Correlation Function and Simplified TBA Equations for XXZ Chain

The calculation of the correlation functions of Bethe ansatz solvable models is very difficult problem. Among these solvable models spin 1/2 XXX chain has been investigated for a long time. Even for this model only the nearest neighbor and the second neighbor correlations were known. In 1990's multiple integral formula for the general correlations is derived. But the integration of this formula is also very difficult problem. Recently these integrals are decomposed to products of one dimensional integrals and at zero temperature, zero magnetic field and isotropic case, correlation functions are expressed by $\log 2$ and Riemann's zeta functions with odd integer argument $\zeta(3),\zeta(5),\zeta(7),...$. We can calculate density sub-matrix of successive seven sites. Entanglement entropy of seven sites is calculated. These methods can be extended to XXZ chain up to $n=4$. Correlation functions are expressed by the generalized zeta functions. Several years ago I derived new thermodynamic Bethe ansatz equation for XXZ chain. This is quite different with Yang-Yang type TBA equations and contains only one unknown function. This equation is very useful to get the high temperature expansion. In this paper we get the analytic solution of this equation at $\Delta=0$.


Introduction
We consider the spin 1/2 XXZ chain Among the solvable models this model has been investigated for a long time. It was believed that the exact calculation of correlation function is impossible except the nearest neighbor correlation function. I derived the second neighbor correlation function for J < 0, T = 0, h = 0, ∆ = 1 using the Lieb-Wu solution of one-dimensional Hubbard model [1]. Details are given in Appendix A. In 1990's multiple integral formula were proposed for h = 0, T = 0 and recently multiple integral formula was extended to h = 0, T = 0. But the factorization of these multiple integrals to the integrals of lower dimension still remains a difficult problem. I explain the present situation of factorization in Section 2.
Since Yang and Yang proposed the thermodynamic Bethe ansatz (TBA) equation for one dimensional bosons [2], the calculation of free energy becomes important for other solvable models. Yang-Yang type integral equations for (1) were proposed in early 1970's [3,4,5]. In this theory infinite number of unknown functions appeared for |∆| ≥ 1 and finite number of unknowns appear for ∆ = cos πν with ν = rational number. The equations change by the value of ∆ and not so convenient for numerical calculations. But some important physical properties at low temperature were investigated using these equations.
Around 1990 the quantum transfer matrix method was applied to this model and numerical results coincide with those of Yang-Yang type equations [6,7,8]. In 2001 I proposed a new TBA equation which contains only one unknown function for (1) [9,10]. This equation is very convenient to do the high-temperature expansions and we get one hundred-th order of high temperature expansion [11]. The traditional cluster expansion method gives up to 22'nd. This method was also applied to the Perk-Schultz model [12].
Unfortunately this equation does not numerically converge at low temperature like T /|J| < 0.07 because the integrand strongly oscillates. Some other numerical method is necessary. I showed that this equation gives the known exact results in Ising limit (∆ → ∞ and J∆ = finite) [9]. In Section 3 I give the analytic solution of this equation for XY case (∆ = 0).

Next nearest-neighbor correlator for XXX
from the ground state energy of the half-filled Hubbard model by Takahashi [1] (1977).
3. The twisted four-body correlation function from the third derivative of transfer matrix by Muramoto and Takahashi [14] (1999).
is represented by n-fold integral. For example the emptiness formation probability (EFP) for XXX chain at T = h = 0 is Other arbitrary correlation functions over successive n-sites have similar n-fold integral representation.

Boos-Korepin method to evaluate integrals
Here I introduce the details of direct factorization of multiple integrals by Boos and Korepin [20,21].
1. Transform the integrand to a certain canonical form without changing the integral value.
2. Perform the integration using the residue theorem. Example: Details of transformation are given in Appendix B.
In canonical form denominator is The n-dimensional integral is decomposed to one and two dimensional integrals. Transform Perform the integration Thus second neighbor correlator was rederived from the integral formula. For P (4) = ρ ++++ 4++++ the integrand is Here we put λ ab ≡ λ a − λ b . Canonical form is P In 2003, we calculated ρ +−+− +−+− by Boos-Korepin method and obtained all the correlation functions on 4 lattice sites [22]. Especially, the third-neighbor correlator is The other correlation functions for n = 4 are . From these results we can reproduce the twisted correlation function in (4).
In a similar way, P (5) was calculated after very tedious calculations [23] But the direct integral of other correlations for five sites is almost impossible.

Algebraic approach and qKZ relation
Next problem is to calculate S z j S z j+4 for XXX model. In principle, it's possible to calculate other five-dimensional integrals by use of Boos-Korepin method. It, however, will take tremendous amount of time.
We propose a different method ("algebraic approach") and obtain analytical form of S z j S z j+4 . This is a generalization of the method by Boos, Korepin, Smirnov (2003) for P (6) [24]. We consider the density matrix ρ of successive n sites of inhomogeneous six vertex model with different spectral parameter z j for j-th site For n = 1,and 2 we have The general element of density matrix must satisfy the following algebraic relations.
• Identity relations If we assume ρ 3 as follows these relations are satisfied. In the homogeneous limit z j → 0 this gives the correct correlation functions of XXX model. In the homogeneous limit each term diverges but we have finite limiting number. Then we can calculate the correlation functions of arbitrary element of density matrix using these algebraic relations, although the calculation become complicated. We have calculated all the inhomogeneous correlation functions up to n ≤ 4 from the multiple integrals and confirmed these relations are fulfilled.
Using this algebraic method we can calculate all the element of density sub-matrix of successive 6-sites. Longer system is quite difficult because of the memory and computing time problem. We can calculate 6-th neighbor and 7-th neighbor correlations using the generation function method. They are represented by long polynomials of ζ a 's. Here we write only numerical results [26] S z j S z j+6 = 0.02444673832795890 . . . , S z j S z j+7 = −0.0224982227633722 . . . .

Calculation by continuous dimensions
In a series of papers Boos, Jimbo, Miwa, Smirnov and Takeyama formulated these algebraic calculation by the trace of continuous dimension of auxiliary space [27,28,29,30] (ρ n ) ǫ 1 ,...,ǫn ǫ 1 ,...,ǫn = vac|(E ǫ 1 ǫ 1 ) 1 · · · (E ǫn ǫn ) n |vac , h n (ǫ 1 , . . . , ǫ n , ǫ n , . . . , ǫ 1 ) = (−1) n n j=1 ǫ j (ρ n ) −ǫ 1 ,...,−ǫn ǫ 1 ,...,ǫn , s n = n j=1 1 2 Density sub-matrix in 2 n dimensional space is mapped to a vector in 2 2n dimensional space. Ω n is an operator in this space. Monodoromy matrix is defined as follows: The trace of any monomial of I, E, H and F is a polynomial of dimension d. One can calculate from the definition (5). For example, The dimension d is replaced by µ − ν. Following [30], the operator Ω n is given by where the integration path should surround all z j counter-clockwise. In the homogeneous limit z j → 0 Ω becomes and the calculation becomes very simple. By this formulation we could calculate all the elements of density sub-matrix at n = 7. Calculating the eigenvalues of matrix, we can calculate the von Neumann entropy (entanglement entropy) up to seven sites, where ω α are eigenvalues of density sub-matrix ρ n . In Table 1 S(n) is given up to n = 7.

Generalization to XXZ model
In 2003, Kato, Shiroishi, Takahashi, Sakai have generalized the Boos-Korepin method to the XXZ models with an anisotropy parameter |∆| ≤ 1 for successive three sites [31]. For example P (n) is represented as follows: Here ∆ = cos(πν). Similar integral representations for any arbitrary correlation function for successive three sites were calculated. Nearest-neighbor correlation functions Next nearest-neighbor correlation functions Here c j := cos πjν, s j := sin πjν, Replacing ν → iη/π, we can also get the correlation functions in the massive region ∆ = cosh η > 1 [32]. Third neighbor correlations is also expressed by functions ζ ν and ζ ′ ν , although the expression becomes more complicated [33]. In Figs. 1 and 2, the nearest neighbor, the second neighbor and the third neighbor correlations are shown as functions of ∆.

Simplif ied thermodynamic Bethe ansatz equation
Simplified thermodynamic Bethe ansatz equation at temperature T [9, 10] is Figure 1. The nearest-neighbor and the next nearest neighbor correlation functions for the XXZ chain.
Free energy per site is In this section we look for analytic solution for XY case ∆ = 0, θ = π/2. For this case equation (7) becomes Putting X = tanh πx/4, Y = tanh πy/4 and using u(−Y ) = u(Y ) we have Consider the Fourier transform of following function Assume that ln u(X) = a 0 + 2 ∞ j=1 a j X 2j . We can show that this satisfies (8). This series is convergent at |X| ≤ 1.

Summary
For J < 0, ∆ = 1, T = h = 0 we obtained the factorized form of density sub-matrix up to n = 7. The entanglement entropy for seven sites is new result of this paper. Up to six sites we published in [26]. The six-th neighbor and the seven-th neighbor correlations are calculated by the generating function method for J < 0, ∆ = 1, T = h = 0 [26].
For arbitrary ∆, T = h = 0 we obtained the factorized form up to n = 4. Correlations are given by two transcendental functions ζ ν (j) and ζ ′ ν (j) with j = 1, 3, 5, . . . defined by (6). For correlations of n ≥ 5 the calculation becomes very tedious and no one has succeeded.
For simplified TBA equation, we obtained the analytic solution for XY limit ∆ = 0. Analytic solution for Ising limit was given in [9].
we can reduce the power of numerator, Thus we have obtained the canonical form for P (3). The derivation of (14) is as follows: