Trigonometric version of quantum-classical duality in integrable systems

We extend the quantum-classical duality to the trigonometric (hyperbolic) case. The duality establishes an explicit relationship between the classical N-body trigonometric Ruijsenaars-Schneider model and the inhomogeneous twisted XXZ spin chain on N sites. Similarly to the rational version, the spin chain data fixes a certain Lagrangian submanifold in the phase space of the classical integrable system. The inhomogeneity parameters are equal to the coordinates of particles while the velocities of classical particles are proportional to the eigenvalues of the spin chain Hamiltonians (residues of the properly normalized transfer matrix). In the rational version of the duality, the action variables of the Ruijsenaars-Schneider model are equal to the twist parameters with some multiplicities defined by quantum (occupation) numbers. In contrast to the rational version, in the trigonometric case there is a splitting of the spectrum of action variables (eigenvalues of the classical Lax matrix). The limit corresponding to the classical Calogero-Sutherland system and quantum trigonometric Gaudin model is also described as well as the XX limit to free fermions.


Introduction
The quantum-classical (QC) duality (correspondence) is an explicit relation between quantum and classical integrable systems of different types. This phenomenon was first observed in [7] for the classical Toda chain. A similar observation was made in [14] for the classical Calogero system and quantum Gaudin model. The classical action variables were assumed to be equal to zero. The case of arbitrary set of action variables was described in [2] using the relation of both models to the KP hierarchy [12]. In a similar way, the QC duality between the classical Ruijsenaars-Schneider (RS) model and the quantum twisted spin chain was proposed in [1,19]. The final version and a direct proof of this relation was presented in [8] via the nested Bethe anzats. Later the duality was extended to the correspondence [18]: it was shown that the RS model is related not to a single quantum model but to a family of supersymmetric spin chains. We do not discuss the supersymmetric case in this paper.
Let us briefly recall the result of [8]. Consider the Lax matrix of the classical N-body RS model 1 [16] L RS ij = νq j q i − q j + ην , i, j = 1 , ... , N (1.1) and quantum transfer matrix of the GL(n) inhomogeneous (generalized) twisted XXX spin chain on N sites 2T In the framework of the algebraic nested Bethe ansatz the spectrum H XXX where {µ a i } is any solution of the Bethe equations. Then the spectrum of the classical Lax matrix (1.1) is given by the twist parameters: (1.6) The multiplicities are defined by the quantum numbers N c . Let us also mention that the QC correspondence appeared also in the framework of gauge theory dualities [15,6,9]. Another relation between classical Lax matrices and quantum R-matrices related to spin chains can be found in [13].
The purpose of this paper is the trigonometric version of the QC duality. We prove an analogue of statement (1.6) for the trigonometric (hyperbolic) RS model and the XXZ twisted inhomogeneous spin chain. We show that in contrast to the rational version, the degeneration of the spectrum of action variables (eigenvalues of the classical Lax matrix) disappears. The identification ην = (1.7) leads to the following eigenvalues of the classical RS Lax matrix (to be compared with (1.6) for the rational case): The eigenvalues of the Lax matrix form "strings" centered at the twist parameters V a .

Trigonometric Ruijsenaars-Schneider model
In this paper we use the following Lax matrix of the trigonometric N-particle RS model: The Hamiltonian is defined as For the velocities we havė Therefore, in terms of velocities the Lax matrix has the form , i, j = 1 , ... , N. (2.5) Here ||C ij || is the trigonometric Cauchy matrix.
It is important for our purpose that the classical Lax matrix (2.1) admits the following factorization (see [10,3]): where P = diag(p 1 , ..., p N ),Ṽ is the (trigonometric) Vandermonde type matrix and The (spectral) parameter ǫ is fictitious -it does not enter the final answer. Notice that where S is the following diagonal matrix: It follows from (2.6) and (2.9) that the eigenvalues of the Lax matrix (2.1) become very simple on the Lagrangian submanifold P = 0 (i.e. p k = 0 for all k = 1, ..., N). The spectrum of (2.1) is then given by the elements of matrix S(ην): The equations of motionṗ j = − ∂H ∂q i RS of the RS model admit the Lax representatioṅ Explicitly, the equations of motion read .
Two special cases of the trigonometric RS model are ην = ±∞ and ην = iπ/2. In the former case the equations of motion simplify to ην = ±∞ :q j = 2 k =jq jqk coth q jk . (2.13) In the latter case they are: (2.14) 3 Inhomogeneous U q (ĝl n ) spin chain The algebraic structure of the Heisenberg XXZ spin chain is based on the quantum affine algebra U q (ĝl n ) [11,5] (see also [4]). The model is defined by the following quantum R-matrix: where z is the spectral parameter, is the anisotropy parameter and e ab denotes the n×n matrix with 1 in the position (a, b) and 0 otherwise.
The transfer matrix of the twisted inhomogeneous Heisenberg XXZ model on N sites is given byT where the diagonal twist matrix acts in the auxiliary n-dimensional vector space labeled by 0. We assume that the parameters q k are in general position, i.e. q j = q k and q j = q k ± for j = k. It follows from the Yang-Baxter equation for the R-matrix that the transfer matrices commute for different values of the spectral parameter: The nested Bethe ansatz gives the following result for eigenvalues of the transfer matrix (3.2): . (3.4) The integer parameters N b (N 0 = N n = 0) are the numbers of Bethe roots µ b β in the b-th group, b = 1 , ... , n−1, β = 1 , ... , N b . They satisfy the system of Bethe equations (BE): .
(3.5) where b = 2, . . . , n − 1. In the last equation it is implied that N n = 0. The BE mean that the eigenvalues (3.4) are regular at z = µ b γ . It is known that the operatorŝ commute with the transfer matrix. The eigenvectors of the latter, built from solutions to the BE, with the number of Bethe roots at level b equal to N b , are also eigenvectors of the operatorsM a with the eigenvalues The transfer matrix (3.2) can be represented as a sum over simple poles at z = q k : (it follows from (3.1) that it is an iπ-periodic function of z). The coefficientŝ where the µ 1 γ 's are taken from a solution to the BE. It is easy to see that i , j = 1 , ... , N and

2)
α, β = 1 , ... , M. For definiteness assume that N ≥ M. Then the following identity holds true:  The proof of (5.4) is performed by successive usage of the determinant identity (4.3) and BE (3.5). Consider the matrix Impose now the BE (3.5). Then we get i.e. L (1) At the next step let us define and again we use (4.3) and (3.5) to get: . . .
The process of the subsequent usage of (4.3) and (3.5) is continued until the last step when equation (3.5) is used: In order to find the characteristic polynomial of the matrix which can be easily proved or taken from [17]. As a result, we get Therefore, we have the following system of polynomial equations for spectrum of the quantum Hamiltonians: N). Here λ i ∈ Spec L are given by (5.3). Setting q i = x i , H XXZ i = H i and tending → 0, these equations become the equations of the universal spectral variety for models of the XXX type [18].
Equations (5.17) at k = N and k = 1 are easy to check without directly appealing to the determinant identity using the side by side products of the BE and the "sum rules" (3.11).
At k = N we have the equation .

(5.19)
The first double product cancels against the product of the C-factors in (5.18). The side by side products of the BE form a chain of identities that yields the right hand side of (5.18).
According to it is exactly the second "sum rule" in (3.11).
6 Limiting cases 6.1 Limit to the Gaudin-Calogero correspondence Calogero-Sutherland model. The Lax matrix of the Calogero-Sutherland model can be represented as with matrices P , V and D defined in (2.6)-(2.8) and velocitieṡ generated by the Hamiltonian H CM = 1 2 tr (L CM ) 2 . The representation (6.2) follows from (2.6) in the non-relativistic limit η → 0: It follows from (2.7) and (2.10) that Therefore, The trigonometric Gaudin model appears in the limit ε → 0 from the inhomogeneous XXZ spin chain with the transfer matrixT XXZ (z; The expansion as ε → 0, defines the commuting Gaudin Hamiltonianŝ aa . (6.10) The commutativity of the Gaudin Hamiltonians follows from commutativity of the transfer matrices, taken into account that the termT 1 (z; {q i }) is central. Their eigenvalues can be found using (3.9) and tending ε → 0. This gives where b = 1 , ... , n−1, N 0 = N n = 0, β = 1 , ... , N b . The matrix v = diag(v 1 , ... , v n ) is the twist matrix of the Gaudin model. Similarly to the (XXZ) spin chain case we use the notation H G i ({q i } N , {µ 1 α } N 1 ) for the function given by the r.h.s. of (6.11). When the set {µ 1 α } N 1 is taken from a solution of the system of BE (6.12) this function is equal to some eigenvalue of the Hamiltonian.
Quantum-classical duality between the classical Calogero-Sutherland system and the quantum Gaudin model is given by the following statement: Theorem 2 Under identification of the parameters ν = (6.16) where H G j are eigenvalues of the quantum Gaudin Hamiltonians corresponding to any common eigenstate, the spectrum of the Lax matrix (6.1) is equal to . . . , v n −(N n−1 −1) , . . . , v n +(N n−1 −1) The proof Theorem 2 is similar to Theorem 1. Similarly to the non-degenerate XXZ case, the eigenvalues of the Lax matrix form "strings" centered at the v a 's. The distance between to subsequent eigenvalues in any string is 2 .

Limit to XX model
The XXZ model has a limit → iπ/2 called the XX model. The latter is often referred to as the free-fermion model, which is due to the fact that the XX Hamiltonian may be mapped to a creation-annihilation form that corresponds to a system of non-interacting fermions on the 1D lattice. As none of the R−matrix entries vanish at = iπ/2, the eigenvalues of the transfer matrix simplify insignificantly: So do the eigenvalues of the quantum Hamiltonians: What is special about the free-fermion point is the simplification of the BE (3.5) due to collapse of one of the two products in the right hand sides caused by periodicity of the sinh-function along the imaginary axis: