diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzcorp" "b/data_all_eng_slimpj/shuffled/split2/finalzzcorp" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzcorp" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nIn recent years significant progress has been achieved in the\nunderstanding of the mathematical structure of the correlation functions\nof the XXZ model and related integrable models.\nFirst of all the ground state correlation functions were studied.\nThey are completely defined through the quantum-mechanical density\nmatrix. An explicit expression for the density matrix of a finite\nsubchain of the infinite XXZ chain in the massive regime was first\nobtained by Jimbo, Miki, Miwa and Nakayashiki \\cite{JMMN92}. They\nexpressed the elements of the density matrix in terms of multiple\nintegrals. Subsequently, extensions of their formulae to the massless\nregime and to a non-vanishing longitudinal magnetic field were obtained\nin \\cite{JiMi96,KMT99b}.\n\nThen it was realized that the multiple integrals can be factorized\n\\cite{BoKo01} and that, utilizing the so-called reduced\nKnizhnik-Zamolodchikov equation, the factorized integrals can be\nwritten in a compact exponential form \\cite{BJMST04a,BJMST04b}. The\nlatter allows one to distinguish between an algebraic part and a physical\npart. The physical part is defined by a small number of transcendental \nfunctions, fixed by the one-point-correlators and by the two-point\nneighbour correlators which depend on the physical parameters like\nanisotropy, temperature, length of the chain, magnetic field, boundary\nconditions etc. The algebraic part is related to the representation\ntheory of the symmetry algebra behind the model, namely the quantum\ngroup $U_q(\\widehat{\\mathfrak{sl}_2})$ in case of the XXZ chain.\n\nIn \\cite{BJMST06b} it was observed that the formula for the correlation\nfunctions looks nicer if the XXZ chain is regularized by introducing an\nadditional parameter, the disorder field~$\\a$. With this new parameter \nit was possible to express the density matrix in terms of special\nfermionic annihilation operators $\\mathbf{b}$ and $\\mathbf{c}$ acting not on states\nof the spin chain, but on the space of (quasi-) local operators on\nthese states. The annihilation operators appeared to be responsible for\nthe algebraic part. The physical part was represented by a transcendental\nfunction $\\om$ determined by a single integral. In \\cite{BJMST08app}\nthe dual fermionic creation operators $\\mathbf{b}^*$, $\\mathbf{c}^*$ and a bosonic\ncreation operator $\\mathbf{t}^*$ were constructed. These operators together\ngenerate a special basis of the space of quasi-local operators. Since\n$\\mathbf{b}^*$, $\\mathbf{c}^*$ are Fermi operators, Wick's theorem applies and\nexpectation values of products of $\\mathbf{b}^*$, $\\mathbf{c}^*$ and $\\mathbf{t}^*$ in an\nappropriately defined vacuum state can be written as determinants, very\nmuch as in the case of free fermions.\n\nThermodynamic properties of integrable lattice models can be\nstudied within the Suzuki-Trotter formalism by considering\nan auxiliary lattice with staggering in the so-called\nTrotter direction \\cite{Suzuki85}. The temperature appears as a result\nof a special limit when the extension of the lattice in\nTrotter direction becomes infinite. Physical quantities are\nexpressed in an efficient way through the solution to certain\nnon-linear integral equations \\cite{Kluemper92}. A detailed discussion\nof this issue and further references can, for instance, be found in\nthe book \\cite{thebook}. \n\nIn papers \\cite{GKS04a,GKS05} the Suzuki-Trotter formalism was used\nin order to generalize the multiple integrals to finite temperature.\nThen their factorization was probed for several examples of correlation\nfunctions, first for the XXX chain \\cite{BGKS06} and later for the\nXXZ chain \\cite{BGKS07,BDGKSW08}. Also a conjecture was formulated\nstating that the above mentioned exponential form is valid with the\nsame fermionic operators (at least as long as they act on spin reversal\ninvariant products of local operators) as for the ground state and\ntwo functions $\\om$, $\\om'$ obtained from an $\\a$-dependent function\nin the limit $\\a\\rightarrow 0$. \n\nUnfortunately, the formulae of \\cite{BGKS07,BDGKSW08} worked only in\nthis limit. The generalization to generic $\\a$ stayed obscure. One of\nthe purposes of the present work is to add to the clarification of this\npoint, starting from a proper multiple integral representation with\ndisorder parameter $\\a$. Here, as we had to learn \\cite{Kitanine08up},\nthe crucial point is that the `Cauchy extraction trick', invented in\n\\cite{IKMT99} and described in more detail in \\cite{KKMST09a}, can be\napplied in the finite temperature case and also in the more general\nsituation of a finite lattice with inhomogeneities in Trotter direction.\n\nImportant new insight came from a recent paper \\cite{JMS08pp} by Jimbo,\nMiwa and Smirnov, where they suggested a purely algebraic\napproach to the problem of calculating the static correlation\nfunctions of the XXZ model.\nThe key idea of \\cite{JMS08pp} is to evaluate a linear functional\nrelated to the partition function within the fermionic basis constructed\nin \\cite{BJMST08app}.\nThe authors of \\cite{JMS08pp} work with a finite lattice, inhomogeneous\nin Trotter direction. In this situation they suggest a new and\nsurprising construction of the function $\\om$ depending on a magnetic\nfield and on the disorder parameter $\\a$.\n\nIn the present paper we discuss the relation of the work by Jimbo,\nMiwa and Smirnov to the approach using non-linear integral equations\nwhich at the moment seems more appropriate e.g.\\ for taking the\nTrotter limit (which was omitted in \\cite{JMS08pp}). In particular,\nwe present an alternative description of the function $\\om$ starting\nfrom the multiple integral and using the explicit factorization\nof the density matrix for two neighbouring lattice sites. We then\ngive a direct proof that our expression, though looking rather\ndifferent than that in \\cite{JMS08pp}, in fact describes the same \nfunction.\n\nAn inhomogeneous lattice in Trotter direction is very general and\nleaves many different options for the realization of physical correlation\nfunctions. Here we shall concentrate on two of them, the correlation\nfunctions of the infinite XXZ chain at finite temperature and magnetic\nfield (temperature case) and the ground state correlation functions\nof a finite chain with twisted periodic boundary conditions (finite\nlength case). Both cases can be treated to a very large extend\nsimultaneously. They are only distinct in that a different distribution\nof inhomogeneity parameters is required and in that for the finite\ntemperature case the Trotter limit has to be performed. Note that\ninstead of the XXZ Hamiltonian we could consider combinations of\nconserved quantities obtained from the transfer matrix of the six-vertex\nmodel within the formalism of non-linear integral equations. For the\nbulk thermodynamic properties this issue was recently studied in\n\\cite{TrKl07}.\n\nThe paper is organized as follows. In the next section we define our\nbasic objects and recall some of their properties. In the third\nsection we show the multiple integral formula for the elements of\nthe ($\\a$-twisted) density matrix for a sub-chain of length~$m$.\nIn section four we consider the simplest case $m = 1$. The fifth\nsection is devoted to applying the factorization technique to the\ndouble integrals for $m = 2$. In section~\\ref{sec:om} we introduce\nthe function $\\om$. We discuss its properties and the relation to \nits realization by Jimbo, Miwa and Smirnov. The content of\nsection~\\ref{sec:top} is some preliminary work on the construction\nof an operator $\\mathbf{t}$, dual to the creation operator $\\mathbf{t}^*$, which\nshould appear in the construction of an exponential form for finite\ntemperature and finite disorder parameter. In the appendices we\nprovide a derivation of the multiple integral formulae, we discuss\nthe limit $\\a \\rightarrow 0$, and we compare with the results of the\npapers \\cite{BGKS07,BDGKSW08}. \n\n\\section{Density matrix and correlation functions}\nThe XXZ quantum spin chain is defined by the Hamiltonian\n\\begin{equation} \\label{xxzham}\n H_N (\\k) = J \\sum_{j=1}^N \\bigl( \\s_{j-1}^x \\s_j^x\n + \\s_{j-1}^y \\s_j^y + \\D (\\s_{j-1}^z \\s_j^z - 1) \\bigr) \\, ,\n\\end{equation}\nwritten here in terms of the Pauli matrices $\\s^x = e_-^+ + e_+^-$,\n$\\s^y = {\\rm i} (e_-^+ - e_+^-)$, $\\s^z = e_+^+ - e_-^-$ (where the $e^\\a_\\be$\nare the elements of the gl(2) standard basis). The two real parameters\n$J$ and $\\D$ control the ground state phase diagram of the model.\nFor simplicity of notation we shall restrict ourselves in the following\nto the critical phase $J > 0$, $|\\D| < 1$. Note, however, that the\nresults of this work can be easily extended to the off-critical\nantiferromagnetic phase $\\D > 1$. We shall also assume without further\nmentioning that the number of lattice sites $N$ is even.\n\nTo fully specify $H_N (\\k)$ we have to define the boundary conditions.\nWe shall consider twisted periodic boundary conditions, when we\nare dealing with the ground state of the finite chain. Then $H_N (\\k)$\ndepends on an additional parameter $\\k$ through\n\\begin{equation} \\label{twistbound}\n \\begin{pmatrix} {e_0}_+^+ & {e_0}_-^+ \\\\\n {e_0}_+^- & {e_0}_-^- \\end{pmatrix} =\n q^{- \\k \\s^z} \\begin{pmatrix} {e_N}_+^+ & {e_N}_-^+ \\\\\n {e_N}_+^- & {e_N}_-^-\n\t\t \\end{pmatrix} q^{\\k \\s^z} \\, .\n\\end{equation}\nHere $q$ is related to $\\D$ as $\\D = (q + q^{-1})\/2$. For the finite\ntemperature case we shall assume periodic boundary conditions for\nthe Hamiltonian. Nevertheless the same parameter $\\k$ will appear\nin that case as a twist parameter of the quantum transfer matrix, having\nthen a rather different physical meaning as an external magnetic field\ncoupling to the spins by a Zeeman term. We shall elaborate on this\nbelow.\n\nThe integrable structure behind the Hamiltonian (\\ref{xxzham}) is\ngenerated by the trigonometric $R$-matrix of the six-vertex model\n\\cite{Babook},\n\\begin{align} \\label{rxxz}\n R(\\la) & = \\begin{pmatrix}\n 1 & 0 & 0 & 0 \\\\\n\t\t 0 & b(\\la) & c(\\la) & 0 \\\\\n\t\t 0 & c(\\la) & b(\\la) & 0 \\\\\n\t\t 0 & 0 & 0 & 1\n\t\t\\end{pmatrix} \\, , \\\\[2ex]\n b(\\la) & = \\frac{{\\rm sh}} \\def\\ch{{\\rm ch}} \\def\\tanh{{\\rm th}(\\la)}{{\\rm sh}} \\def\\ch{{\\rm ch}} \\def\\tanh{{\\rm th}(\\la + \\h)} \\, , \\qd\n c(\\la) = \\frac{{\\rm sh}} \\def\\ch{{\\rm ch}} \\def\\tanh{{\\rm th}(\\h)}{{\\rm sh}} \\def\\ch{{\\rm ch}} \\def\\tanh{{\\rm th}(\\la + \\h)} \\, , \\label{defbc}\n\\end{align}\nacting on ${\\mathbb C}^2 \\otimes {\\mathbb C}^2$. As presented here it\nsatisfies the Yang-Baxter equation in additive form. To facilitate the\ncomparison with \\cite{BJMST08app,JMS08pp}, where the multiplicative form\nwas preferred, we set $q = {\\rm e}^\\h$ and $\\z = {\\rm e}^\\la$. Then for arbitrary\ncomplex inhomogeneity parameters $\\be_j$, $j = 1, \\dots, N$, the definition\n\\begin{equation} \\label{defmono}\n T_a (\\z) = R_{a, N} (\\la - \\be_N) \\dots R_{a, 1} (\\la - \\be_1)\n\\end{equation}\nof the monodromy matrix makes sense, where, as usual, the indices\n $1, \\dots,N$ refer to the spin chain while $a$ refers to an additional\nsite defining the so-called auxiliary space. We also set $T_a (\\z, \\k) =\nT_a (\\z) q^{\\k \\s^z_a}$ and introduce the twisted transfer matrix\n\\begin{equation}\n t(\\z, \\k) = {\\rm tr}_a \\bigl( T_a (\\z, \\k) \\bigr) \\, .\n\\end{equation}\n\nIn \\cite{JMS08pp} a six vertex-model with $N$ horizontal rows and\nan arbitrary distribution of the inhomogeneities $\\tau_j = {\\rm e}^{\\be_j}$\non these rows was considered. Here we would like to point out that two\nspecific distributions are of particular interest in physical\napplications. Moreover, in both cases the special functions that enter\nthe representations of the transfer matrix eigenvalues and correlation\nfunctions have nice descriptions in terms of solutions of linear and\nnon-linear integral equations.\n\nThe first case relates to the ground state of the Hamiltonian\n(\\ref{xxzham}). We call it the finite length case. In this case we choose\n\\begin{equation} \\label{tdistr}\n \\be_j = \\h\/2 \\, , \\qd j = 1, \\dots, N \\, .\n\\end{equation}\nThen\n\\begin{equation} \\label{hamfromt}\n H_N (\\k) = 2 J {\\rm sh}} \\def\\ch{{\\rm ch}} \\def\\tanh{{\\rm th}(\\h) \\,\n \\partial_\\la \\ln \\bigl( t^{-1}(q^\\frac{1}{2}} \\def\\4{\\frac{1}{4},\\k) \\, t(\\z,\\k) \\bigr)%\n\t \\big|_{\\la = \\h\/2} \\, ,\n\\end{equation}\nwith twisted boundary conditions (\\ref{twistbound}) if we identify\n$\\D = \\ch (\\h)$. The critical regime $|\\D| < 1$ corresponds to purely\nimaginary $\\h = {\\rm i} \\g$, $\\g \\in [0, \\p)$. In this case the physical twist\nangle or flux $\\PH \\in [0,2\\p)$ is $\\PH = - \\k \\g$, whence $\\k$ should\nbe real. If we stick to the vertex model picture of \\cite{JMS08pp},\nthen $t(\\z,\\k)$ is the vertical or column-to-column transfer matrix in\nthis case.\n\nThe second case is determined by an alternating choice\n\\begin{equation} \\label{qtmdistr}\n \\be_j = \\begin{cases} \\be_{2j-1} = \\h - \\frac{\\be}{N} \\\\\n \\be_{2j} = \\frac{\\be}{N}\n \\end{cases} \\, , \\qd j = 1, \\dots, N\/2 \\, ,\n\\end{equation}\nof inhomogeneity parameters. This case will be called the finite\ntemperature case as it relates to the quantum transfer matrix, whose\nmonodromy matrix is\n\\begin{multline}\n T^{QTM}_a (\\z) = \\\\ R_{a, N} (\\la - \\be\/N)\n R_{N-1, a}^{t_1} (- \\be\/N - \\la) \\dots\n R_{a, 2} (\\la - \\be\/N)\n R_{1, a}^{t_1} (- \\be\/N - \\la) \\, .\n\\end{multline}\nHere the superscript `$t_1$' indicates transposition with respect to the\nfirst space. In fact, setting $Y = \\prod_{j=1}^{N\/2} \\s_{2j-1}^y$ and\nusing the crossing symmetry\n\\begin{equation}\n \\s_j^y R_{a, j} (\\la - \\h) \\s_j^y = b(\\la - \\h) R_{j, a}^{t_1} (- \\la)\n\\end{equation}\nof the $R$-matrix we find that\n\\begin{equation} \\label{ttqtm}\n T^{QTM}_a (\\z) = Y T_a (\\z) Y\n \\prod_{j=1}^{N\/2} \\frac{1}{b(\\la - \\be_{2j-1})} \\, .\n\\end{equation}\nThe quantum transfer matrix is by definition\n\\begin{equation}\n t^{QTM} (\\z,\\k) = {\\rm tr}_a \\bigl( T^{QTM}_a (\\z, \\k) \\bigr) \\, ,\n\\end{equation}\nwhere $T^{QTM}_a (\\z, \\k) = T^{QTM}_a (\\z) q^{\\k \\s^z_a}$.\n\nAgain, within the vertex model picture, $t^{QTM} (\\z,\\k)$, or $t (\\z,\\k)$\nwith the choice (\\ref{qtmdistr}) of inhomogeneity parameter, corresponds\nto the vertical transfer matrix. There is an important difference,\nthough, that has been explained at several occasions \\cite{Kluemper92,%\nGKS04a}. In the finite length case the Hamiltonian can be derived\nfrom the vertical transfer matrix. In particular, the vertical\ntransfer matrix and the Hamiltonian (\\ref{xxzham}) have the same\neigenstates. In the finite temperature case, on the other hand,\nwith a lattice which is homogeneous in horizontal direction, say, the\nHamiltonian is related to the horizontal transfer matrix with purely\nperiodic boundary conditions. It is then also periodic and will be\ndenoted $H_L (0)$, where $L$ is the horizontal extension of the lattice.\nIn this case the vertical transfer matrix eigenstates are different\nfrom those of the Hamiltonian. In particular, the eigenstate with\nthe largest modulus determines the state of thermodynamic equilibrium\nin the thermodynamic limit, i.e.\\ the free energy of the XXZ chain\nand all its static correlation functions \\cite{GKS04a}. Also the\nphysical interpretation of the parameter $\\k$ is rather different in\nthis case. It corresponds to a magnetic field coupling to the spin chain\nthrough a Zeeman term (see e.g.\\ \\cite{GKS04a}).\n\nUsing a lattice of finite extension $L$ in horizontal direction we can\nexpress the partition function of the homogeneous XXZ chain of length $L$\nas\n\\begin{equation} \\label{zustand}\n Z_L = {\\rm tr}_{1, \\dots, L} {\\rm e}^{- \\be H_L (0) + h S_{[1,L]}\/T}\n = \\lim_{N \\rightarrow \\infty} {\\rm tr}_{1, \\dots, N}\n\t \\bigl( t^{QTM} (1,h\/(2 \\h T)) \\bigr)^L \\, .\n\\end{equation}\nHere $T$ is the temperature and $h$ is a longitudinal magnetic field.\n$\\be$ must be chosen as $\\be = 2J {\\rm sh}} \\def\\ch{{\\rm ch}} \\def\\tanh{{\\rm th}(\\h)\/T$. Furthermore\n\\begin{equation}\n S_{[1,L]} = \\tst{\\frac{1}{2}} \\def\\4{\\frac{1}{4}} \\sum_{j=1}^L \\s_j^z\n\\end{equation}\nis the conserved $z$-component of the total spin. Equation (\\ref{zustand})\nbecomes efficient in the thermodynamic limit $L \\rightarrow \\infty$,\nsince then a single eigenvalue $\\La^{QTM} (1, \\k)$ of $t^{QTM} (1,\\k)$\nof largest modulus dominates the large-$L$ asymptotics of $Z_L$ in the\nTrotter limit $N \\rightarrow \\infty$. We shall refer to this eigenvalue\nas the dominant one.\n\nWe would like to remark that in our understanding the quantum transfer\nmatrix is, in general, more appropriate for studying integrable spin\nmodels on the infinite lattice than the usual transfer matrix. In\ngeneral there is no crossing symmetry, and the quantum transfer matrix\nand the usual transfer matrix are not related by a similarity\ntransformation like in (\\ref{ttqtm}). Also within the quantum transfer\nmatrix formulation the density matrix directly takes its `natural form' in\nterms of monodromy matrix elements (see below). No solution of a quantum\ninverse problem as in \\cite{KMT99b} is required. In our particular case\nwe do have the crossing symmetry, and the quantum transfer matrix\nand the usual transfer matrix with staggered inhomogeneities\n(\\ref{qtmdistr}) give an equivalent description of the density matrix\n(see below). Still, the largest eigenvalue of $t(\\z,\\k)$ with the\ndistribution (\\ref{qtmdistr}) of inhomogeneities diverges in the Trotter\nlimit as can be seen from (\\ref{ttqtm}).\n\nLet us come back to the situation of arbitrarily distributed\ninhomogeneity parameters $\\be_j$. Following \\cite{JMS08pp} we shall\nassume that for a certain spectral parameter $\\z_0$ and any\n$\\k \\in {\\mathbb C}$ the transfer matrix $t(\\z_0,\\k)$ has a unique\neigenvector $|\\k\\>$ with eigenvalue $\\La (\\z_0, \\k)$ of largest modulus.\nThis is certainly true for the two special cases considered above.\nIn the finite length case $\\z_0 = q^{1\/2}$, while $\\z_0 = 1$ in the\nfinite temperature case. We fix a set of `vertical inhomogeneity\nparameters' $\\n_1, \\dots, \\n_m$ and set $\\x_j = {\\rm e}^{\\n_j}$. Then we\ncan define the object of our main interest, the density matrix\nwith matrix elements\n\\begin{equation} \\label{defdens}\n {D_N}^{\\e_1' \\dots \\e_m'}_{\\e_1 \\dots \\e_m}\n (\\x_1, \\dots, \\x_m|\\k, \\a) =\n\t\\frac{\\<\\k + \\a| T^{\\e_1'}_{\\e_1} (\\x_1, \\k) \\dots\n\t T^{\\e_m'}_{\\e_m} (\\x_m, \\k) |\\k\\>}\n {\\<\\k + \\a|\\prod_{j=1}^m t (\\x_j,\\k)|\\k\\>} \\, ,\n\\end{equation}\nwhich is in fact an inhomogeneous and `$\\a$-twisted' version of the\nusual density matrix.\n\nIn the finite length case (\\ref{tdistr}) with twist angle $\\PH$ the\nexpectation value in the ground state $|\\PH\\>$ of any operator\n$X_{[1,m]}$ acting non-trivially only on the first $m$ lattice\nsites is \\cite{DGHK07}\n\\begin{equation}\n \\frac{\\<\\PH|X_{[1,m]}|\\PH\\>}{\\<\\PH|\\PH\\>} =\n \\lim_{\\a \\rightarrow 0}\\, \\lim_{\\n_j \\rightarrow \\h\/2}\n\t{\\rm tr}_{1, \\dots, m} \\bigl\\{\n\t {D_N} (\\x_1, \\dots, \\x_m|- \\PH\/\\g, \\a)\\, X_{[1,m]}\n\t \\bigr\\} \\, .\n\\end{equation}\nIn the finite temperature case (\\ref{qtmdistr}) we use that the\nright hand side of (\\ref{defdens}) stays form invariant under\nthe transformation (\\ref{ttqtm}) which replaces all objects\nrelating to the ordinary transfer matrix with the corresponding\nobjects relating to the quantum transfer matrix. Hence, from\n\\cite{GKS05},\n\\begin{multline}\n \\_{T, h} =\n \\lim_{L \\rightarrow \\infty}\n\t\\frac{{\\rm tr}_{1, \\dots, L} \\bigl\\{ {\\rm e}^{- \\be H_L (0) + h S_{[1,L]}\/T}%\n\t X_{[1,m]} \\bigr\\}}{Z_L} \\\\[1ex] =\n \\lim_{\\a \\rightarrow 0}\\, \\lim_{\\n_j \\rightarrow 0}\n \\lim_{N \\rightarrow \\infty}\n\t{\\rm tr}_{1, \\dots, m} \\bigl\\{\n\t {D_N} (\\x_1, \\dots, \\x_m|h\/(2\\h T), \\a)\\, X_{[1,m]}\n\t \\bigr\\} \\, .\n\\end{multline}\n\nThe density matrix (\\ref{defdens}) allows for reduction from the\nleft and from the right expressed by\n\\begin{subequations}\n\\label{redu}\n\\begin{align}\n {\\rm tr}_1 \\bigl\\{ D_N (\\x_1, \\dots, \\x_m|\\k, \\a) q^{\\a \\s_1^z} \\bigr\\} & =\n \\r (\\x_1) D_N (\\x_2, \\dots, \\x_m|\\k, \\a) \\, , \\\\[1ex]\n {\\rm tr}_m \\bigl\\{ D_N (\\x_1, \\dots, \\x_m|\\k, \\a) \\bigr\\} & =\n D_N (\\x_1, \\dots, \\x_{m-1}|\\k, \\a) \\, ,\n\\label{reductionD}\n\\end{align}\n\\end{subequations}\nwhere\n\\begin{equation} \\label{defrho}\n \\r (\\z) = \\frac{\\La (\\z, \\k + \\a)}{\\La (\\z, \\k)} \\, .\n\\end{equation}\nThe function $\\r$ plays an important role in \\cite{JMS08pp}. As we\nshall see below it is also important for the formulation of a\nmultiple integral formula for the density matrix and is the only\nnon-trivial one-point function for finite $\\a$. In the temperature case\nwith $\\k = h\/(2\\h T)$ we have\n\\enlargethispage{3ex}\n\\begin{equation}\n \\r(1) = 1 + m(T, h)2 \\h \\a + {\\cal O} (\\a^2) \\, ,\n\\end{equation}\nwhere $m(T, h)$ is the magnetization.\n\nIn the temperature case as well as in the finite length case and\nin certain inhomogeneous generalizations of both cases the function\n$\\r$ can be expressed in terms of an integral over certain auxiliary\nfunctions (see e.g.\\ \\cite{GKS04a,DGHK07}),\n\\begin{equation} \\label{rhoint}\n \\r(\\z) = q^\\a \\exp \\biggl\\{\n \\int_{\\cal C} \\frac{{\\rm d} \\m}{2 \\p {\\rm i}} \\: {\\rm e} (\\m - \\la)\n\t\t \\ln \\biggl[ \\frac{1 + \\mathfrak{a} (\\m, \\k + \\a)}\n\t\t {1 + \\mathfrak{a} (\\m, \\k )} \\biggr] \\biggr\\}\n\t\t\t\t \\, .\n\\end{equation}\nHere ${\\rm e}(\\la)$ is the `bare energy'\n\\begin{equation}\n {\\rm e}(\\la) = \\cth(\\la) - \\cth(\\la + \\h)\n\\end{equation}\nand $\\mathfrak{a}(\\la, \\k)$ is the solution of a non-linear integral equation\nwith integration kernel\n\\begin{equation} \\label{kernel}\n K(\\la) = \\cth(\\la - \\h) - \\cth(\\la + \\h) \\, .\n\\end{equation}\nIn the finite length case this equation reads\n\\begin{multline} \\label{nliefin}\n \\ln (\\mathfrak{a} (\\la, \\k)) = \\\\ (N - 2\\k) \\h + \\sum_{j=1}^N\n\t\t \\ln \\biggl[ \\frac{{\\rm sh}} \\def\\ch{{\\rm ch}} \\def\\tanh{{\\rm th} (\\la - \\be_j)}\n\t\t {{\\rm sh}} \\def\\ch{{\\rm ch}} \\def\\tanh{{\\rm th} (\\la - \\be_j + \\h)} \\biggr]\n - \\int_{\\cal C} \\frac{{\\rm d} \\m}{2 \\p {\\rm i}}\n\t\t\t K(\\la - \\m) \\ln (1 + \\mathfrak{a} (\\m, \\k )) \\, .\n\\end{multline}\nEquations (\\ref{rhoint}) and (\\ref{nliefin}) are still valid if the\n$\\be_j$ are not precisely those of equation (\\ref{tdistr}), but are close\nto $\\h\/2$ with $\\Im \\be_j = \\g\/2$. The contour of integration to be used\nin (\\ref{rhoint}) and (\\ref{nliefin}) is shown in figure \\ref{fig:cont}.\n\\begin{figure}\n \\centering\n \\includegraphics{cont10.eps}\n \\caption{\\label{fig:cont} The canonical contour ${\\cal C}$ surrounds\n the real axis in counterclockwise manner inside the\n strip $- \\frac{\\g}{2} < \\Im \\la < \\frac{\\g}{2}$.} \n\\end{figure}\nIn the temperature case the non-linear integral equation has a similar\nstructure, but the driving term is different. Suppose that for $j = 1,\n\\dots, N\/2$ the $\\be_{2j-1}$ are close to $\\h$, whereas the $\\be_{2j}$\nare close to $0$. Then\n\\begin{multline} \\label{nlietem}\n \\ln (\\mathfrak{a} (\\la, \\k)) = - 2 \\k \\h \\\\ + \\sum_{j=1}^{N\/2}\n\t \\ln \\biggl[ \\frac{{\\rm sh}} \\def\\ch{{\\rm ch}} \\def\\tanh{{\\rm th}(\\la - \\be_{2j})\n\t {\\rm sh}} \\def\\ch{{\\rm ch}} \\def\\tanh{{\\rm th}(\\la - \\be_{2j-1} + 2\\h)}\n\t\t\t {{\\rm sh}} \\def\\ch{{\\rm ch}} \\def\\tanh{{\\rm th} (\\la - \\be_{2j} + \\h)\n\t\t\t {\\rm sh}} \\def\\ch{{\\rm ch}} \\def\\tanh{{\\rm th}(\\la - \\be_{2j-1} + \\h)} \\biggr]\n - \\int_{\\cal C} \\frac{{\\rm d} \\m}{2 \\p {\\rm i}}\n\t\t\t K(\\la - \\m) \\ln (1 + \\mathfrak{a} (\\m, \\k )) \\, .\n\\end{multline}\n\nWe presented both equations (\\ref{nliefin}) and (\\ref{nlietem}) in\ninhomogeneous form, since we shall need this later, when comparing\nwith \\cite{JMS08pp}. Note, however, that the homogeneous limit is\ntrivial in both cases and that, moreover, the Trotter limit\ncan be performed in (\\ref{nlietem}). Then\n\\begin{equation} \\label{nliefinhom}\n \\ln (\\mathfrak{a} (\\la, \\k)) = (N - 2\\k) \\h\n + N \\ln \\biggl[ \\frac{{\\rm sh}} \\def\\ch{{\\rm ch}} \\def\\tanh{{\\rm th} (\\la - \\h\/2)}\n\t {{\\rm sh}} \\def\\ch{{\\rm ch}} \\def\\tanh{{\\rm th} (\\la + \\h\/2)} \\biggr]\n - \\int_{\\cal C} \\frac{{\\rm d} \\m}{2 \\p {\\rm i}}\n\t\t K(\\la - \\m) \\ln (1 + \\mathfrak{a} (\\m, \\k )) \\, .\n\\end{equation}\nin the finite length case and\n\\begin{equation} \\label{nlietemhom}\n \\ln (\\mathfrak{a} (\\la, \\k)) = - 2\\k \\h - \\frac{2J {\\rm sh}} \\def\\ch{{\\rm ch}} \\def\\tanh{{\\rm th}(\\h) {\\rm e} (\\la)}{T}\n - \\int_{\\cal C} \\frac{{\\rm d} \\m}{2 \\p {\\rm i}}\n\t\t K(\\la - \\m) \\ln (1 + \\mathfrak{a} (\\m, \\k ))\n\\end{equation}\nin the temperature case and in the Trotter limit. Equations\n(\\ref{nliefinhom}) and (\\ref{nlietemhom}) are what we call the\n$\\mathfrak{a}$-form of the non-linear integral equation. There is another\nso-called $\\mathfrak{b} \\overline{\\mathfrak{b}}$-form \\cite{Kluemper92,DGHK07} which is more\nconvenient for an accurate calculation of the numerical values\nof the functions.\n\n\\section{The multiple integral representation of the density matrix}\n\\label{sec:multint}\nIn appendix \\ref{app:dermult} we derive the following multiple integral\nrepresentation for the elements of the density matrix.\n\\begin{multline} \\label{multint}\n {D_N}^{\\e_1' \\dots \\e_m'}_{\\e_1 \\dots \\e_m}\n (\\x_1, \\dots, \\x_m|\\k, \\a) =\n\t \\biggl[ \\prod_{j=1}^p \\int_{\\cal C} {\\rm d} m(\\la_j) \\:\n\t F^+_{\\ell_j} (\\la_j) \\biggr]\n\t \\biggl[ \\prod_{j=p+1}^m \\int_{\\cal C} {\\rm d} \\overline{m}(\\la_j) \\:\n\t F^-_{\\ell_j} (\\la_j) \\biggr] \\\\[1ex]\n \\frac{\\det_{j, k = 1, \\dots, m} \\bigl[- G(\\la_j, \\n_k) \\bigr]}\n\t {\\prod_{1 \\le j < k \\le m} {\\rm sh}} \\def\\ch{{\\rm ch}} \\def\\tanh{{\\rm th}(\\la_j - \\la_k - \\h)\n\t\t {\\rm sh}} \\def\\ch{{\\rm ch}} \\def\\tanh{{\\rm th}(\\n_k - \\n_j)} \\, ,\n\\end{multline}\nwhere we have used the notation\n \n \n\t\t \n \n \n \n\\begin{align}\n {\\rm d} m(\\la) &\n = \\frac{{\\rm d} \\la}{2 \\p {\\rm i} \\, \\r(\\z) (1 + \\mathfrak{a} (\\la, \\k))} \\, , \\qd\n {\\rm d} \\overline{m} (\\la)\n = \\mathfrak{a} (\\la, \\k) {\\rm d} m(\\la) \\, , \\\\[2ex] \\notag\n F_{\\ell_j}^\\pm (\\la) &\n = \\prod_{k=1}^{\\ell_j - 1} {\\rm sh}} \\def\\ch{{\\rm ch}} \\def\\tanh{{\\rm th}(\\la - \\n_k)\n\t \\prod_{k=\\ell_j + 1}^m {\\rm sh}} \\def\\ch{{\\rm ch}} \\def\\tanh{{\\rm th}(\\la - \\n_k \\mp \\h) \\, , \\qd\n \\ell_j = \\begin{cases}\n \\e_j^+ & j = 1, \\dots, p \\\\\n\t\t \\e_{m - j + 1}^- & j = p + 1, \\dots, m\n \\end{cases}\n\\end{align}\nwith $\\e_j^+$ the $j$th plus in the sequence $(\\e_j)_{j=1}^m$, $\\e_j^-$\nthe $j$th minus sign in the sequence $(\\e_j')_{j=1}^m$ and $p$ the\nnumber of plus signs in $(\\e_j)_{j=1}^m$. The function $G$ is new here.\nIt is defined as the solution of the linear integral equation\n\\begin{equation} \\label{newg}\n G(\\la, \\n) = q^{-\\a} \\cth(\\la - \\n - \\h) - \\r (\\x) \\cth (\\la - \\n) \n + \\int_{\\cal C} {\\rm d} m(\\m) K_\\a (\\la - \\m) G(\\m, \\n) \\, ,\n\\end{equation}\nwhere $\\x = {\\rm e}^\\n$, and the kernel\n\\begin{equation}\n K_\\a (\\la) = q^{- \\a} \\cth (\\la - \\h) - q^\\a \\cth (\\la + \\h)\n\\end{equation}\nis a deformed version of (\\ref{kernel}).\n\nEquation (\\ref{multint}) is a generalization to finite $\\a$ of the multiple\nintegral formulae first derived in \\cite{GKS05,DGHK07}. To simplify\nthe notation we shall sometimes suppress the dependence of the density\nmatrix elements on $\\k$ and $\\a$.\n\\section{The case m = 1}\nFor $m = 1$ there are only two non-vanishing density matrix elements.\nThey are related to the function $\\r$ by the reduction relations\n(\\ref{redu}) which imply that\n\\begin{equation}\n \\begin{pmatrix} D^+_+ (\\x) \\\\ D^-_- (\\x) \\end{pmatrix}\n = \\frac{1}{q^\\a - q^{-\\a}}\n\t \\begin{pmatrix}\n\t - q^{-\\a} & \\mspace{14.mu} 1 \\\\ q^\\a & - 1\n\t \\end{pmatrix}\n\t \\begin{pmatrix} 1 \\\\ \\r(\\x) \\end{pmatrix} \\, .\n\\end{equation}\nWhen we insert equation (\\ref{multint}) for $m = 1$ here, we do not\nobtain an independent equation, but rather an interesting identity\nfor $\\r$ (recall that $\\r$ appears in the measure),\n\\begin{equation} \\label{grhoid}\n \\r (\\x) = q^{-\\a} - (q^\\a - q^{-\\a}) \\int_{\\cal C} {\\rm d} m(\\m) G(\\m, \\n)\n \\, .\n\\end{equation}\nIt allows us to calculate the asymptotic behaviour of the\nfunction $G$,\n\\begin{equation}\n \\lim_{{\\rm Re\\,}} \\def\\Im{{\\rm Im\\,} \\la \\rightarrow \\pm \\infty} G(\\la, \\n) = 0 \\, .\n\\end{equation}\n\\section{Factorization of the density matrix for m = 2}\nThe factorization of the multiple integrals for the ground state\ndensity matrix was discovered in \\cite{BoKo01}. In that case the\nintegrand consists of explicit functions whose analytic properties\nwere used in the calculation. In the finite temperature case a\ndifferent factorization technique had to be invented. As was demonstrated \nin \\cite{BGKS06} the linear integral equation for the function G,\nappropriately used under the multiple integral, can be viewed as the\nsource of the factorization, at least for the special case of the\nisotropic chain at $\\a = 0$. For the XXZ chain outside the isotropic\npoint and without the disorder parameter $\\a$, however, that trick\ndoes not work anymore. Here we shall see that a finite $\\a$ allows us\nto perform the factorization of the density matrix in much the same\nway as in \\cite{BGKS06}.\n\nLet us consider $m = 2$ in (\\ref{multint}). There are six non-vanishing\nmatrix elements in this case, one for $p = 0$, four for $p = 1$ and one\nfor $p = 2$. We shall concentrate on the case $p = 1$, since the matrix\nelements for $p = 0$ or $ 2$ can be obtained from those for $p = 1$ by\nmeans of the reduction relation (\\ref{redu}). After substituting\n$w_j = {\\rm e}^{2 \\m_j}$ and $\\x_j = {\\rm e}^{\\n_j}$, $j = 1, 2$, the corresponding\nintegrals are all of the form\n\\begin{equation} \\label{intm2}\n {\\cal I} = \\frac{1}{\\x_2^2 - \\x_1^2}\n \\int_{\\cal C} {\\rm d} m(\\m_1) \\int_{\\cal C} {\\rm d} \\overline{m} (\\m_2)\n\t\\det \\bigl[ G(\\m_j, \\n_k) \\bigr] r(w_1, w_2) \\, ,\n\\end{equation}\nwhere\n\\begin{equation}\n r(w_1, w_2) = \\frac{p(w_1, w_2)}{w_1 - q^2 w_2} \\, , \\qd\n p(w_1, w_2) = c_0 w_1 w_2 + c_1 w_1 + c_2 w_2 + c_3 \\, .\n\\end{equation}\nThe coefficients $c_j$ are different for the four different matrix\nelements. They are listed in table \\ref{tab:pcoeff}.\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{table}[t]\n\\begin{minipage}{\\linewidth}\n \\centering\n \\begin{tabular}{ccccc}\n \\toprule\n\t$\\begin{smallmatrix} \\e_1' & \\e_2' \\\\ \\e_1 & \\e_2\n\t \\end{smallmatrix}$\n & $c_0$ & $c_1$ & $c_2$ & $c_3$ \\\\\n \\midrule\n\t$\\begin{smallmatrix} + & - \\\\ + & - \\end{smallmatrix}$\n\t& 1 & $- \\x_1^2$ & $- q^2 \\x_2^2$ & $q^2 \\x_1^2 \\x_2^2$ \\\\\n\t$\\begin{smallmatrix} - & + \\\\ - & + \\end{smallmatrix}$\n\t& $q^2$ & $- \\x_2^2$ & $- q^2 \\x_1^2$ & $\\x_1^2 \\x_2^2$ \\\\\n\t$\\begin{smallmatrix} + & - \\\\ - & + \\end{smallmatrix}$\n\t& $q \\x_2\/\\x_1$ & $- q \\x_1 \\x_2$ & $- q \\x_1 \\x_2$\n\t& $q \\x_1^3 \\x_2$ \\\\\n\t$\\begin{smallmatrix} - & + \\\\ + & - \\end{smallmatrix}$\n\t& $q \\x_1\/\\x_2$ & $- q^{-1} \\x_1 \\x_2$ & $- q^3 \\x_1 \\x_2$\n\t& $q \\x_1 \\x_2^3$ \\\\\n \\bottomrule\n \\end{tabular}\n \\caption{\\label{tab:pcoeff} The coefficients of the polynomial $p$.}\n\\end{minipage}\n\\end{table}\n\\renewcommand{\\arraystretch}{1.1}\n\nInserting\n\\begin{equation}\n {\\rm d} \\overline{m} (\\m) = \\frac{{\\rm d} \\la}{2 \\p {\\rm i} \\r({\\rm e}^\\m)}\n - {\\rm d} m(\\m)\n\\end{equation}\ninto (\\ref{intm2}) and taking into account that $\\r ({\\rm e}^\\m)$ is\nanalytic and non-zero inside $\\cal C$ we obtain\n\\begin{multline} \\label{ij}\n {\\cal I} (\\x_2^2 - \\x_1^2) = - \\int_{\\cal C} {\\rm d} m(\\m) \\,\n \\det \\begin{pmatrix} G(\\m, \\n_1) & G(\\m, \\n_2) \\\\\n\t r(w, \\x_1^2) & r(w, \\x_2^2) \\end{pmatrix}\n\t\t\t \\\\[1ex]\n - \\int_{\\cal C} {\\rm d} m(\\m_1) \\int_{\\cal C} {\\rm d} m (\\m_2)\n\t \\det \\bigl[ G(\\m_j, \\n_k) \\bigr] r(w_1, w_2) \\, ,\n\\end{multline}\nwhere $w = {\\rm e}^{2 \\m}$. Here the first integral is already factorized.\nUnder the second integral the integration measures now appear\nsymmetrically. Hence, we may replace $r(w_1, w_2)$ by $(r(w_1, w_2) -\nr(w_2, w_1))\/2$.\n\nFollowing \\cite{BGKS06} we want to use the integral equation (\\ref{newg})\nunder the second integral in (\\ref{ij}). This is possible if rational\nfunctions $F(w_1, w_2)$ and $g(w)$ exist, such that\n\\begin{equation} \\label{decr}\n r(w_1, w_2) - r(w_2, w_1) = F(w_1, w_2) +\n g(w_1) K_\\a (\\m_1 - \\m_2) - g(w_2) K_\\a (\\m_2 - \\m_1) \\, ,\n\\end{equation}\nand the antisymmetric function $F(w_1, w_2)$ is a sum of factorized\nfunctions in $w_1$ and $w_2$. Then $F$ considered as a function of\n$w_1$ cannot have poles whose position depends on $w_2$. In particular,\nthe residue at $w_1 = q^2 w_2$ must vanish. Using this in (\\ref{decr})\nwith the explicit forms of $r$ and $K_\\a$ inserted we obtain a\ndifference equation for $g$,\n\\begin{equation} \\label{diffg}\n g(q^2 w) y^{-1} - g(w) y = \\frac{p(q^2 w, w)}{2 q^2 w} \\, .\n\\end{equation}\nHere $y = q^\\a$. Clearly this equation has a solution of the form\n\\begin{equation}\n g(w) = g_+ w + g_0 + \\frac{g_-}{w} \\, .\n\\end{equation}\nThe coefficients are easily obtained by substituting the latter expression\ninto (\\ref{diffg}),\n\\begin{equation}\n g_+ = \\frac{c_0 y}{2(q^2 - y^2)} \\, , \\qd\n g_- = \\frac{c_3 y}{2(1 - q^2 y^2)} \\, , \\qd\n g_0 = \\frac{(c_1 + q^{-2} c_2) y}{2(1 - y^2)} \\, .\n\\end{equation}\nSubstituting $g$ back into (\\ref{decr}) we obtain $F(w_1, w_2) = f(w_1)\n- f(w_2)$, where\n\\begin{equation}\n f(w) = (y - y^{-1})\\Bigl(g_+ w - \\frac{g_-}{w}\\Bigr) \\, .\n\\end{equation}\nConsequentially\n\\begin{equation}\n r(w_1, w_2) = f(w_1) + g(w_1) K_\\a (\\m_1 - \\m_2)\n + \\text{symmetric function.}\n\\end{equation}\n\nWith this we can factorize the second integral in (\\ref{ij}) by means\nof the integral equation (\\ref{newg}),\n\\begin{multline} \\label{ifirstfac}\n \\int_{\\cal C} {\\rm d} m(\\m_1) \\int_{\\cal C} {\\rm d} m (\\m_2)\n \\det \\bigl[ G(\\m_j, \\n_k) \\bigr] r(w_1, w_2) \\\\\n = (y - y^{-1})\n\t \\det \\begin{pmatrix}\n\t g_+ \\ph_+ (\\n_1) - g_- \\ph_- (\\n_1) &\n\t g_+ \\ph_+ (\\n_2) - g_- \\ph_- (\\n_2) \\\\\n\t \\ph_0 (\\n_1) & \\ph_0 (\\n_2)\n\t \\end{pmatrix} \\\\[1ex] +\n \\int_{\\cal C} {\\rm d} m(\\m) \\,\n\t \\det \\begin{pmatrix}\n\t G(\\m, \\n_1) & G(\\m, \\n_2) \\\\\n\t g(w) H(\\m, \\n_1; y^{-1}) & g(w) H(\\m, \\n_2; y^{-1})\n\t \\end{pmatrix} \\, ,\n\\end{multline}\nwhere\n\\begin{subequations}\n\\begin{align}\n \\ph_j (\\n) & = \\int_{\\cal C} {\\rm d} m(\\m) \\, w^j G(\\m, \\n) \\, , \\qd\n j = +, 0, - \\, , \\\\\n H(\\m, \\n; y^{-1}) &\n = \\r(\\x) \\cth (\\m - \\n) - y^{-1} \\cth(\\m - \\n - \\h) \\, .\n\\end{align}\n\\end{subequations}\nFinally we substitute (\\ref{ifirstfac}) into (\\ref{ij}) and further\nsimplify the resulting expression using the identities\n\\begin{subequations}\n\\begin{align}\n g(w) H(\\m, \\n; y^{-1}) & = g(\\x^2) H(\\m, \\n; y)\n - \\frac{p(q^2 \\x^2, \\x^2)}{2 q^2 \\x^2} \\cth(\\m - \\n - \\h)\n\t\\notag \\\\\n\t& \\mspace{36.mu} - \\ph_0 (\\n) \\bigl(f(w) - f(\\x^2)\\bigr)\n\t + \\frac{y^{-1}}{y - y^{-1}} \\bigl(f(\\x^2) - f(q^2 \\x^2)\\bigr)\n\t \\, , \\\\[1ex]\n r(w, \\x^2) & = \\frac{p(q^2 \\x^2, \\x^2)}{2 q^2 \\x^2} \\cth(\\m - \\n - \\h)\n\t\t - \\frac{p(- q^2 \\x^2, \\x^2)}{2 q^2 \\x^2} \\, .\n\\end{align}\n\\end{subequations}\nThen\n\\begin{multline} \\label{innerfac}\n {\\cal I} =\n \\frac{g(\\x_2^2) \\Ps (\\x_2, \\x_1) - g(\\x_1^2) \\Ps (\\x_1, \\x_2)}\n {\\x_2^2 - \\x_1^2}\n + \\frac{(c_1 - q^{-2} c_2)(\\r(\\x_1) - \\r(\\x_2))}\n\t {2(\\x_2^2 - \\x_1^2)(y - y^{-1})} \\\\[1ex]\n + \\frac{(y^{-1} - \\r(\\x_1))(y - \\r(\\x_2)) f(\\x_2^2)\n\t - (y^{-1} - \\r(\\x_2))(y - \\r(\\x_1)) f(\\x_1^2)}\n\t {(\\x_2^2 - \\x_1^2)(y - y^{-1})^2} \\, ,\n\\end{multline}\nwhere\n\\begin{equation} \\label{Psi}\n \\Ps (\\x_1, \\x_2) =\n \\int_{\\cal C} {\\rm d} m(\\m) G(\\m, \\n_2)\n\t\\bigl(q^\\a \\cth(\\m - \\n_1 - \\h) - \\r(\\x_1) \\cth(\\m - \\n_1) \\bigr)\n\t\\, .\n\\end{equation}\n\nEquation (\\ref{innerfac}) determines the four density matrix elements\nfor $p = 1$ in factorized form. Note that the matrix elements depend\non only two transcendental functions $\\r$ and $\\Ps$. The remaining\ntwo non-zero density matrix elements for $m = 2$ follow from\n(\\ref{innerfac}) by means of the reduction relations (\\ref{redu}),\n\\begin{subequations}\n\\begin{align}\n D^{++}_{++} (\\x_1, \\x_2) &\n = \\frac{\\r(\\x_1) - y^{-1}}{y - y^{-1}}\n\t - D^{+-}_{+-} (\\x_1, \\x_2) \\, , \\\\\n D^{--}_{--} (\\x_1, \\x_2) &\n = \\frac{y - \\r(\\x_1)}{y - y^{-1}} - D^{-+}_{-+} (\\x_1, \\x_2) \\, .\n\\end{align}\n\\end{subequations}\nWe shall give a fully explicit matrix representation of the factorized\ndensity matrix for $m = 2$ below, after we have introduced the function\n$\\om$.\n\n\\section{The function $\\om$} \\label{sec:om}\nIn the recent work \\cite{JMS08pp} is was shown that the correlation\nfunctions defined by the inhomogeneous and $\\a$-twisted density\nmatrix (\\ref{defdens}) factorize and can all be expressed in terms\nof only two transcendental functions, the function $\\r$ entering the\nreduction relations (\\ref{redu}) and another function $\\om$ which\nin \\cite{JMS08pp} was defined as the expectation value of a product\nof two creation operators and was represented by a determinant formula.\nThe approach of \\cite{JMS08pp} is slightly different from ours here\nin that the lattice used in \\cite{JMS08pp} is homogeneous in `horizontal\ndirection' (all the $\\x$s in (\\ref{defdens}) are taken to be 1 from\nthe outset). For the ground state both cases lead to the same function\n$\\om$ (see section 5.3 and 5.4 of \\cite{BJMST08app}). In particular,\nin the inhomogeneous case, following sections 5.1 and 5.3 of\n\\cite{BJMST08app}, we have\\footnote{More precisely this function\nwas denoted $(\\om_0 - \\om)(\\x_1\/\\x_2, \\a)$ in \\cite{BJMST08app}.}\n\\begin{equation} \\label{defom}\n \\om (\\x_1, \\x_2)\n = - \\bigl\\langle\n\t \\mathbf{c}^*_{[1,2]} (\\x_2,\\a) \\mathbf{b}^*_{[1,2]}(\\x_1,\\a - 1) (1)\n\t \\bigr\\rangle \\, .\n\\end{equation}\n\nReplacing the vacuum expectation value by the expectation value\ncalculated with the density matrix (\\ref{defdens}) we take (\\ref{defom})\nas our definition of the function $\\om$. In our case $\\om$ depends on\ntwo twist parameters $\\k$ and $\\a$. We indicate this by writing \n$\\om (\\x_1, \\x_2| \\k, \\a)$. The construction of the operators\n$\\mathbf{b}^*_{[1,2]}$ and $\\mathbf{c}^*_{[1,2]}$ is explained in \\cite{BJMST08app}.\nFor the product needed in (\\ref{defom}) we find the explicit expression\n\\begin{align} \\label{c2b11}\n \\x^{-\\a} \\mathbf{c}^*_{[1,2]} & (\\x_2, \\a) \\mathbf{b}^*_{[1,2]} (\\x_1,\\a - 1) (1) =\n \\notag \\\\[1ex] &\n \\biggl( \\frac{q^{\\a - 1} \\x^{-1}}{q \\x - q^{-1} \\x^{-1}} -\n \\frac{q^{1 - \\a} \\x^{-1}}{q^{-1} \\x - q \\x^{-1}} +\n \\frac{q^\\a - q^{- \\a}}{2} \\biggr) \\s^z \\otimes \\s^z\n \\notag \\\\[1ex] & +\n \\frac{q^\\a - q^{- \\a}}{2}\n \\biggl( \\frac{q^{-1} \\x^{-1}}{q \\x - q^{-1} \\x^{-1}} -\n \\frac{q \\x^{-1}}{q^{-1} \\x - q \\x^{-1}} \\biggr)\n \\bigl( I_2 \\otimes \\s^z - \\s^z \\otimes I_2 \\bigr)\n \\notag \\\\[1ex] & +\n 2 \\biggl( \\frac{q^\\a}{q \\x - q^{-1} \\x^{-1}} -\n \\frac{q^{- \\a}}{q^{-1} \\x - q \\x^{-1}} \\biggr)\n \\bigl( \\s^+ \\otimes \\s^- + \\s^- \\otimes \\s^+ \\bigr)\n \\notag \\\\[1ex] & +\n (q^\\a - q^{- \\a})\n \\biggl( \\frac{1}{q \\x - q^{-1} \\x^{-1}} +\n \\frac{1}{q^{-1} \\x - q \\x^{-1}} \\biggr)\n \\bigl( \\s^+ \\otimes \\s^- - \\s^- \\otimes \\s^+ \\bigr) \\, ,\n\\end{align}\nwhere $\\x = \\x_1\/\\x_2$. Inserting this into (\\ref{defom}) and calculating\nthe average with the factorized two-site density matrix of the previous\nsection we obtain\n\\begin{equation} \\label{ompsi}\n \\om(\\x_1, \\x_2|\\k, \\a) = 2 \\x^\\a \\Ps(\\x_1, \\x_2) - \\D \\ps(\\x)\n + 2 \\bigl( \\r(\\x_1) - \\r(\\x_2) \\bigr) \\ps(\\x) \\, .\n\\end{equation}\nHere we adopted the notation from \\cite{BJMST08app},\n\\begin{equation} \\label{defpsi}\n \\ps(\\x) = \\frac{\\x^\\a (\\x^2 + 1)}{2(\\x^2 - 1)} \\, ,\n\\end{equation}\nand $\\D$ is the difference operator whose action on a function $f$\nis defined by $\\D f(\\x) = f(q \\x) - f(q^{-1} \\x)$.\n\nThe remaining part of this section is devoted to the exploration of\nthe properties of~$\\om$. First of all we substitute $\\om$ back into\nthe equation for the two-site density matrix, which can then be\nexpressed entirely in terms of $\\om$ and a function\n\\begin{equation}\n \\ph (\\z|\\k, \\a) = \\frac{\\ch (\\a \\h) - \\r(\\z)}{{\\rm sh}} \\def\\ch{{\\rm ch}} \\def\\tanh{{\\rm th}(\\a \\h)}\n\\end{equation}\nwhich is sometimes more convenient than the function $\\r$ itself.\nWe obtain\n\\begin{align}\n D_N & (\\x_1, \\x_2|\\k, \\a) = \\4 I_2 \\otimes I_2 \\notag \\\\\n & - \\frac{1}{4(q^{\\a - 1} - q^{1 - \\a})}\n \\biggl( \\frac{\\x^{1 - \\a} \\om_{12} - \\x^{\\a - 1} \\om_{21}}\n {\\x - \\x^{-1}} +\n \\frac{\\ph_1 \\ph_2 (q^\\a - q^{- \\a})}{2} \\biggr)\n \\notag \\\\ & \\qd\n \\biggl( \\frac{q - q^{-1}}{2} I_2 \\otimes \\s^z\n - \\frac{q + q^{-1}}{2} \\s^z \\otimes \\s^z\n + \\x^{-1} \\, \\s^+ \\otimes \\s^- + \\x \\, \\s^- \\otimes \\s^+\n \\biggr) \\notag \\\\[1ex]\n & - \\frac{1}{4(q^{\\a + 1} - q^{- \\a - 1})}\n \\biggl( \\frac{\\x^{- \\a - 1} \\om_{12} - \\x^{\\a + 1} \\om_{21}}\n {\\x - \\x^{-1}} +\n \\frac{\\ph_1 \\ph_2 (q^\\a - q^{- \\a})}{2} \\biggr)\n \\notag \\\\ & \\qd\n \\biggl( - \\frac{q - q^{-1}}{2} I_2 \\otimes \\s^z\n - \\frac{q + q^{-1}}{2} \\s^z \\otimes \\s^z\n + \\x \\, \\s^+ \\otimes \\s^- + \\x^{-1} \\, \\s^- \\otimes \\s^+\n \\biggr) \\notag \\\\[1ex]\n & - \\frac{\\x^{-\\a} \\om_{12} - \\x^\\a \\om_{21}}\n {4(\\x - \\x^{-1})(q^\\a - q^{- \\a})}\n \\bigl( (\\x + \\x^{-1}) \\s^z \\otimes \\s^z\n - (q + q^{-1})(\\s^+ \\otimes \\s^- + \\s^- \\otimes \\s^+)\n \\bigr) \\notag \\\\[1ex]\n & - \\4 \\bigl(\\ph_1 \\, \\s^z \\otimes I_2\n + \\ph_2 \\, I_2 \\otimes \\s^z \\bigr)\n - \\frac{q - q^{-1}}{4(\\x - \\x^{-1})} (\\ph_1 - \\ph_2)\n (\\s^+ \\otimes \\s^- - \\s^- \\otimes \\s^+) \\, ,\n\\end{align}\nwhere we introduced the abbreviations $\\om_{jk} = \\om(\\x_j, \\x_k| \\k, \\a)$\nand $\\ph_j = \\ph(\\x_j| \\k, \\a)$.\n\nFor the limit $\\a \\rightarrow 0$ the properties of the functions $\\ph$\nand $\\om$ with respect to negating $\\k$ and $\\a$ are important. They\nfollow from the fact that the $R$-matrix is invariant under spin reversal,\n\\begin{equation} \\label{rspinrev}\n R(\\la) = (\\s^x \\otimes \\s^x) R(\\la) (\\s^x \\otimes \\s^x) \\, .\n\\end{equation}\nIntroducing the spin reversal operator $J = \\s_1^x \\dots \\s_N^x$\nwe conclude with (\\ref{rspinrev}) that\n\\begin{equation}\n T_a (\\z, - \\k) = \\s_a^x J \\, T_a (\\z, \\k) \\, J \\s_a^x \\, .\n\\end{equation}\nIt follows that $t(\\z, -\\k) = J t(\\z, \\k) J$. Hence,\n\\begin{subequations}\n\\begin{align}\n J |\\k\\> & = |-\\k\\> \\, , \\\\[1ex] \\La (\\z, \\k) & = \\La (\\z, -\\k) \\, .\n \\label{evinv}\n\\end{align}\n\\end{subequations}\nThe latter two equations used in the definition (\\ref{defdens}) of\nthe $\\a$-twisted density matrix imply that\n\\begin{equation} \\label{densrevers}\n D_N (\\x_1, \\dots, \\x_m|- \\k, - \\a) = (\\s^x)^{\\otimes m} \\,\n D_N (\\x_1, \\dots, \\x_m|\\k, \\a) \\, (\\s^x)^{\\otimes m} \\, .\n\\end{equation}\n\nFrom (\\ref{defrho}), (\\ref{evinv}) we obtain the relation\n\\begin{equation}\n \\ph(\\z|- \\k, - \\a) = - \\ph(\\z| \\k, \\a) \\, .\n\\end{equation}\nEquation (\\ref{densrevers}) together with (\\ref{defom})-(\\ref{ompsi})\nand the expressions for the density matrix elements of the previous\nsection implies that\n\\begin{equation}\n \\om(\\x_1, \\x_2|\\k, \\a) = \\om(\\x_2, \\x_1|- \\k, - \\a) \\, .\n\\end{equation}\n\nOur next step is to verify that the function $\\om$ given by the formula\n(\\ref{ompsi}) satisfies a property called the 'normalization condition'\nby the authors of \\cite{JMS08pp} (see equation (6.10) there). So we come\nback to the case of finite Trotter number $N$ with arbitrary inhomogeneity\nparameters $\\be_j$, $j = 1, \\dots, N$ as it is written in (\\ref{defmono}).\nWe shall also use multiplicative parameters $\\tau_j=e^{\\be_j}$.\n\nWe consider the normalization condition in the following form \n\\begin{multline} \\label{norm}\n \\bigl(\\om(\\z,\\xi|\\k, \\a)\n + {\\overline{D}}_{\\z}{\\overline D}_{\\xi}\\Delta_{\\z}^{-1}\n \\psi(\\z\/\\xi)\\bigr)\\bigr|_{\\z=\\tau_j} + \\\\\n +\\rho(\\tau_j) \\bigl(\\om(\\z,\\xi|\\k, \\a)\n +{\\overline D}_{\\z}{\\overline D}_{\\xi}\\Delta_{\\z}^{-1}\n \\psi(\\z\/\\xi)\\bigr)\\bigr|_{\\z=q^{-1}\\tau_j} = 0 \\, ,\n\\end{multline}\n$j = 1, \\dots N$, which can be obtained from the integral in (6.10) of\n\\cite{JMS08pp} by taking the residues and using the TQ-relation (4.2)\nof that paper. Also let us recall the definition \n\\begin{equation}\n {\\overline{D}}_{\\z} g(\\z) = g(q\\z)+g(q^{-1}\\z)-2\\rho(\\z)g(\\z) \\, .\n\\label{D}\n\\end{equation}\nActually, (6.10) of \\cite{JMS08pp} comprises one more equation related\nto the residue at $\\z^2 = 0$. This case needs separate treatment and will\nbe discussed below.\n\nFirst we use the following difference equation for the function \n$\\Psi$ defined by (\\ref{Psi}),\n\\begin{align} \\label{eqPsi}\n \\Psi&(\\xi_1,\\xi_2)+\\rho(\\xi_1)q^{-\\a}\\Psi(q^{-1}\\xi_1, \\x_2)=\n \\frac{G(\\nu_1,\\nu_2)}{1+\\bar\\mathfrak{a}(\\nu_1,\\kappa)}\n -\\rho(\\xi_1)q^{-\\a}\\frac{G(\\nu_1-\\eta,\\nu_2)}\n {1+\\mathfrak{a}(\\nu_1-\\eta,\\kappa)} \\notag \\\\[1ex]\n &+\\rho(\\xi_2)\\cth(\\nu_1-\\nu_2)-q^{-\\a}\\cth(\\nu_1-\\nu_2-\\eta)\n \\notag \\\\[1ex]\n &- q^{- \\a} \\bigl(\\rho(\\xi_1)\\rho(q^{-1}\\xi_1)-1\\bigr)\n \\int_{\\cal C} {\\rm d} m(\\mu) G(\\mu,\\nu_2)\\cth(\\mu-\\nu_1+\\eta) \\, ,\n\\end{align}\nwhere $\\overline{\\mathfrak{a}} = 1\/\\mathfrak{a}$ by definition.\nThis equation is the result of an analytical continuation defined for\n$\\Psi(q^{-1} \\x_1, \\x_2)$ through an appropriate deformation of the\nintegration contour in (\\ref{Psi}). Some simplifications occur in the\nlimit $\\nu_1 \\rightarrow \\be_j$ or equivalently $\\xi_1 \\rightarrow \\tau_j$,\nnamely, since $\\mathfrak{a}(\\be_j,\\kappa)=\\bar\\mathfrak{a}(\\be_j-\\eta,\\kappa)=0$ or\n$\\bar\\mathfrak{a}(\\be_j,\\kappa)=\\mathfrak{a}(\\be_j-\\eta,\\kappa)=\\infty$, the first two terms\nin the right hand side of (\\ref{eqPsi}) do not contribute. Then we have\n\\[\n \\rho(\\tau_j)\\rho(q^{-1}\\tau_j) = \n \\frac{Q^-(q^{-1}\\tau_j;\\kappa+\\a)Q^+(\\tau_j;\\kappa)}\n {Q^-(\\tau_j;\\kappa+\\a)Q^+(q^{-1}\\tau_j;\\kappa)}\\cdot\n \\frac{Q^-(\\tau_j;\\kappa+\\a)Q^+(q^{-1}\\tau_j;\\kappa)}\n {Q^-(q^{-1}\\tau_j;\\kappa+\\a)Q^+(\\tau_j;\\kappa)} = 1\n\\]\nwith the $Q$-functions $Q^\\pm$ defined in \\cite{JMS08pp}. This means\nthat also the last term in the right hand side of (\\ref{eqPsi}) does\nnot contribute. Hence, we obtain\n\\begin{equation} \\label{eqPsi1}\n \\Psi(\\tau_j, \\x_2) + \\rho(\\tau_j)q^{-\\a}\\Psi(q^{-1}\\tau_j, \\x_2) =\n \\rho(\\xi_2) \\cth(\\be_j-\\nu_2)-q^{-\\a} \\cth(\\be_j-\\nu_2-\\eta) \\, .\n\\end{equation}\nNote that the right hand side is, up to the sign, equal to the driving\nterm in the integral equation (\\ref{newg}) for $G$.\n\nIf we take the formula (\\ref{ompsi}) and use (\\ref{eqPsi1}) then, after\nsome algebra, we obtain \n\\begin{align} \\label{eqom}\n \\om(&\\tau_j,\\xi_2|\\kappa,\\a)\n +\\rho(\\tau_j)\\om(q^{-1}\\tau_j,\\xi_2|\\kappa,\\a) = \\notag \\\\[1ex]\n &-{\\bigl(\\Delta_{\\z}\\psi(\\z\/\\xi_2)\\bigr)}\\bigr|_{\\z=\\tau_j}-\n \\rho(\\tau_j){\\bigl(\\Delta_{\\z}\\psi(\\z\/\\xi_2)\\bigr)}\n \\bigr|_{\\z=q^{-1}\\tau_j} \\notag \\\\[1ex]\n &+2(\\rho(\\tau_j)+\\rho(\\xi_2))\\;\\psi(\\tau_j\/\\xi_2)\n -2(1+\\rho(\\tau_j)\\rho(\\xi_2))\\;\\psi(q^{-1}\\tau_j\/\\xi_2) \\, .\n\\end{align}\nNow we need to check that this equation is equivalent to (\\ref{norm}).\nTo this end we should verify the following equality \n\\begin{multline} \\label{equal}\n \\bigl( {\\overline{D}}_{\\z} {\\overline D}_{\\xi}\n \\Delta_{\\z}^{-1}\\psi(\\z\/\\xi)\\bigr)\\bigr|_{\\z=\\tau_j}\n +\\rho(\\tau_j) \\bigl({\\overline D}_{\\z}{\\overline D}_{\\xi}\n \\Delta_{\\z}^{-1}\\psi(\\z\/\\xi) \\bigr) \\bigr|_{\\z=q^{-1}\\tau_j}\n = \\\\[1ex]\n \\bigl( \\Delta_{\\z}\\psi(\\z\/\\xi_2) \\bigr) \\bigr|_{\\z=\\tau_j} +\n \\rho(\\tau_j) \\bigl( \\Delta_{\\z} \\psi(\\z\/\\xi_2) \\bigr)\n \\bigr|_{\\z=q^{-1}\\tau_j} \\\\\n -2(\\rho(\\tau_j)+\\rho(\\xi_2)) \\psi(\\tau_j\/\\xi_2)\n +2(1+\\rho(\\tau_j)\\rho(\\xi_2))\\psi(q^{-1}\\tau_j\/\\xi_2) \\, .\n\\end{multline}\nUsing the definition (\\ref{D}) we come after a little algebra to the\nfollowing expression for an arbitrary function $g(\\z)$ \n\\begin{multline} \\label{actdbar}\n {\\overline{D}}_{\\z}{\\overline D}_{\\xi}\\;g(\\z\/\\xi)=\n \\Delta_{\\z}^2\\;g(\\z\/\\xi)\n + 4(1-\\rho(\\z))\\;(1-\\rho(\\xi))\\;g(\\z\/\\xi) \\\\\n -2(\\rho(\\z)+\\rho(\\xi))\\;(g(q\\z\/\\xi)+g(q^{-1}\\z\/\\xi)-2g(\\z\/\\xi)) \\, .\n\\end{multline}\nNow take \n\\begin{multline}\n \\bigl({\\overline{D}}_{\\z} {\\overline D}_{\\xi}\n \\;g(\\z\/\\xi)\\bigr)\\bigr|_{\\z=\\tau_j}+\n \\rho(\\tau_j) \\bigl( {\\overline D}_{\\z} {\\overline D}_{\\xi}\n \\;g(\\z\/\\xi)\\bigr) \\bigr|_{\\z=q^{-1}\\tau_j} = \\\\[1ex]\n \\bigl(\\Delta_{\\z}^2\\;g(\\z\/\\xi)\\bigr)\\bigr|_{\\z=\\tau_j}\n +\\bigl(\\Delta_{\\z}^2 \\;g(\\z\/\\xi)\\bigr)\\bigr|_{\\z=q^{-1}\\tau_j} -\n 2(\\rho(\\tau_j)+\\rho(\\xi)) \\bigl(\\Delta_{\\z}\\;g(\\z\/\\xi)\\bigr)\n \\bigr|_{\\z=\\tau_j} \\\\[1ex] + 2(1+\\rho(\\tau_j)\\rho(\\xi))\n \\bigl(\\Delta_{\\z}\\;g(\\z\/\\xi)\\bigr) \\bigr|_{\\z=q^{-1}\\tau_j} \\, .\n\\end{multline}\nIf we substitute $g(\\z\/\\xi)=\\Delta_{\\z}^{-1}\\psi(\\z\/\\xi)$ and take\n$\\xi=\\xi_2$, then we immediately arrive at the equality (\\ref{equal}).\n\nAs was mentioned above, there is one more case to be considered,\ncorresponding to the contour $\\G_0$, i.e.\\ to the residue at $\\z^2 = 0$\nin equation (6.10) of \\cite{JMS08pp} which has to vanish. Its vanishing\nfollows from\n\\begin{multline}\n \\lim_{\\x_1 \\rightarrow 0} \\x^{- \\a} \\bigl( \\om (\\x_1, \\x_2) +\n \\overline{D}_{\\x_1} \\overline{D}_{\\x_2} \\D^{-1}_{\\x_1} \\ps (\\x)\n \\bigr) = \\\\\n \\frac{2 q^{- \\k}}{q^\\k + q^{- \\k}} \\biggl[\n \\r(\\x_2) - q^{- \\a} + (q^\\a - q^{- \\a})\n \\int_{\\cal C} {\\rm d} m(\\m) G(\\m, \\n_2) \\biggr] = 0 \\, .\n\\end{multline}\nHere we have used (\\ref{Psi}), (\\ref{ompsi}), (\\ref{defpsi}),\n(\\ref{actdbar}) as well as the fact that $\\lim_{\\n \\rightarrow - \\infty}\n\\r (\\x) = (q^{\\a + \\k} + q^{- \\a - \\k})\/(q^\\k + q^{- \\k})$ in the first\nequation and the identity (\\ref{grhoid}) in the second equation.\n\nThe normalization condition just shown to be satisfied by our function\n$\\om$ defined in (\\ref{defom}) is the main ingredient in our proof\nthat $\\om$ is in fact the same function as introduced in equation (7.2)\nof \\cite{JMS08pp}. Let us consider $\\om$ as a function of $\\x_1$. As\nwas shown in \\cite{JMS08pp} the function $\\r (\\x_1)$ depends only on\n$\\x_1^2$. The same is then true for $\\Psi (\\x_1, \\x_2)$ from (\\ref{Psi}).\nUsing (\\ref{ompsi}) we conclude that $\\x^{- \\a} \\om(\\x_1, \\x_2|\\k, \\a)$\nis a function of $\\x_1^2$. From its definition (\\ref{defom}) and from\n(\\ref{defdens}), (\\ref{c2b11}) we see that $\\om$ is rational in $\\x_1^2$\nof the form $P(\\x_1^2)\/Q(\\x_1^2)$, where $P$ and $Q$ are polynomials.\nClearly both of them are at most of degree $N + 2$. The zeros of $Q$ are\nthe $N$ zeros of the transfer matrix eigenvalue $\\La (\\x_1, \\k)$ plus\ntwo zeros at $q^{\\pm 2} \\x_2^2$ stemming from the two simple poles of\n$\\x^{-\\a} \\mathbf{c}^*_{[1,2]} (\\x_2, \\a) \\mathbf{b}^*_{[1,2]} (\\x_1,\\a - 1) (1)$.\nComparing now with the definition (7.2) of \\cite{JMS08pp} we see that\nthe functions there has precisely the same structure. It is rational\nof the form $\\tilde P (\\x_1^2)\/ \\tilde Q (\\x_1^2)$ with two polynomials\n$\\tilde P$, $\\tilde Q$ at most of degree $N + 2$. $Q$ and $\\tilde Q$\nhave the same zeros. We may therefore assume that they are identical.\nIn order to show that $P$ and $\\tilde P$ also agree we have to provide\n$N + 3$ relations. $N + 1$ of them are given by the normalization\ncondition above. Another two come from the residues at the two trivial\npoles.\n\nSince they are outside the canonical contour, we have to consider\nagain the analytic continuation of the integral (\\ref{Psi}) defining\n$\\Psi$ with respect to $\\x_1$. There are four regions depending on the\n\\begin{figure}\n \\centering\n \\includegraphics{anacontpsi}\n \\caption{\\label{fig:anacontpsi}\n Four cases to be considered for the analytic continuation\n of $\\Psi (\\x_1, \\x_2)$ with respect to $\\n_1$. Here $\\cal C$\n is the canonical contour of figure~\\ref{fig:cont}.}\n\\end{figure}\nlocation of $\\n_1$ relative to the contour (see figure\n\\ref{fig:anacontpsi}). Using (\\ref{Psi}) we obtain\n\\begin{multline} \\label{anacontpsi}\n \\Ps (\\x_1, \\x_2) =\n \\int_{\\cal C} {\\rm d} m(\\m) G(\\m, \\n_2)\n\t\\bigl(q^\\a \\cth(\\m - \\n_1 - \\h) - \\r(\\x_1) \\cth(\\m - \\n_1) \\bigr)\n \\\\[1ex] - \\begin{cases}\n \\dst{\\frac{G(\\n_1, \\n_2)}{1 + \\mathfrak{a} (\\n_1, \\k)}}\n & \\text{case (I)} \\\\[2ex]\n 0 & \\text{case (II)} \\\\[1ex]\n \\dst{\\frac{G(\\n_1, \\n_2)}{1 + \\mathfrak{a} (\\n_1, \\k)} +\n \\frac{q^\\a G(\\n_1 + \\h, \\n_2)}\n {(1 + \\mathfrak{a} (\\n_1 + \\h, \\k)) \\r(q \\x_1)}}\n & \\text{case (III)} \\\\[2ex]\n \\dst{\\frac{G(\\n_1, \\n_2)}{1 + \\mathfrak{a} (\\n_1, \\k)}}\n & \\text{case (IV)} \\, .\n \\end{cases}\n\\end{multline}\nThen, e.g.\\ be means of the integral equation (\\ref{newg})\n\\begin{subequations}\n\\label{restrivpsi}\n\\begin{align} \\label{resa}\n \\res_{\\x_1^2 = q^2 \\x_2^2} & \\Psi (\\x_1, \\x_2) = \\notag \\\\\n & - \\frac{2 \\x_2^2 q^{2 - \\a}}\n {(1 + \\mathfrak{a} (\\n_2 + \\h, \\k))(1 + \\overline{\\mathfrak{a}}(\\n_2, \\k))} =\n - \\frac{2 \\x_2^2 q^{2 - \\a} a(\\x_2 q) d(\\x_2)}\n {\\La (\\x_2 q, \\k) \\La(\\x_2, \\k)} \\, , \\\\[2ex]\n \\res_{\\x_1^2 = q^{- 2} \\x_2^2} & \\Psi (\\x_1, \\x_2) = \\notag \\\\\n \\label{resb}\n & \\frac{2 \\x_2^2 q^{\\a - 2}}\n {(1 + \\mathfrak{a} (\\n_2, \\k))(1 + \\overline{\\mathfrak{a}}(\\n_2 - \\h, \\k))} =\n \\frac{2 \\x_2^2 q^{\\a - 2} a(\\x_2) d(\\x_2 q^{- 1})}\n {\\La (\\x_2, \\k) \\La(\\x_2 q^{-1}, \\k)} \\, ,\n\\end{align}\n\\end{subequations}\nwhere $a$ and $d$ are the vacuum expectation values of the diagonal\nelements of $T(\\z)$. Since the ratios on the right hand side are\ninvariant under changing the normalization of the $R$-matrix, we can\ndirectly compare the residues obtained from (\\ref{ompsi}),\n(\\ref{restrivpsi}) with those obtained from equation (7.2) of\n\\cite{JMS08pp}. We find agreement, which completes the proof.\n\nWhat if we consider (\\ref{ompsi}) as the definition of $\\om$? Then,\nin addition, we have to show that there are no poles other than the\ntwo trivial ones and those at the location of the zeros of\n$\\La(\\x_1, \\k)$. But this is immediately clear from (\\ref{anacontpsi}).\nThe integral has only poles at the zeros of $\\La(\\x_1, \\k)$. In case\n(I) there is one additional pole at $\\n_1 = \\n_2 + \\h$ with residue\n(\\ref{resa}). The simple poles of $G (\\n_1, \\n_2)$ at $\\la_j + \\h$,\nwhere the $\\la_j$ are the Bethe roots (see appendix \\ref{app:dermult}),\nare canceled by the simple poles of $\\mathfrak{a} (\\n_1, \\k)$ (see equation\n(\\ref{auxes})). In cases (II) and (IV) there is nothing to show. In\ncase (III) we have one additional pole at $\\n_1 = \\n_2 - \\h$ with residue\n(\\ref{resb}). The simple poles at $\\la_j - \\h$ have vanishing\nresidue due to (\\ref{auxes}) and since\n\\begin{equation}\n \\res_{\\n_1 = \\la_j - \\h} G(\\n_1, \\n_2) =\n - \\frac{q^\\a G(\\la_j, \\n_2)}{\\r (\\z_j) \\mathfrak{a}' (\\la_j)} \\, .\n\\end{equation}\n\n\\section{The exponential form -- preliminary remarks}\n\\label{sec:top}\nThe main result of \\cite{JMS08pp} is the formula (1.12). It makes\nthe calculation of arbitrary correlation functions possible, because\nthe operators $\\mathbf{t}^*,\\mathbf{b}^*,\\mathbf{c}^*$ generate a basis of the space of\nquasi-local operators \\cite{BJMS09app}. Although this formula proves the\nfactorization of the correlation functions and allows, in principle,\nalso for their direct numerical evaluation, it may be sometimes\npreferable to avoid the creation operators and to have an explicit\nformula for the correlation functions in the standard basis generated\nby the local operators ${e_j}_\\e^{\\e'}$. We believe that some form of\nthe exponential formula discussed in the previous papers\n\\cite{BJMST08app,BGKS06,BGKS07,BDGKSW08} must be valid in case of\ntemperature, disorder and magnetic fields as well. Unfortunately, the\nproblem of constructing all operators that appear in this formula remains\nstill open. We hope to come back to it in a future publication. Here we\nformulate the general properties we expect for these operators and show\nby examples how they should look like for short distances.\n\nFrom now on we shall use the notation and the terminology of the paper\n\\cite{BJMST08app}. In particular we shall be dealing with the space\n${\\cal W}^{(\\a)}$ of quasi-local operators of the form $q^{2\\a S(0)}{\\cal O}} \\def\\CP{{\\cal P}$\nintroduced there. First we define a density operator $D_N^{\\ast}:\n{\\cal W}^{(\\a)} \\rightarrow {\\mathbb C}$ which generalizes that one\ndefined by the formulae (33), (34) of \\cite{BGKS07}, namely, for any\nquasi-local operator ${\\cal O}} \\def\\CP{{\\cal P}$ we define\n\\begin{equation}\n D_N^\\ast ({\\cal O}} \\def\\CP{{\\cal P}) = \\< {\\cal O}} \\def\\CP{{\\cal P} \\>_{T,\\a,\\kappa}\n\\end{equation}\nin such a way that \n\\begin{equation}\n D_N^\\ast \\bigl( {e_1}^{\\e_1}_{\\e_1'} \\dots {e_m}^{\\e_m}_{\\e_m'}\n \\bigr) =\n {D_N}^{\\e_1' \\dots \\e_m'}_{\\e_1 \\dots \\e_m}(\\x_1, \\dots, \\x_m|\\k, \\a)\n\\end{equation}\nwhere ${D_N}$ is the density matrix defined in (\\ref{defdens}).\n\nWe expect that as before\n\\begin{equation} \\label{exponential}\n D_N^\\ast ({\\cal O}} \\def\\CP{{\\cal P}) = \\mathbf{tr}^{\\a} \\bigl\\{ \\exp(\\Omega)\n \\bigl( q^{2 \\a S(0)} \\mathcal{O} \\bigr) \\bigr\\} \\, ,\n\\end{equation}\nwhere $\\mathbf{tr}^\\a$ is the $\\a$-trace defined in \\cite{BJMST08app}\nand where the operator $\\Omega$ consists of two terms like that one\nconstructed in \\cite{BGKS07}\n\\begin{equation} \\label{Omega}\n \\Omega = \\Omega_1 + \\Omega_2 \\, .\n\\end{equation}\nIn fact, the first term follows from \\cite{BJMST08app,JMS08pp}. It must\nbe of the form\n\\begin{equation} \\label{Omega1}\n \\Omega_1 = \\int\\frac{d\\z_1^2}{2\\pi i\\z_1^2}\n \\int\\frac{d\\z_2^2}{2\\pi i\\z_2^2}\n \\bigl(\\om_0(\\z_1\/\\z_2|\\a) - \\om(\\z_1,\\z_2|\\k,\\a) \\bigr)\n \\mathbf{b}(\\z_1)\\mathbf{c}(\\z_2) \\, ,\n\\end{equation}\nwhere the function $\\om_0$ was defined in \\cite{BJMST08app},\n\\begin{equation} \\label{om0}\n \\om_0(\\z|\\a) = - \\biggl( \\frac{1-q^{\\a}}{1+q^{\\a}} \\biggr)^2\n \\Delta_{\\z} \\psi(\\z) \\, .\n\\end{equation}\nThe second part in the right hand side of (\\ref{Omega}) should be of the\nform \n\\begin{equation}\n \\Omega_2 = \\int\\frac{d\\z^2}{2\\pi i\\z^2}\\log(\\rho(\\z))\\mathbf{t}(\\z)\n\\label{Omega2}\n\\end{equation}\nwhere the operator $\\mathbf{t}$ is yet to be determined. In some sense\nit must be the conjugate of the operator $\\mathbf{t}^*$. The integration\ncontour for both, $\\Om_1$ and $\\Om_2$, is taken around all simple\npoles $\\z_1,\\z_2,\\z=\\xi_j$ with $j=1,\\dots,m$ in anti-clockwise\ndirection. The number $m$ is the length of locality of the operator\n${\\cal O}} \\def\\CP{{\\cal P}$.\n\nLet us list some of the most important expected properties of the\noperator $\\mathbf{t}$. First, we expect that like $\\mathbf{t} ^*(\\z )$ the operator\n$\\mathbf{t}(\\z)$ is block diagonal,\n\\[\n \\mathbf{t}(\\z ):\\ \\ \\mathcal{W}_{\\a ,s} \\rightarrow \\mathcal{W}_{\\a ,s} \\, ,\n\\]\nwhere, as was explained in \\cite{BJMST08app}, $\\mathcal{W}_{\\a ,s}\n\\subset \\mathcal{W}^{(\\a)}$ is the space of quasi-local operators of\nspin $s$. We will deal below mostly with the sector $s=0$.\n\nThen we expect $\\mathbf{t}(\\z)$ to have simple poles at $\\z=\\xi_j$. Let us define\n\\begin{equation} \\label{tbj}\n \\mathbf{t}_j = \\res_{\\z=\\xi_j} \\mathbf{t}(\\z) \\frac{d\\z^2}{\\z^2}\n\\end{equation}\nwhile\n\\begin{equation}\n \\mathbf{t}^*_j = \\mathbf{t}^*(\\xi_j) \\, .\n\\label{tb*j}\n\\end{equation}\nIn contrast to (\\ref{tbj}) the operator $\\mathbf{t}^*_j$ is well defined only\nif it acts on the states $X_{[k,l]}$ with $l