T-W relation and free energy of the Heisenberg chain at a finite temperature

A new nonlinear integral equation (NLIE) describing the thermodynamics of the Heisenberg spin chain is derived based on the t-W relation of the quantum transfer matrices. The free energy of the system in a magnetic field is thus obtained by solving the NLIE. This method can be generalized to other lattice quantum integrable models. Taking the SU(3)-invariant quantum spin chain as an example, we construct the corresponding NLIEs and compute the free energy. The present results coincide exactly with those obtained via other methods previously.


Introduction
Quantum integrable systems (or exactly solvable models [1]) play important roles in investigating some nonpertubative properties of quantum field/string theory such as the planar N = 4 super-symmetric Yang-Mills (SYM) theory and the planar AdS/CFT [2,3] (see also references therein). They also enhance our understanding of quantum phase transitions and critical phenomena in statistical physics [4,5], condensed matter physics [6] and cold atom systems [7]. In the past decades, several theoretical methods [1,[8][9][10][11][12][13][14][15][16] have been developed to approach eigenvalue problem of quantum integrable models. A method to approach thermodynamic properties of quantum integrable models was first achieved by Yang and Yang for the quantum Bose gas [17,18] based on the Bethe ansatz solution [19,20]. Later, the method (now known as thermodynamic Bethe ansatz (TBA)) was extended by Gaudin [21] and Takahashi [22,23] to investigate the thermodynamics of the Heisenberg spin chain. With their methods, the free energy was finally found to be encoded by a set of infinitely many nonlinear integral equations (NLIEs). The numerical studies of these equations need some kind of truncation scheme [24][25][26][27]. An alternative approach, the so-called quantum transfer matrix (QTM) method [28][29][30][31][32][33][34] has also been proposed. In the QTM formalism, a one-dimensional quantum system at a finite temperature can be mapped into a classical system on two-dimensional inhomogeneous lattice by the Trotter-Suzuki mapping [28]. The free energy of the quantum system can be expressed by the largest eigenvalue of the quantum transfer matrix and the next-largest eigenvalue provides the correlation length [34].
Recently, a novel t − W method has been proposed to calculate physical properties of quantum integrable systems with or without U (1) symmetry [65,66]. The key point of this method lies in that a single t − W relation determines the whole spectrum of the transfer matrix and the roots possess well-defined patterns.
In this paper, we will construct the t − W relation of the quantum transfer matrix. By analysing the root patterns of the quantum transfer matrix, a new NLIE describing the thermodynamics can be derived straightforwardly based on the t − W relation.
Let us consider the Hamiltonian of the periodic Heisenberg spin chain in anti-ferromagnetic regime (J > 0): where σ α 1+L = σ α 1 , for α = x, y, z, (1.2) and σ x , σ y , σ z are Pauli matrices. The model is one of the best studied paradigmatic models in quantum integrable systems and still remains a source of inspiration and fascinating new progress of quantum integrable systems.
The paper is organized as follows. Section 2 serves as an introduction of our notations and some basic ingredients. We also briefly review that the partition function of the Heisenberg chain at a finite temperature is expressed in the QTM formalism. In Section 3, we derive the t − W relation of the transfer matrix via the fusion technique. With the help of the resulting t−W relation and some asymptotical behaviors of eigenvalues of the transfer matrices, we obtain the Bethe-ansatz-like equations (BAEs), which may completely determine eigenvalues. In Section 4, based on the root distributions of eigenvalues corresponding to the state with the maximus |Λ (Q) (0)|, we derive a new nonlinear integral equation (NLIE) and the analytic properties, which enable us to obtain the partition function (and free energy). In Section 5, we have succeeded in giving the associated t − W relations among the transfer matrices, which allow one to derive the associated NLIEs.
Taking the SU (3)-invariant spin chain an example, we apply our method to obtain the corresponding free energy. In Section 6, we summarize our results and give some discussions. Some supporting materials are given in Appendices A-F.

Heisenberg chain and the associated QTM
The integrability of the model (1.1)-(1.2) is associated with the well-known rational six-vertex R-matrix where u is the spectral parameter and the crossing parameter η = i. The R-matrix satisfies the quantum Yang-Baxter equation (QYBE) 2) and the properties: Unitarity relation : Crossing relation : Fusion condition : R 12 (±η) = ±ηP PT-symmetry : Here R 21 (u) = P 12 R 12 (u)P 12 with P 12 being the usual permutation operator and t i denotes transposition in the i-th space. Throughout this paper we adopt the standard notations: for any matrix A ∈ End(C 2 ), A j is an embedding operator in the tensor space C 2 ⊗ C 2 ⊗ · · ·, which acts as A on the j-th space and as identity on the other factor spaces; R ij (u) is an embedding operator of R-matrix in the tensor space, which acts as identity on the factor spaces except for the i-th and j-th ones.
Let us introduce the transfer matrix t(u) of the XXX closed chain [10] where tr 0 denotes trace over the "auxiliary space" 0. The expression (2.1) of the R-matrix R(u), the definition (2.9) of the transfer matrix imply that Moreover, the Hamiltonian described by (1.1) and (1.2) can be expressed in terms of the transfer matrix which implies that for a small u the transfer matrix has the expansion where t (L) (0) = η L P 1L · · · P 12 . The above relation and the crossing-symmetry (2.5) of the R-matrix allow one to introduce a quantum transfer matrix t (Q) (u) [31,32], where the positive real parameter β is related to the temperature T of the system as β = 1 T . For a very large even integer N , the partition function Z(β) of the spin-1 2 XXX closed chain described by the Hamiltonian (1.1) and (1.2) at a temperature T can be expressed in terms of the quantum transfer matrix t (Q) (u) by the QTM method (for details the reader is referred to Ref. [34]), Here Λ (Q) (0) max is the eigenvalue corresponding to the state with the maximus value |Λ (Q) (0)|. Moreover, it was shown [31,32,34] that in the limit of N → ∞ Λ (Q) (0) max is gaped from the others eigenvalues of Λ (Q) (0).

T-W relation and eigenvalues of the transfer matrix
Similarly as the quantum transfer matrix (2.11), for a large even positive integer N , let us introduce another where {θ j |j = 1, · · · , N } are some generic complex number, which are called the inhomogeneous parameters (for the special choice of the inhomogeneous parameters, one can recover the quantum transfer matrix (2.11)).
The expression (2.1) of the R-matrix R(u), the definition (3.1) of the transfer matrix t(u) imply that Moreover with the help of the fusion of R-matrix [70], we can derive that the transfer matrix t(u) satisfies where the functions a(u) and d(u) are given by and W(u) (given by below (A.10)), as a function of u, is an operator-valued polynomial of degree N , which actually is some fused transfer matrix of the fundamental one. The details of the proof the t − W relation (3.3) will be given in Appendix A.
It is easy to shown that the transfer matrices t(u) and W(u) commute with each other, namely, which implies that they have common eigenstates. Let |Ψ be a common eigenstate of the transfer matrices with eigenvalues Λ(u) and W (u), namely, The operator identity (3.3) of the transfer matrices then gives rise to the corresponding relation for their The expansion expression (3.2) and (3.7) allow us to express any eigenvalue Λ(u) of the transfer matrix (or W (u) of the fused one) in terms of its N zero points {z j |j = 1, · · · , N } (or {w j |j = 1, · · · , N }) as follow Taking u at the 2N points {z j |j = 1, · · · , N } and {w j |j = 1, · · · , N }, we have the associated BAEs Then 2N parameters {z j |j = 1, · · · , N } and {w j |j = 1, · · · , N }, which are related to the roots of the eigenvalues Λ(u) and W (u), can be determined completely by the above BAEs.
In order to investigate the thermodynamics of the spin-1 2 XXX closed chain described by the Hamiltonian (1.1) and (1.2), let us focus on the quantum transfer matrix t (Q) (u) given by (2.11) for a large even N and denote its eigenvalue by Λ (Q) (u). In this case the inhomogeneous parameters are specially chosen by (2.11) and the associated functions a(u) and d(u) become (3.13) The free energy per site f (β) is given in terms of the partition function (2.12) by Hence it is sufficient to calculate Λ (Q) (u) of the eigenstate with |Λ (Q) (0)| max . Eigenvalues of the QTM can be also obtained by the algebraic Bethe ansatz method [10] alternatively, where Λ (Q) (u) is given in terms of a homogeneous T − Q relation, namely, where the functions a(u) and d(u) are given in (3.13). The parameters {λ j |j = 1, · · · , M ; M = 0, · · · , N } satisfy the BAEs It was shown [31,32,34] that the eigenvalue of the eigenstate with |Λ (Q) (0)| max belongs to the sector of M = N 2 with all the Bethe roots being real. For the simplicity, let us introduce M = N 2 in the following part of the paper, and introduce a parameter τ (a positive real number ) associated with the temperature as and a normalized eigenvalueΛ (Q) (u) The T − Q relation (3.15) allows us to expressΛ (Q) (u) as where the real Bethe roots satisfy the associated BAEs where the imaginary parts of u (±) j are close to zero for a large N (namely, Im(u (±) j ) ∼ 0) and τ is given in (3.17). The Bethe ansatz solution (3.15) also shows that the roots of Λ (Q) (u) of the state with |Λ (Q) (0)| max indeed has the distribution (3.21). With the help of the t−W relation (3.7), we can derive that the eigenvalue W (Q) (u) of the state with |Λ (Q) (0)| max has the decomposition where Im(w (±) j ) ∼ 0 for a large N . Then the t − W relation (3.7) for the state with the |Λ (Q) (0)| max becomes where the functions q(u) andw(u) are It is remarked that the functionǭ(u) satisfies the analytic property: ǫ(u) is analytic except some singularities on the axis Im(u) = ± 3 2 and lim

Nonlinear integral equations and the free energy
The decomposition (3.21) and the very t − W relation (3.23) allow us to give an integral representation of where the closed integral contour C 1 is surrounding the axis of Im(v) = 1 2 , while C 2 is surrounding the axis of Im(v) = − 1 2 . With the help of the t − W relation (3.23) and the integral representation (4.1), we can derive a NLIE of the functionǭ(u) ln(q(u) + (4 cosh 2 hβ 2 − 1)e −βǭ(u) ) = 2 ln 2 cosh hβ 2 Due to the fact that the roots and the poles ofΛ (Q) (u) locate nearly on the two lines with imaginary parts close to ±1 (see the decomposition (3.21)), we can use the Fourier transformation to obtain another where we have used η = i. Let us introduce the dressing energy function ǫ(u) It is believed that the analytic property (3.26) and the NLIE (4.2) and the asymptotical behavior (4.4) might completely determine the functionǭ(u).
Finally we obtain the free energy of the XXX chain described by the Hamiltonian (1.1)-(1. 2) as Using the numerical iterative procedure in Appendix B, we obtain the free energy f variation with temperature T in different magnetic fields as shown in figure 2. From the figure, we find that our result coincides well with the those of [31,32] and [60] obtained with different approaches. Moreover, the analytic property (3.26) and the NLIE (4.2) allow us to give the HTE of the free energy as which recovers that of [56] obtained previously with a different approach. The details of the derivation of (4.6) will be given in Appendix C.
Besides the QYBE, the R-matrix satisfies the properties: Unitarity relation : The corresponding QTM can be constructed as follow [52] t (Q) where the diagonal matrix S 0 = diag(µ 1 , µ 2 , ..., µ n ) is related to the external field and τ is given in (3.17).
In the case of the SU (3) invariant spin chain, we have S 0 = S z 0 = diag(1, 0, −1). The expression (5.2) of the R-matrix R(u), the definition (5.8) of the QTM imply that With the help of the fusion [69,70] of the R-matrix we can introduce some fused quantum transfer matrices 3  i (u)|i = 1, · · ·, n − 1}. Using the method developed in [71] we can derive that the fused transfer matrices satisfy the associated t − W relations The operators t n (u) and the functions a m (u) are given by 3 It is remarked that the fused transfer matrices {t (Q) i (u)|i = 2, · · ·, n − 1} in this paper correspond to those {τ The proof of the relations (5.10) will be given in Appendix A.
Some remarks are in order. We have introduced the 2n − 3 extra (or auxiliary) fused transfer matrices i (u)|i = 1, · · ·, n − 1}. Hence in order to determine the eigenvalue Λ (Q) (u) of the original quantum transfer matrix t (Q) (u), we need to further introduce 2n − 3 auxiliary functions (c.f., 2 n − 2 auxiliary functions for the SU (n) case [52,72]) which correspond to the eigenvalues of the resulting fused transfer matrices.

T-W relations of the SU(3)-variant chain
Taking the SU (3)-invariant spin chain as an example, we shall show how our method works in the following parts of the section. The Hamiltonian of the SU (3)-invariant closed spin chain is given by with the periodic boundary condition 14) The associated R-matrix reads For the periodic SU (3) model with an external field h, the corresponding QTM can be constructed as follow [52] t (Q) where the operator S z 0 = diag(1, 0, −1). The expression (5.15) of the R-matrix R(u), the definition (5.16) of the QTM imply that The corresponding t − W relations 4 (5.10) read For later calculative and notational convenience, we shift the spectral parameter u of the transfer matrix t The resulting transfer matrices t 2 (u), as the functions of u, are three operatorvalued polynomials of degree N . In addition, the transfer matrices t (5.20) The commutativity (5.20) of the transfer matrices t with different spectral parameters implies that they have common eigenstates. Let |Ψ be a common eigenstate of the QTMs with the eigenvalues Λ The operator identities (5.18) and (5.19) of the QTMs then give rise to the corresponding relations for their eigenvalues  j |j = 1, · · · , N }(i = 1, 2), we have the associated BAEs Similarly, the normalized eigenvaluesΛ where the functionsλ 1 (u),λ 2 (u),w 1 (u) andw 2 (u) arē where the closed integral contour C ′ 1 (or C ′ 2 ) is surrounding the axis of Im(v)= 1 2 (or − 1 2 ), while C ′ 3 ( or C ′ 4 ) is surrounding the axis of Im ln(e −βε1(u) + (b(β) − 1)e −βε4(u) ) = ln b(β) Finally we can obtain the free energy of the periodic SU (3) chain described by the Hamiltonian (5.13) with an external field h as Using the numerical iterative procedure in Appendix E, we obtain the free energy f variation with temperature T in different magnetic fields as shown in figure 5. From this figure, we can conclude that our result coincides well that of [52] with a different approach. Moreover, we can obtain the HTE of the free energy of the SU (3)-invariant spin chain described by the Hamiltonian (5.13)-(5.14) (the details of the derivation is given in Appendix F) (5.46)

Conclusions
In this paper, we have studied the thermodynamics of the Heisenberg chain at a finite temperature in antiferromagnetic regime via the recent developed t − W method [65,66]. A novel nonlinear integral equation The QYBE (2.2) and the fusion condition (2.6) allow us to derive the relation Direct calculation shows that where the fused R-matrix R Keeping (3.1) in mind, let us introduce one-row monodromy matrix The where the functions a(u) and d(u) are given by (3.4), and the fused monodromy matrix T Let us take the product of the transfer matrix t(u) and t(u − η) given by (3.1) Using the similar fusion procedure as that we have done for the SU (2) case, we can also derive the associated t − W relations (5.10) among the fused transfer matrices {t Let us introduce two small positive parameters δ and ∆ such that 0 < δ < ∆ < 1 2 , and we can deform the integral contours in (4.2) without changing values of the resulting integrals as follows. The decomposition (the first identity of (3.23)) implies that the function ξ(u) has singularities only on the straight lines Im(u) = ± 1 2 , ± 3 2 and vanishes asymptotically, i.e., lim u→∞ ξ(u) = 0, which allows us to deform the integral contour C 1 along the straight line Im(v) = ( 1 2 − δ) and the line Im(v) = ( 1 2 + δ) ( the contour C 2 along the straight line Im(v) = −( 1 2 + δ) and the line Im(v) = −( 1 2 − δ) anti-clockwise without changing the integral values in (4.2). Namely, we can have the integral representation ln(q(u) + (4 cosh 2 hβ 2 − 1)e −βǭ(u) ) = 2 ln 2 cosh hβ 2 The above new integral representation allows us to compute the values ofǭ(u±( 1 2 +∆)η) with u ∈ R provided that the values of ξ(u) on the four straight lines Im(v) = ±( 1 2 − δ), ±( 1 2 + δ) are known. With the help of the analytical property (3.26) of the functionǭ(u) and the Cauchy's theorem, we can compute the values ǫ(u) on the four straight lines Im(u) = ±( 1 2 − δ), ±( 1 2 + δ) if we know its values on the two straight lines Im(v) = ±( 1 2 + ∆). Namely, we have Now our numerical strategy can be constructed as follows. Starting fromǭ (n) (u ± ( 1 2 + ∆)η), we can compute

Appendix C: High-temperature expansion of the Heisenberg chain
For β → 0, the functionw(u) becomes independent of u since the integrand in relation (4.2) has no poles in the area surrounded by the contours C 1 and C 2 . Insertingǭ(u) ∼ 0 into the integral in relation (4.5) leads to the correct high-temperature entropy −βf = ln Λ(0) ∼ ln 2.
For small values of β, we seekǭ(u) as the series expansion ǫ(u) =ǭ 1 (u) + βǭ 2 (u) + · · ·. (C.1) With regard to the expansion formula ln[q(u) + ae −βǭ ] = ln(a + 1) where we have set a = 4 cosh 2 ( hβ 2 ) − 1 and the integral equation (4.2) transforms itself into an infinite sequence of coupled equations for the expansion functions {ǭ j (u)}: etc. Note that the contour integrals of {ǭ j (u)} in RHS vanishes because the functionǭ(u) is analytic except some singularities on the axis Im(u) = ± 3 2 . Therefore, Eq. (C.3) have two poles of q(u) inside the contour C 1 and C 2 respectively. Using the residue theorem, we obtain the expression ofǭ 1 (u) (C.5) (C.6) The above expressions allow us to obtain the HTE (4.6) of the free energy.