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arXiv:hep-th/0503130 · Journal of Physics A Mathematical and General

A Q-operator identity for the correlation functions of the infinite XXZ spin-chain

Christian Korff

Published Jul 13, 2005• 31 citations
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Abstract

An explicit construction for Q-operators of the finite XXZ spin chain with twisted boundary conditions is presented. The massless and the massive regimes are considered as well as the root of unity case. It is explained how these results yield an alternative expression for the trace function employed in the description of the correlation functions of the inhomogeneous XXZ model on the infinite lattice by Boos, Jimbo, Miwa, Smirnov and Takeyama.

DOI
10.1088/0305-4470/38/30/002

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An explicit construction for Q-operators of the finite XXZ spin chain with twisted boundary conditions is presented. The massless and the massive regimes are considered as well as the root of unity case. It is explained how these results yield an alternative expression for the trace function employed in the description of the correlation functions of the inhomogeneous XXZ model on the infinite lattice by Boos, Jimbo, Miwa, Smirnov and Takeyama.
Taking the isotropic limit Δ → 1 in a recent representation theoretic construction of Baxter's Q -operators for the XXZ model with quasi-periodic boundary conditions we obtain new results for the XXX model. We show that quasi-periodic boundary conditions are needed to ensure convergence of the Q -operator construction and derive a quantum Wronskian relation which implies two different sets of Bethe ansatz equations, one above, the other below the 'equator' of total spin S z = 0. We discuss the limit to periodic boundary conditions at the end and explain how this construction relates to the trace functional introduced by Boos et al in the context of correlation functions on the infinite lattice. We also identify a special subclass of solutions to the quantum Wronskian and numerically verify them up to spin chains of ten sites. This special type of solutions might persist for longer chains.
The scalar products, form factors, and correlation functions of the XXZ spin chain with a twisted (or antiperiodic) boundary condition are obtained based on the inhomogeneous T–Q relation and the Bethe states constructed via the off-diagonal Bethe ansatz. It is shown that the scalar product of two off-shell Bethe states, the form factors, and the two-point correlation functions can be expressed as the summation of certain determinants. The corresponding homogeneous limits are studied. The results are also checked by the numerical calculations.
The half-infinite XXZ spin chain with a triangular boundary is considered in the massive regime. Two integral representations of correlation functions are proposed using bosonization. Sufficient conditions such that the expressions for triangular boundary conditions coincide with those for diagonal boundary conditions are identified. As an application, summation formulae of the boundary expectation values 〈σ1a〉 with a=z,± are obtained. Exploiting the spin-reversal property, relations between n-fold integrals of elliptic theta functions are extracted.
We explain how to compute correlation functions at zero temperature within the framework of the quantum version of the Separation of Variables (SoV) in the case of a simple model: the XXX Heisenberg chain of spin 1/2 with twisted (quasi-periodic) boundary conditions. We first detail all steps of our method in the case of anti-periodic boundary conditions. The model can be solved in the SoV framework by introducing inhomogeneity parameters. The action of local operators on the eigenstates are then naturally expressed in terms of multiple sums over these inhomogeneity parameters. We explain how to transform these sums over inhomogeneity parameters into multiple contour integrals. Evaluating these multiple integrals by the residues of the poles outside the integration contours, we rewrite this action as a sum involving the roots of the Baxter polynomial plus a contribution of the poles at infinity. We show that the contribution of the poles at infinity vanishes in the thermodynamic limit, and that we recover in this limit for the zero-temperature correlation functions the multiple integral representation that had been previously obtained through the study of the periodic case by Bethe Ansatz or through the study of the infinite volume model by the q-vertex operator approach. We finally show that the method can easily be generalized to the case of a more general non-diagonal twist: the corresponding weights of the different terms for the correlation functions in finite volume are then modified, but we recover in the thermodynamic limit the same multiple integral representation than in the periodic or anti-periodic case, hence proving the independence of the thermodynamic limit of the correlation functions with respect to the particular form of the boundary twist.
We provide two methods of producing the QQQ-operator of XXZ spin chain of higher spin, one for NNNth root-of-unity qqq with odd NNN and another for a general qqq, as the generalization of those known in the six-vertex model. In the root-of-unity case, we discuss the functional relations involving the constructed QQQ-operator for the symmetry study of the theory. The QQQ-operator of XXZ chain of higher spin for a generic qqq is constructed by extending Baxter's argument in spin-1/2 case for the six-vertex QQQ-operator.
We extend the recent approach of M. Jimbo, K. Miki, T. Miwa, and A. Nakayashiki to derive an integral formula for the N-point correlation functions of arbitrary local operators of the antiferromagnetic spin-1 XXZ model. For this, we realize the quantum affine symmetry algebra Uq(su(2)2)U_q(su(2)_2)Uq​(su(2)2​) of level 2 and its corresponding type I vertex operators in terms of a deformed bosonic field free of a background charge, and a deformed fermionic field. Up to GSO type projections, the Fock space is already irreducible and therefore no BRST projections are involved. This means that no screening charges with their Jackson integrals are required. Consequently, our N-point correlation functions are given in terms of usual classical integrals only, just as those derived by Jimbo et al in the case of the spin-1/2 XXZ model through the Frenkel-Jing bosonization of Uq(su(2)1)U_q(su(2)_1)Uq​(su(2)1​).