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Communications in Theoretical Physics

Integrability of a New Type of Deformed XXZ Model

Like Cao, Ke San-Min, Rui‐Hong Yue

Published Oct 1, 2007• 0 citations
Publisher

Abstract

A new type of deformed XXZ model was constructed and diagonalized by the coordinate Bethe ansatz method. We obtained the energy and the Bethe ansatz equations of the model and also discussed some thermodynamics of the model.
DOI
10.1088/0253-6102/48/4/021

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In this paper, we review a few known facts on the coordinate Bethe ansatz. We present a detailed construction of the Bethe ansatz vector $\psi $ and energy $\Lambda $, which satisfy $V\psi =\Lambda \psi $, where VVV is the transfer matrix of the six-vertex model on a finite square lattice with periodic boundary conditions for weights a=b=1a=b=1a=b=1 and c>0c>0c>0. We also show that the same vector $\psi $ satisfies $H\psi =E\psi $, where HHH is the Hamiltonian of the XXZ model (which is the model for which the Bethe ansatz was first developed), with a value EEE computed explicitly. Variants of this approach have become central techniques for the study of exactly solvable statistical mechanics models in both the physics and mathematics communities. Our aim in this paper is to provide a pedagogically-minded exposition of this construction, aimed at a mathematical audience. It also provides the opportunity to introduce the notation and framework which will be used in a subsequent paper by the authors [5] that amounts to proving that the random-cluster model on Z2\mathbb{Z}^{2}Z2 with cluster weight q>4q>4q>4 exhibits a first-order phase transition.
Based on the inhomogeneous T – Q relation constructed via the off-diagonal Bethe Ansatz, a systematic method for retrieving the Bethe-type eigenstates of integrable models without obvious reference state is developed by employing certain orthogonal basis of the Hilbert space. With the XXZ spin torus model and the open XXX spin- 1 2 chain as examples, we show that for a given inhomogeneous T – Q relation and the associated Bethe Ansatz equations, the constructed Bethe-type eigenstate has a well-defined homogeneous limit.
Based on the inhomogeneous T–Q relation constructed via the off-diagonal Bethe Ansatz, a systematic method for retrieving the Bethe-type eigenstates of integrable models without obvious reference state is developed by employing certain orthogonal basis of the Hilbert space. With the XXZ spin torus model and the open XXX spin- chain as examples, we show that for a given inhomogeneous T–Q relation and the associated Bethe Ansatz equations, the constructed Bethe-type eigenstate has a well-defined homogeneous limit.