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arXiv:math-ph/0511022 · Journal of Physics A Mathematical and General

AQ-operator for the twisted XXX model

Christian Korff

Published Mar 15, 2006• 25 citations

Abstract

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.

DOI: 10.1088/0305-4470/39/13/002

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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.

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.

A continuous family of quasilocal exact conservation laws is constructed in the anisotropic Heisenberg (XXZ) spin-1/2 chain for periodic (or twisted) boundary conditions and for a set of commensurate anisotropies densely covering the entire easy plane interaction regime. All local conserved operators follow from the standard (Hermitian) transfer operator in fundamental representation (with auxiliary spin s=1/2), and are all even with respect to a spin flip operation. However, the quasilocal family is generated by differentiation of a non-Hermitian highest weight transfer operator with respect to a complex auxiliary spin representation parameter s and includes also operators of odd parity. For a finite chain with open boundaries the time derivatives of quasilocal operators are not strictly vanishing but result in operators localized near the boundaries of the chain. We show that a simple modification of the non-Hermitian transfer operator results in exactly conserved, but still quasilocal operators for periodic or generally twisted boundary conditions. As an application, we demonstrate that implementing the new exactly conserved operator family for estimating the high-temperature spin Drude weight results, in the thermodynamic limit, in exactly the same lower bound as for almost conserved family and open boundaries. Under the assumption that the bound is saturating (suggested by agreement with previous thermodynamic Bethe ansatz calculations) we propose a simple explicit construction of infinite time averages of local operators such as the spin current.

Baxter's Q-operator for the quantum transfer matrix of the XXZ spin-chain is constructed employing the representation theory of quantum groups. The spectrum of this Q-operator is discussed and novel functional relations which describe the finite temperature regime of the XXZ spin-chain are derived. For non-vanishing magnetic field the previously known Bethe ansatz equations can be replaced by a system of quadratic equations which is an important advantage for numerical studies. For vanishing magnetic field and rational coupling values it is argued that the quantum transfer matrix exhibits a loop algebra symmetry closely related to the one of the classical six-vertex transfer matrix at roots of unity.