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Journal of Physics A Mathematical and Theoretical

A τ-function on the inhomogeneous lattice and the classical isotropic Heisenberg spin chain

Atsushi Mukaihira

Published Oct 20, 2008• 4 citations
Publisher

Abstract

A τ-function on the inhomogeneous lattice is introduced by a deformation of that of the Toda lattice hierarchy. An associated discrete-time integrable system is constructed in Sato and Lax formalisms. It is shown that a one-dimensional reduction of the system gives a Bäcklund transformation of the classical isotropic Heisenberg spin chain, which is equivalent to the R II -chain proposed by Spiridonov and Zhedanov.

DOI
10.1088/1751-8113/41/47/475201

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Abstract For an arbitrary generalized quantum integrable spin chain we introduce a “masterT-operator” which represents a generating function for commuting quantum transfer matrices constructed by means of the fusion procedure in the auxiliary space. We show that the functional relations for the transfer matrices are equivalent to an infinite set of model-independent bilinear equations of the Hirota form for the masterT-operator, which allows one to identify it withτ-function of an integrable hierarchy of classical soliton equations. In this paper we consider spin chains with rationalGL(N)-invariantR-matrices but the result is independent of a particular functional form of the transfer matrices and directly applies to quantum integrable models with more general (trigonometric and elliptic)R-matrices and to supersymmetric spin chains.
Exploiting the quantum integrability condition we construct an ancestor model associated with a new underlying quadratic algebra. This ancestor model represents an exactly integrable quantum lattice inhomogeneous anisotropic model and at its various realizations and limits can generate a wide range of integrable models. They cover quantum lattice as well as field models associated with the quantum RRR-matrix of trigonometric type or at the undeformed qto1q \\to 1qto1 limit similar models belonging to the rational class. The classical limit likewise yields the corresponding classical discrete and field models. Thus along with the generation of known integrable models in a unifying way a new class of inhomogeneous models including variable mass sine-Gordon model, inhomogeneous Toda chain, impure spin chains etc. are constructed.