Integral equations for thermodynamics of the osp(1|2s) integrable spin chain

We propose a system of nonlinear integral equations (NLIE), which gives the free energy of the osp(1|2s) integrable spin chain at finite temperatures. In contrast with usual thermodynamic Bethe ansatz equations, our new NLIE contain only a finite number of unknown functions. On deriving NLIE, we use our osp(1|2s) version of the T-system and the quantum transfer matrix method. Based on our NLIE, we also calculate the high temperature expansion of the free energy and the specific heat.


Introduction
In recent years, thermodynamics of solvable lattice models related to superalgebras have been studied by thermodynamic Bethe ansatz (TBA) method, and TBA equations are examined from various point of view (see, section 1 in [1] for a comment on the present status of TBA equations for models related to superalgebras). In our previous papers, we have derived TBA equations for the osp(1|2) model [2,3,4] and the osp(1|2s) model [1] from the string hypothesis and the T -system [5] of the quantum transfer matrix (QTM). These TBA equations contain an infinite number of unknown functions, and thus are not always easily treated. It is important to reduce the TBA equations to tractable integral equations which contain only a finite number of unknown functions.
As for the XXZ spin chain, which is related to the algebra U q (A (1) 1 ) of rank one, Takahashi proposed [6] a nonlinear integral equation (NLIE) recently. This NLIE contains only one unknown function. Due to its simplicity, we can calculate [7] the high temperature expansion of physical quantities to very high order from this NLIE. The purpose of this letter is to derive a system of NLIE with only a finite number (the number of rank s) of unknown functions from our osp(1|2s) version of the T -system [5]. This letter is the first attempt to derive this type of NLIE for a vertex model associated with an algebra of arbitrary rank 1 .
In section 2, we briefly mention the T -system for the osp(1|2s) model and the QTM method [13,14,15,16,17]. This section overlaps with section 2 and section 4 in [1], but normalization of the fused QTM is different from [1]. In section 3, we derive the NLIE. The normalized QTM defined in section 2 play the role of the unknown functions of these NLIE. These NLIE contain a parameter m, which corresponds to the fusion degree of the model. For m = 1, these NLIE form a closed set of equations, which contains only a finite number of unknown functions. On the other hand for m ≥ 2, they couple with the ones for m = 1, and contain an infinite number of unknown functions. This type of equations (for m ≥ 2) has never been considered before even in the case of Takahashi's NLIE for the XXZ spin chain [6]. These NLIE relate to traditional TBA equations through the dependant variable transformation (5.1). Using our new NLIE, we calculate the high temperature expansion of the free energy in section 4. Section 5 are devoted to discussions. 1 In the case of the deformation parameter q of an underlining quantum affine algebra is root of unity (|q| = 1, q = 1), the T -system becomes a finite set of difference equations. Thus the corresponding TBA equation becomes a finite set of integral equations. See, [8] for TBA analysis of integrable field theories related to osp(1|2s). We also note that different types of NLIE with finite numbers of unknown functions for different algebras of arbitrary rank were considered in [9,10] in rather different contexts.
The QTM is defined as where R cd ab (v) =Ř cd ba (v); R jk (v) = t k R kj (v) (t k is the transposition in the kth space); N is the Trotter number and assumed to be even. By using the largest eigenvalue T where we set u = − J T N (T : temperature); the Boltzmann constant is set to 1. One can obtain the eigenvalue formulae of the QTM (2.3) by replacing the vacuum part of the dressed vacuum form (DVF) for the row-to-row transfer matrix with that of the QTM. This DVF is imbedded into a DVF for a fusion hierarchy of the QTM. It reads as follows.

Nonlinear integral equations
In [22], Takahashi's NLIE for the XXZ-model [6] was derived from the Tsystem of the QTM. Here we derive our new NLIE from our T -system (2.10). A numerical analysis for finite N, u, s indicates that a two-string solution (for every color) in the sector N = M 1 = M 2 = · · · = M s of the BAE (2.8) provides the largest eigenvalue of the QTM (2.3) at v = 0 [1]. From now on, we consider only this two-string solution. In this case, the phase factors are ε a = ζ a = 1 for any a. Moreover, we expect the following conjecture is valid for this two-string solution [1].
We are considering the case without an external field. If an external field exists, (3.1) will be deformed.
We may assume m,j,k ∈ C are given as follows Here the contour C         1,k ). Now we shall take the Trotter limit N → ∞ in (3.6).
Here the contour C   (v)} 1≤a≤s by using a Jacobi-Trudi determinant formula [5]. In this sense, we need not consider (3.8) for m ∈ Z ≥2 in practical calculations. However, consideration on (3.8) for m ∈ Z ≥2 manifests the relation between our new NLIE and the traditional TBA equations (see, section 5). The free energy per site is given as Solving (3.10) iteratively, we can obtain the free energy through (3.11). For s = 1 case, we find (3.10) converges numerically at least for |T /J| > 0.37.

High Temperature Expansion: osp(1|2) case
It was pointed out [7] that Takahashi's NLIE for the XXX-model [6] is very useful to calculate the high temperature expansion of the free energy. In this section, we shall calculate the high temperature expansion of the free energy (3.11) for s = 1 by our new NLIE (3.10). We assume the following expansion for large T : Due to the shift of the argument y of T 1 (y) in (3.10), we have to take into account the residues from the coefficients {a n (v)}, which contrasts with the XXX-model case [7]. Thus the derivation is not so easy as the XXX-model case. But we still expect that it is easier than to use the traditional TBA equation which contains an infinite number of unknown functions (as for the high temperature expansion for the XXX-model from the traditional TBA equation, see for example, p.124 in [29]). To calculate the coefficients {a n (v)} efficiently, we need further assumption for n ∈ Z ≥1 : a n (v) = where b n,j , c n,j ∈ C are independent of v. Substituting (4.1) into (3.10), we obtain the coefficients {a n (v)} up to order 12. For example, we have a 0 (v) = log 3, Using {a n (0)}, we obtain We have plotted the high temperature expansion of the specific heat (4.5) in figure 1. For comparison, we also plotted a pade approximation for (4.5). For large T , (4.5) agrees with our TBA analysis (see, Fig.1 in [4]). This indicates the validity of our new NLIE (3.10).

Discussion
In this letter, we have derived a system of NLIE with a finite number of unknown functions, which describes thermodynamics of the osp(1|2s) integrable spin chan. This type of NLIE for arbitrary rank is derived for the first time. We shall point out a relation between our new NLIE and usual TBA equations. The functions { T This relation is also valid in the Trotter limit.
In closing this letter, we shall enumerate future problems: • There are T -systems for other algebras [24,23,25,26,27,28,5]. Using these T -systems, we can derive NLIE similar to the ones in this letter, which will be reported elsewhere [30].
• In this letter, we have considered the case without an external field. When an external field exists, a fugacity expansion of the free energy can be calculated recursively through (3.10) (as for the fugacity expansion of the XXXmodel from the traditional TBA equation, see for example, p.127 in [29]).
• One will be able to extend (3.10) to trigonometric or elliptic case. In this case, one must take into account the periodicity in the summation in (3.3).