = 0$, then the vector field $\\tau$ is called a parallel vector field.\\\\\nIn 2017, Chen \\cite{chen1} introduced a new vector field called torqued vector field. If the vector field $\\tau$ staisfies \\eqref{2.1} with $\\alpha(\\tau) = 0$, then $\\tau$ is called torqued vector field. Also in this case, $\\phi$ is known as the torqued function and the 1-form $\\alpha$ is the torqued form of $\\tau$.\\\\\nFrom \\cite{Friedmann}, \\cite{Hayden}, \\cite{kyano2}, the relation between the semi-symmetric metric connection $\\bar{\\nabla}$ and the connection $\\nabla$ of $M$ is given by:\n\\begin{equation}\\label{2.2}\n \\bar{\\nabla}_X Y = \\nabla_X Y + \\pi(Y)X - g(X, Y)\\rho,\n\\end{equation}\nwhere $\\pi(X) = g(X, \\rho), \\forall X \\in \\chi(M)$, the Lie algebra of vector fields of $M$.\\\\\nAlso the Riemannian curvature tensor $\\bar{R}$, Ricci tensor $\\bar{S}$ and the scalar curvature $\\bar{r}$ of $M$ associated with the semi-symmetric metric connection $\\bar{\\nabla}$ are given by \\cite{Friedmann}:\n\\begin{equation}\\label{2.2new}\n \\bar{R}(X,Y)Z=R(X,Y)Z-P(Y,Z)X+P(X,Z)Y-g(Y,Z)LX+g(X,Z)LY,\n\\end{equation}\n\\begin{equation}\\label{2.3}\n \\bar{S}(Y, Z) = S(Y, Z) - (n - 2)P(Y, Z) - ag(Y, Z),\n\\end{equation}\n\\begin{equation}\\label{2.4}\n \\bar{r} = r - 2(n - 1)a,\n\\end{equation}\nwhere $P$ is a (0,2) tensor field given by: $$P(X, Y) =g(LX,Y)= (\\nabla_X \\pi)(Y) - \\pi(X)\\pi(Y) +\\frac{1}{2}\\pi(\\rho)g(X, Y), \\forall X,Y \\in \\chi(M)$$ and $a=\\Tr(P)$.\\\\\nFrom \\cite{Zhao}, the relation between projective semi-symmetric connection $\\tilde{\\nabla}$ and the connection $\\nabla$ is given by:\n\\begin{equation}\\label{2.5}\n \\tilde{\\nabla}_X Y = \\nabla_X Y + \\psi(Y)X + \\psi(X)Y + \\mu(Y)X - \\mu(X)Y,\n\\end{equation}\nwhere the 1-forms $\\psi$ and $\\mu$ are given by:\n$$\\psi(X) = \\frac{n-1}{2(n+1)}\\pi(X),$$\n $$\\mu(X) = \\frac{1}{2}\\pi(X)$$.\\\\\nAlso the Riemannian curvature tensor $\\tilde{R}$, Ricci tensor $\\tilde{S}$ and the scalar curvature $\\tilde{r}$ of $M$ associated with the projective semi-symmetric connection $\\tilde{\\nabla}$ are given by \\cite{Zhao}, \\cite{shaikh}:\n\\begin{equation}\\label{2.5new}\n \\tilde{R}(X,Y)Z=R(X,Y)Z+\\theta(X,Y)Z+\\omega(X,Z)Y-\\omega(Y,Z)X,\n\\end{equation}\n\\begin{equation}\\label{2.6}\n \\tilde{S}(Y,Z)=S(Y, Z) + \\theta(Y, Z) - (n - 1)\\omega(Y, Z),\n\\end{equation}\n\\begin{equation}\\label{2.7}\n \\tilde{r}= r + \\Tr(\\theta) - (n - 1)\\Tr(\\omega),\n\\end{equation}\nwhere\n$$\\theta(X,Y)=\\frac{1}{2} [(\\nabla_Y \\pi)(X) - (\\nabla_X\\pi)(Y)],$$\n$$\\omega(X,Y)= \\frac{n - 1}{2(n + 1)} (\\nabla_X\\pi)(Y)+\\frac{1}{2}(\\nabla_Y \\pi)(X)-\\frac{n^2}{(n+1)^2}\\pi(X)\\pi(Y),$$\n$\\forall X,Y,Z \\in \\chi(M).$\n\\vspace {0.3cm}\n\\section{\\textbf{Application of torse forming vector field on conformal Yamabe soliton}}\nLet $(g,\\tau,\\lambda)$ be a conformal Yamabe soliton on $M$ with respect to the Riemannian connection $\\nabla$. Then from \\eqref{1.4} we have,\n\\begin{equation}\\label{3.1}\n (\\pounds_\\tau g)(X,Y)+\\Big[2\\lambda -2r - \\Big(p + \\frac{2}{n}\\Big)\\Big]g(X,Y)=0.\n\\end{equation}\nNow using\\eqref{2.1}, we obtain,\n\\begin{eqnarray}\\label{3.2}\n (\\pounds_\\tau g)(X,Y) &=& g(\\nabla_X \\tau,Y)+g(X,\\nabla_Y \\tau) \\nonumber\\\\\n &=& 2\\phi g(X,Y)+\\alpha(X)g(\\tau,Y)+\\alpha(Y)g(\\tau,X),\n\\end{eqnarray}\nfor all $X,Y \\in M$.\\\\\nThen using \\eqref{3.2}, \\eqref{3.1} becomes,\n\\begin{equation}\\label{3.3}\n \\Big[r-\\phi-\\lambda+\\frac{1}{2}\\Big(p + \\frac{2}{n}\\Big)\\Big]g(X,Y)=\\frac{1}{2}\\Big[\\alpha(X)g(\\tau,Y)+\\alpha(Y)g(\\tau,X)\\Big].\n\\end{equation}\nTaking contraction of \\eqref{3.3} over X and Y, we have,\n\\begin{equation}\\label{3.4}\n \\Big[r-\\phi-\\lambda+\\frac{1}{2}\\Big(p + \\frac{2}{n}\\Big)\\Big]n=\\alpha(\\tau),\n\\end{equation}\nwhich leads to,\n\\begin{equation}\\label{3.5}\n \\lambda=r-\\phi-\\frac{\\alpha(\\tau)}{n}+\\frac{1}{2}\\Big(p + \\frac{2}{n}\\Big).\n\\end{equation}\nSo we can state the following theorem:\\\\\\\\\n\\textbf{Theorem 3.1.} {\\em Let $(g,\\tau,\\lambda)$ be a conformal Yamabe soliton on $M$ with respect to the Riemannian connection $\\nabla$. Then the vector field $\\tau$ is torse-forming if $\\lambda=r-\\phi-\\frac{\\alpha(\\tau)}{n}+\\frac{1}{2}(p + \\frac{2}{n})$, is constant and the soliton is expanding, steady, shrinking according as $r-\\phi-\\frac{\\alpha(\\tau)}{n}+\\frac{1}{2}(p + \\frac{2}{n}) \\gtreqqless 0$. }\\\\\\\\\nNow in \\eqref{3.5}, if the 1-form $\\alpha$ vanishes identically then $\\lambda=r-\\phi+\\frac{1}{2}(p + \\frac{2}{n}).$\\\\\\\\\nIf the 1-form $\\alpha$ vanishes identically and the function $\\phi = 1$ in \\eqref{3.5}, then $\\lambda=r-1+\\frac{1}{2}(p + \\frac{2}{n}).$\\\\\\\\\nIn \\eqref{3.5}, if the function $\\phi = 0$, then $\\lambda=r-\\frac{\\alpha(\\tau)}{n}+\\frac{1}{2}(p + \\frac{2}{n}).$\\\\\\\\\nIf $\\phi = \\alpha = 0$ in \\eqref{3.5}, then $\\lambda=r+\\frac{1}{2}(p + \\frac{2}{n}).$\\\\\\\\\nFinally in \\eqref{3.5}, if $\\alpha(\\tau) = 0$, then $\\lambda=r-\\phi+\\frac{1}{2}(p + \\frac{2}{n}).$\\\\\nThen we have,\\\\\\\\\n\\textbf{Corollary 3.2.} {\\em Let $(g,\\tau,\\lambda)$ be a conformal Yamabe soliton on $M$ with respect to the Riemannian connection $\\nabla$. Then the vector field $\\tau$ is\\\\\\\\\n(i)concircular if $\\lambda=r-\\phi+\\frac{1}{2}(p + \\frac{2}{n})$, is constant and the soliton is expanding, steady, shrinking according as $r-\\phi+\\frac{1}{2}(p + \\frac{2}{n})\\gtreqqless 0$.\\\\\\\\\n(ii)concurrent if $\\lambda=r-1+\\frac{1}{2}(p + \\frac{2}{n})$, is constant and the soliton is expanding, steady, shrinking according as $r-1+\\frac{1}{2}(p + \\frac{2}{n})\\gtreqqless 0$.\\\\\\\\\n(iii)recurrent if $\\lambda=r-\\frac{\\alpha(\\tau)}{n}+\\frac{1}{2}(p + \\frac{2}{n})$, is constant and the soliton is expanding, steady, shrinking according as $r-\\frac{\\alpha(\\tau)}{n}+\\frac{1}{2}(p + \\frac{2}{n}) \\gtreqqless 0$.\\\\\\\\\n(iv)parallel if $\\lambda=r+\\frac{1}{2}(p + \\frac{2}{n})$, is constant and the soliton is expanding, steady, shrinking according as $r+\\frac{1}{2}(p + \\frac{2}{n}) \\gtreqqless 0$.\\\\\\\\\n(v)torqued if $\\lambda=r-\\phi+\\frac{1}{2}(p + \\frac{2}{n})$, is constant and the soliton is expanding, steady, shrinking according as $r-\\phi+\\frac{1}{2}(p + \\frac{2}{n}) \\gtreqqless 0$.}\\\\\\\\\nLet us now consider $(g,\\tau,\\lambda)$ as a conformal Yamabe soliton on $M$ with respect to the semi-symmetric metric connection $\\bar{\\nabla}$. Then we have,\n\\begin{equation}\\label{3.6}\n (\\bar{\\pounds}_\\tau g)(X,Y)+[2\\lambda -2\\bar{r} - (p + \\frac{2}{n})]g(X,Y)=0,\n\\end{equation}\nwhere $\\bar{\\pounds}_\\tau$ is the Lie derivative along $\\tau$ with respect to $\\bar{\\nabla}$.\\\\\nNow using \\eqref{2.2}, we get,\n\\begin{eqnarray}\\label{3.7}\n (\\bar{\\pounds}_\\tau g)(X,Y) &=& g(\\bar{\\nabla}_X \\tau,Y)+g(X,\\bar{\\nabla}_Y \\tau) \\nonumber\\\\\n &=& g(\\nabla_X \\tau + \\pi(\\tau)X - g(X, \\tau)\\rho, Y)\\nonumber\\\\\n &+& g(X,\\nabla_Y \\tau + \\pi(\\tau)Y - g(Y, \\tau)\\rho)\\nonumber\\\\\n &=&(\\pounds_\\tau g)(X,Y)+2\\pi(\\tau)g(X,Y)\\nonumber\\\\\n &-&[g(x,\\tau)\\pi(Y)+g(Y,\\tau)\\pi(X)].\n\\end{eqnarray}\nUsing \\eqref{3.2} in \\eqref{3.7}, we obtain,\n\\begin{eqnarray}\\label{3.8}\n (\\bar{\\pounds}_\\tau g)(X,Y)&=& 2\\phi g(X,Y)+\\alpha(X)g(\\tau,Y)+\\alpha(Y)g(\\tau,X) +2\\pi(\\tau)g(X,Y)\\nonumber\\\\\n &-& [g(x,\\tau)\\pi(Y)+g(Y,\\tau)\\pi(X)].\n\\end{eqnarray}\nFrom \\eqref{2.4} and \\eqref{3.8}, \\eqref{3.6} becomes,\n\\begin{multline}\\label{3.9}\n \\Big[\\phi+\\pi(\\tau)-r+2(n-1)a+\\lambda-\\frac{1}{2} \\Big(p + \\frac{2}{n}\\Big)\\Big] g(X,Y)\\\\\n + \\frac{1}{2}\\Big[\\Big\\{\\alpha(X)-\\pi(X)\\Big\\} g(\\tau,Y)+\\Big\\{\\alpha(Y)-\\pi(Y)\\Big\\} g(\\tau,X)\\Big] =0.\n \\end{multline}\nTaking contraction of \\eqref{3.9} over X and Y, we have,\n\\begin{equation}\\label{3.10}\n \\Big[\\phi-r+2(n-1)a+\\lambda-\\frac{1}{2} \\Big(p + \\frac{2}{n}\\Big)\\Big]n+(n-1)\\pi(\\tau)+\\alpha(\\tau)=0,\n\\end{equation}\nwhich leads to,\n\\begin{equation}\\label{3.11}\n \\lambda=r-\\phi-2(n-1)a+\\frac{1}{2}\\Big(p + \\frac{2}{n}\\Big)-\\frac{n-1}{n}\\pi(\\tau)-\\frac{\\alpha(\\tau)}{n}.\n\\end{equation}\nHence we can state the following:\\\\\\\\\n\\textbf{Theorem 3.3} {\\em Let $(g,\\tau,\\lambda)$ be a conformal Yamabe soliton on $M$ with respect to the semi-symmetric metric connection $\\bar{\\nabla}$. Then the vector field $\\tau$ is torse-forming if $\\lambda=r-\\phi-2(n-1)a+\\frac{1}{2} (p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau)-\\frac{\\alpha(\\tau)}{n}$, is constant and the soliton is expanding, steady, shrinking according as $r-\\phi-2(n-1)a+\\frac{1}{2} (p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau)-\\frac{\\alpha(\\tau)}{n} \\gtreqqless 0$. }\\\\\\\\\nNow in \\eqref{3.11}, if the 1-form $\\alpha$ vanishes identically then $\\lambda=r-\\phi-2(n-1)a+\\frac{1}{2} (p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau).$\\\\\\\\\nIf the 1-form $\\alpha$ vanishes identically and the function $\\phi = 1$ in \\eqref{3.11}, then $\\lambda=r-1-2(n-1)a+\\frac{1}{2} (p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau).$\\\\\\\\\nIn \\eqref{3.11}, if the function $\\phi = 0$, then $\\lambda=r-2(n-1)a+\\frac{1}{2} (p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau)-\\frac{\\alpha(\\tau)}{n}.$\\\\\\\\\nIf $\\phi = \\alpha = 0$ in \\eqref{3.11}, then $\\lambda=r-2(n-1)a+\\frac{1}{2} (p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau).$\\\\\\\\\nFinally in \\eqref{3.11}, if $\\alpha(\\tau) = 0$, then $\\lambda=r-\\phi-2(n-1)a+\\frac{1}{2} (p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau).$\\\\\nThen we have,\\\\\\\\\n\\textbf{Corollary 3.4.} {\\em Let $(g,\\tau,\\lambda)$ be a conformal Yamabe soliton on $M$ with respect to the semi-symmetric metric connection $\\bar{\\nabla}$. Then the vector field $\\tau$ is\\\\\\\\\n(i)concircular if $\\lambda=r-\\phi-2(n-1)a+\\frac{1}{2} (p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau)$, is constant and the soliton is expanding, steady, shrinking according as $r-\\phi-2(n-1)a+\\frac{1}{2} (p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau) \\gtreqqless 0.$\\\\\\\\\n(ii)concurrent if $\\lambda=r-1-2(n-1)a+\\frac{1}{2} (p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau)$, is constant and the soliton is expanding, steady, shrinking according as $r-1-2(n-1)a+\\frac{1}{2} (p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau) \\gtreqqless 0.$\\\\\\\\\n(iii)recurrent if $\\lambda=r-2(n-1)a+\\frac{1}{2} (p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau)-\\frac{\\alpha(\\tau)}{n}$, is constant and the soliton is expanding, steady, shrinking according as $r-2(n-1)a+\\frac{1}{2} (p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau)-\\frac{\\alpha(\\tau)}{n} \\gtreqqless 0.$\\\\\\\\\n(iv)parallel if $\\lambda=r-2(n-1)a+\\frac{1}{2} (p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau)$, is constant and the soliton is expanding, steady, shrinking according as $r-2(n-1)a+\\frac{1}{2} (p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau) \\gtreqqless 0.$\\\\\\\\\n(v)torqued if $\\lambda=r-\\phi-2(n-1)a+\\frac{1}{2} (p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau)$, is constant and the soliton is expanding, steady, shrinking according as $r-\\phi-2(n-1)a+\\frac{1}{2} (p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau) \\gtreqqless 0.$}\\\\\\\\\nNow we consider $(g,\\tau,\\lambda)$ as a conformal Yamabe soliton on $M$ with respect to the projective semi-symmetric connection $\\tilde{\\nabla}$.\nThen we have,\n\\begin{equation}\\label{3.12}\n (\\tilde{\\pounds}_\\tau g)(X,Y)+[2\\lambda -2\\tilde{r} - (p + \\frac{2}{n})]g(X,Y)=0,\n\\end{equation}\nwhere $\\tilde{\\pounds}_\\tau$ is the Lie derivative along $\\tau$ with respect to $\\tilde{\\nabla}$.\\\\\nNow from \\eqref{2.5}, we have,\n\\begin{eqnarray}\\label{3.13}\n (\\tilde{\\pounds}_\\tau g)(X,Y) &=& g(\\tilde{\\nabla}_X \\tau,Y)+g(X,\\tilde{\\nabla}_Y \\tau) \\nonumber\\\\\n &=& g(\\nabla_X \\tau + \\psi(\\tau)X + \\psi(X)\\tau + \\mu(\\tau)X - \\mu(X)\\tau,Y) \\nonumber\\\\\n &+&g(X,\\nabla_Y \\tau + \\psi(\\tau)Y + \\psi(Y)\\tau + \\mu(\\tau)Y - \\mu(Y)\\tau)\\nonumber\\\\\n &=& (\\pounds_\\tau g)(X,Y)+\\frac{1}{n+1}[2n \\pi(\\tau)g(X,Y)-\\pi(X)g(\\tau,X) \\nonumber\\\\\n &-&\\pi(Y)g(\\tau,X)].\n\\end{eqnarray}\nUsing \\eqref{3.2} in \\eqref{3.13}, we get,\n\\begin{eqnarray}\\label{3.14}\n (\\tilde{\\pounds}_\\tau g)(X,Y)&=& 2\\phi g(X,Y)+\\alpha(X)g(\\tau,Y)+\\alpha(Y)g(\\tau,X) \\nonumber \\\\\n &+& \\frac{1}{n+1}[2n \\pi(\\tau)g(X,Y)-\\pi(X)g(\\tau,X)-\\pi(Y)g(\\tau,X)].\\nonumber\\\\\n\\end{eqnarray}\nNow from \\eqref{2.7} and \\eqref{3.14}, \\eqref{3.12} becomes,\n\\begin{multline}\\label{3.15}\n \\Big[\\phi+\\frac{n}{n+1}\\pi(\\tau)-r-\\Tr(\\theta)+(n-1)\\Tr(\\omega)+\\lambda-\\frac{1}{2}\\Big(p + \\frac{2}{n}\\Big)\\Big]g(X,Y) \\\\\n +\\frac{1}{2}\\Big[\\Big\\{\\alpha(X)-\\frac{\\pi(X)}{n+1}\\Big\\}g(\\tau,Y)+\\Big\\{\\alpha(Y)-\\frac{\\pi(Y)}{n+1}\\Big\\}g(\\tau,X)\\Big]=0. \\\\\n\\end{multline}\nTaking contraction of \\eqref{3.15} over X and Y, we have,\n\\begin{equation}\\label{3.16}\n \\Big[\\phi-r-\\Tr(\\theta)+(n-1)\\Tr(\\omega)+\\lambda-\\frac{1}{2}\\Big(p + \\frac{2}{n}\\Big)\\Big]n+(n-1)\\pi(\\tau)+\\alpha(\\tau)=0,\n\\end{equation}\nwhich leads to,\n\\begin{equation}\\label{3.17}\n \\lambda=r-\\phi+\\Tr(\\theta)-(n-1)\\Tr(\\omega)+\\frac{1}{2}\\Big(p + \\frac{2}{n}\\Big)-\\frac{n-1}{n}\\pi(\\tau)-\\frac{\\alpha(\\tau)}{n}.\n\\end{equation}\nSo we can state the following theorem:\\\\\\\\\n\\textbf{Theorem 3.5.} {\\em Let $(g,\\tau,\\lambda)$ be a conformal Yamabe soliton on $M$ with respect to the projective semi-symmetric connection $\\tilde{\\nabla}$. Then the vector field $\\tau$ is torse-forming if $ \\lambda=r-\\phi+\\Tr(\\theta)-(n-1)\\Tr(\\omega)+\\frac{1}{2}(p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau)-\\frac{\\alpha(\\tau)}{n}$, is constant and the soliton is expanding, steady, shrinking according as $r-\\phi+\\Tr(\\theta)-(n-1)\\Tr(\\omega)+\\frac{1}{2}(p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau)-\\frac{\\alpha(\\tau)}{n} \\gtreqqless 0$. }\\\\\\\\\nNow in \\eqref{3.17}, if the 1-form $\\alpha$ vanishes identically then $ \\lambda=r-\\phi+\\Tr(\\theta)-(n-1)\\Tr(\\omega)+\\frac{1}{2}(p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau)$.\\\\\\\\\nIf the 1-form $\\alpha$ vanishes identically and the function $\\phi = 1$ in \\eqref{3.17}, then $\\lambda=r-1+\\Tr(\\theta)-(n-1)\\Tr(\\omega)+\\frac{1}{2}(p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau)$.\\\\\\\\\nIn \\eqref{3.17}, if the function $\\phi = 0$, then $ \\lambda=r+\\Tr(\\theta)-(n-1)\\Tr(\\omega)+\\frac{1}{2}(p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau)-\\frac{\\alpha(\\tau)}{n}$.\\\\\\\\\nIf $\\phi = \\alpha = 0$ in \\eqref{3.17}, then $ \\lambda=r+\\Tr(\\theta)-(n-1)\\Tr(\\omega)+\\frac{1}{2}(p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau)$.\\\\\\\\\nFinally in \\eqref{3.17}, if $\\alpha(\\tau) = 0$, then $ \\lambda=r-\\phi+\\Tr(\\theta)-(n-1)\\Tr(\\omega)+\\frac{1}{2}(p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau)$.\\\\\nThen we have,\\\\\\\\\n\\textbf{Corollary 3.6.} {\\em Let $(g,\\tau,\\lambda)$ be a conformal Yamabe soliton on $M$ with respect to the projective semi-symmetric connection $\\tilde{\\nabla}$. Then the vector field $\\tau$ is\\\\\\\\\n(i)concircular if $\\lambda=r-\\phi+\\Tr(\\theta)-(n-1)\\Tr(\\omega)+\\frac{1}{2}(p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau)$, is constant and the soliton is expanding, steady, shrinking according as $r-\\phi+\\Tr(\\theta)-(n-1)\\Tr(\\omega)+\\frac{1}{2}(p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau) \\gtreqqless 0.$\\\\\\\\\n(ii)concurrent if $\\lambda=r-1+\\Tr(\\theta)-(n-1)\\Tr(\\omega)+\\frac{1}{2}(p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau)$, is constant and the soliton is expanding, steady, shrinking according as $r-1+\\Tr(\\theta)-(n-1)\\Tr(\\omega)+\\frac{1}{2}(p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau) \\gtreqqless 0.$\\\\\\\\\n(iii)recurrent if $\\lambda=r+\\Tr(\\theta)-(n-1)\\Tr(\\omega)+\\frac{1}{2}(p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau)-\\frac{\\alpha(\\tau)}{n}$, is constant and the soliton is expanding, steady, shrinking according as $r+\\Tr(\\theta)-(n-1)\\Tr(\\omega)+\\frac{1}{2}(p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau)-\\frac{\\alpha(\\tau)}{n} \\gtreqqless 0.$\\\\\\\\\n(iv)parallel if $\\lambda=r+\\Tr(\\theta)-(n-1)\\Tr(\\omega)+\\frac{1}{2}(p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau)$, is constant and the soliton is expanding, steady, shrinking according as $r+\\Tr(\\theta)-(n-1)\\Tr(\\omega)+\\frac{1}{2}(p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau) \\gtreqqless 0.$\\\\\\\\\n(v)torqued if $\\lambda=r-\\phi+\\Tr(\\theta)-(n-1)\\Tr(\\omega)+\\frac{1}{2}(p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau)$, is constant and the soliton is expanding, steady, shrinking according as $r-\\phi+\\Tr(\\theta)-(n-1)\\Tr(\\omega)+\\frac{1}{2}(p + \\frac{2}{n})-\\frac{n-1}{n}\\pi(\\tau) \\gtreqqless 0.$}\\\\\n\\vspace {0.3cm}\n\\section{\\textbf{Application of torse forming vector field on $*$-Yamabe soliton}}\nLet $(g,\\tau,\\lambda)$ be a $*$-Yamabe soliton on $M$ with respect to the Riemannian connection $\\nabla$. Then from \\eqref{1.7}, we get,\n\\begin{equation}\\label{4.1}\n \\frac{1}{2}(\\pounds_\\tau g)(X,Y) = (r^*-\\lambda)g(X,Y).\n\\end{equation}\nUsing \\eqref{3.2}, \\eqref{4.1} becomes,\n\\begin{equation}\\label{4.2}\n (r^*-\\lambda-\\phi)g(X,Y)=\\frac{1}{2}[\\alpha(X)g(\\tau,Y)+\\alpha(Y)g(\\tau,X)].\n\\end{equation}\nTaking contraction of \\eqref{4.2} over X and Y, we have,\n\\begin{equation}\\label{4.3}\n (r^*-\\lambda-\\phi)n=\\alpha(\\tau),\n\\end{equation}\nwhich leads to,\n\\begin{equation}\\label{4.4}\n \\lambda=r^*-\\phi-\\frac{\\alpha(\\tau)}{n}.\n\\end{equation}\nHence we can state the following:\\\\\\\\\n\\textbf{Theorem 4.1} {\\em Let $(g,\\tau,\\lambda)$ be a $*$-Yamabe soliton on $M$ with respect to the Riemannian connection $\\nabla$. Then the vector field $\\tau$ is torse-forming if $\\lambda=r^*-\\phi-\\frac{\\alpha(\\tau)}{n}$, is constant and the soliton is expanding, steady, shrinking according as $r^*-\\phi-\\frac{\\alpha(\\tau)}{n} \\gtreqqless 0$. }\\\\\\\\\nNow in \\eqref{4.4}, if the 1-form $\\alpha$ vanishes identically then $\\lambda=r^*-\\phi.$\\\\\nIf the 1-form $\\alpha$ vanishes identically and the function $\\phi = 1$ in \\eqref{4.4}, then $\\lambda=r^*-1.$\\\\\nIn \\eqref{4.4}, if the function $\\phi = 0$, then $\\lambda=r^*-\\frac{\\alpha(\\tau)}{n}.$\\\\\nIf $\\phi = \\alpha = 0$ in \\eqref{4.4}, then $\\lambda=r^*.$\\\\\nFinally in \\eqref{4.4}, if $\\alpha(\\tau) = 0$, then $\\lambda=r^*-\\phi.$\\\\\nThen we have,\\\\\\\\\n\\textbf{Corollary 4.2.} {\\em Let $(g,\\tau,\\lambda)$ be a $*$-Yamabe soliton on $M$ with respect to the Riemannian connection $\\nabla$. Then the vector field $\\tau$ is\\\\\\\\\n(i)concircular if $\\lambda=r^*-\\phi$, is constant and the soliton is expanding, steady, shrinking according as $r^*-\\phi \\gtreqqless 0$.\\\\\\\\\n(ii)concurrent if $\\lambda=r^*-1$, is constant and the soliton is expanding, steady, shrinking according as $r^*-1 \\gtreqqless 0$.\\\\\\\\\n(iii)recurrent if $\\lambda=r^*-\\frac{\\alpha(\\tau)}{n} $, is constant and the soliton is expanding, steady, shrinking according as $r^*-\\frac{\\alpha(\\tau)}{n} \\gtreqqless 0$.\\\\\\\\\n(iv)parallel if $\\lambda=r^*$, is constant and the soliton is expanding, steady, shrinking according as $r^* \\gtreqqless 0$.\\\\\\\\\n(v)torqued if $\\lambda=r^*-\\phi$, is constant and the soliton is expanding, steady, shrinking according as $r^*-\\phi \\gtreqqless 0$.}\\\\\n\\vspace {0.3cm}\n\\section{\\textbf{Example}}\nLet $M = \\{(x, y, z) \\in \\mathbb{R}^3, z \\neq 0\\}$ be a manifold of dimension 3, where $(x, y, z)$ are standard coordinates in $\\mathbb{R}^3$. The vector fields,\n\\begin{equation}\n e_1 = z^2 \\frac{\\partial}{\\partial x}, \\quad e_2= z^2 \\frac{\\partial}{\\partial y}, \\quad e_3= \\frac{\\partial}{\\partial z} \\nonumber\n\\end{equation}\nare linearly independent at each point of $M$.\\\\\nLet $g$ be the Riemannian metric defined by\n\\begin{equation}\n g(e_1,e_2)=g(e_2,e_3)=g(e_3,e_1)=0,\\nonumber\n\\end{equation}\n\\begin{equation}\n g(e_1,e_1) = g(e_2,e_2) = g(e_3,e_3) =1.\\nonumber\n\\end{equation}\nLet $\\nabla$ be the Levi-Civita connection with respect to the Riemannian metric $g$. Then we have,\n $$ [e_1,e_2] =0, \\quad [e_1,e_3] =-\\frac{2}{z} e_1, \\quad [e_2,e_3]= -\\frac{2}{z} e_2.$$\nThe connection $\\nabla$ of the metric $g$ is given by,\n\\begin{eqnarray}\n 2g(\\nabla_X Y,Z) &=& Xg(Y,Z)+Yg(Z,X)-Zg(X,Y)\\nonumber \\\\\n &-& g(X, [Y,Z])-g(Y, [X, Z]) + g(Z, [X, Y ]),\\nonumber\n\\end{eqnarray}\nwhich is known as Koszul's formula.\\\\\nUsing Koszul's formula, we can easily calculate,\n$$\\nabla_{e_1} e_1 =\\frac{2}{z} e_3, \\quad \\nabla_{e_1} e_2 =0 ,\\quad \\nabla_{e_1} e_3 =- \\frac{2}{z} e_1,$$\n$$\\nabla_{e_2} e_1 =0, \\quad \\nabla_{e_2} e_2 =\\frac{2}{z} e_3, \\quad \\nabla_{e_2} e_3 =-\\frac{2}{z} e_2,$$\n$$\\nabla_{e_3} e_1 =0, \\quad \\nabla_{e_3} e_2 = 0, \\quad \\nabla_{e_3} e_3 =0.$$\nAlso, the Riemannian curvature tensor $R$ is given by,\n$$R(X, Y )Z = \\nabla_X\\nabla_Y Z - \\nabla_Y \\nabla_X Z - \\nabla_{[X,Y]} Z.$$\nHence,\n$$R(e_1,e_2)e_1 =\\frac{4}{z^2} e_2, \\quad R(e_1,e_2)e_2 = -\\frac{4}{z^2} e_1, \\quad R(e_1,e_3)e_1 =\\frac{6}{z^2} e_3,$$\n$$R(e_1,e_3)e_3 = -\\frac{6}{z^2}e_1, \\quad R(e_2,e_3)e_2 = \\frac{6}{z^2}e_3, \\quad R(e_2,e_3)e_3 = -\\frac{6}{z^2}e_2.$$\n$$R(e_1,e_2)e_3 = 0, \\quad R(e_2,e_3)e_1 = 0, \\quad R(e_3,e_1)e_2 = 0.$$\nThen, the Ricci tensor $S$ is given by,\n$$S(e_1,e_1) = -\\frac{10}{z^2}, \\quad S(e_2,e_2) = -\\frac{10}{z^2}, \\quad S(e_3,e_3)= -\\frac{12}{z^2}.$$\nHence the scalar curvature is, $r=-\\frac{32}{z^2}$.\\\\\nSince $\\{e_1, e_2, e_3\\}$ forms a basis then any vector field $X, Y,W \\in \\chi(M)$ can be written as:\n$$X = a_1e_1 + b_1e_2 + c_1e_3, \\quad Y = a_2e_1 + b_2e_2 + c_2e_3,\\quad W = a_3e_1 + b_3e_2 + c_3e_3,$$\nwhere $a_i, b_i, c_i \\in \\mathbb{R}^+$ for $i = 1, 2, 3$ such that\n$$\\frac{a_1a_2 + b_1b_2}{c_1} + c_1\\left(\\frac{b_2}{b_1}-\\frac{a_2}{a_1}- 1\\right) \\neq 0.$$\nNow we choose the 1-form $\\alpha$ by $\\alpha(U)=g(U,\\frac{2}{z}e_3)$ for any $U \\in \\chi(M)$ and the smooth function $\\phi$ as:\n$$\\phi=\\frac{2}{z}\\left\\{\\frac{a_1a_2 + b_1b_2}{c_1} + c_1\\left(\\frac{b_2}{b_1}-\\frac{a_2}{a_1}- 1\\right)\\right\\}.$$\nThen the relation\n\\begin{equation}\\label{5.1}\n \\nabla_X Y =\\phi X +\\alpha(X)Y\n\\end{equation}\nholds. Hence $Y$ is a torse-forming vector field.\\\\\nThen from \\eqref{5.1}, we obtain,\n\\begin{eqnarray}\\label{5.2}\n (\\pounds_Y g)(X, W) &=& g(\\nabla_X Y,W)+g(X,\\nabla_W Y) \\nonumber\\\\\n &=& 2\\phi g(X,W)+\\alpha(X)g(Y,W)+\\alpha(W)g(Y,X).\n\\end{eqnarray}\nAlso we have,\n\\begin{eqnarray}\\label{5.3}\n g(X,Y)&=&a_1a_2 + b_1b_2 + c_1c_2, \\nonumber\\\\\n g(Y,W)&=&a_2a_3 + b_2b_3 + c_2c_3, \\nonumber\\\\\n g(X,W)&=&a_1a_3 + b_1b_3 + c_1c_3,\n\\end{eqnarray}\nand\n\\begin{equation}\\label{5.4}\n \\alpha(X)=\\frac{2c_1}{z}, \\quad \\alpha(Y)= \\frac{2c_2}{z}, \\quad \\alpha(W)=\\frac{2c_3}{z}.\n\\end{equation}\nFrom \\eqref{5.3} and \\eqref{5.4}, \\eqref{5.2} becomes,\n\\begin{eqnarray}\\label{5.5}\n (\\pounds_Y g)(X, W) &=& \\frac{2}{z}\\Big[\\Big\\{\\frac{2(a_1a_2 + b_1b_2)}{c_1} + 2c_1\\Big(\\frac{b_2}{b_1}-\\frac{a_2}{a_1}- 1\\Big)\\Big\\}(a_1a_3 + b_1b_3 + c_1c_3) \\nonumber\\\\\n &+& c_1(a_2a_3 + b_2b_3 + c_2c_3)+c_3(a_1a_2 + b_1b_2 + c_1c_2)\\Big],\n\\end{eqnarray}\nand\n\\begin{equation}\\label{5.6}\n \\Big[2\\lambda-2r-\\Big(p + \\frac{2}{3}\\Big)\\Big]g(X,W)=2\\Big[\\lambda+\\frac{32}{z^2}-\\frac{1}{2}\\Big(p + \\frac{2}{3}\\Big)\\Big](a_1a_3 + b_1b_3 + c_1c_3).\n\\end{equation}\nLet us assume that $a_1a_3 + b_1b_3 + c_1c_3 \\neq 0$ and\n\\begin{equation}\\label{5.7}\n 3c_1(a_2a_3 + b_2b_3 + c_2c_3)+3c_3(a_1a_2 + b_1b_2 + c_1c_2)-2c_2(a_1a_3 + b_1b_3 + c_1c_3)=0.\n\\end{equation}\nHence $(g, Y, \\lambda)$ is a conformal Yamabe soliton on $M$, i.e.\n$$(\\pounds_Y g)(X, W)-2rg(X,W)+\\Big[2\\lambda-\\Big(p + \\frac{2}{3}\\Big)\\Big]g(X,W) = 0,$$\nprovided,\n\\begin{eqnarray}\n \\lambda &=& -\\frac{32}{z^2}-\\frac{2}{z}\\Big\\{\\frac{a_1a_2 + b_1b_2}{c_1} + c_1\\Big(\\frac{b_2}{b_1}-\\frac{a_2}{a_1}- 1\\Big)\\Big\\} \\nonumber\\\\\n &-& \\frac{c_1(a_2a_3 + b_2b_3 + c_2c_3)+c_3(a_1a_2 + b_1b_2 + c_1c_2)}{z(a_1a_3 + b_1b_3 + c_1c_3)}+\\frac{1}{2}\\Big(p + \\frac{2}{3}\\Big)\\nonumber \\\\\n &=& r-\\phi-\\frac{1}{3}\\alpha(Y)+\\frac{1}{2}\\Big(p + \\frac{2}{3}\\Big)\\quad (using ~ \\eqref{5.4} ~ and ~ \\eqref{5.7}) \\nonumber \\\\\n &=& constant.\\nonumber\n\\end{eqnarray}\nHence the condition of existence of the conformal Yamabe soliton $(g,Y,\\lambda)$ on a 3-dimensional Riemannian manifold $M$ with potential vector field $Y$ as torse forming in Theorem:3.1 is satisfied.\\\\\n\\vspace {0.3cm}\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}