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Journal of Physics A Mathematical and General

Hamiltonian of a ID quantum chain for Belavin's zn× znmodel

Hua Wei, ZHOU YU-KUI, Bo-Yu Hou

Published Jul 7, 1989• 3 citations
Publisher

Abstract

The Hamiltonian of a 1D quantum chain corresponding to Belavin's z n × z n symmetric model is derived. This Hamiltonian is a multicomponent generalisation of the XYZ model with n 2 coupling constants in general. In the limit tau to i infinity it reduces to the generalised XXZ model with (n-1) n-fold-degenerate coupling constants and n non-degenerate ones. The Hamiltonian is Hermitian only for n=2 or for n > 2 with a crossing parameter w restricted to integers. There exists a domain of parameters in which the Boltzmann weights are positive but the corresponding Hamiltonian is non-Hermitian. The relations of the Hamiltonian to other models are discussed.

DOI
10.1088/0305-4470/22/13/008

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The Hamiltonian of a 1D quantum chain corresponding to Belavin's z n × z n symmetric model is derived. This Hamiltonian is a multicomponent generalisation of the XYZ model with n 2 coupling constants in general. In the limit tau to i infinity it reduces to the generalised XXZ model with (n-1) n-fold-degenerate coupling constants and n non-degenerate ones. The Hamiltonian is Hermitian only for n=2 or for n > 2 with a crossing parameter w restricted to integers. There exists a domain of parameters in which the Boltzmann weights are positive but the corresponding Hamiltonian is non-Hermitian. The relations of the Hamiltonian to other models are discussed.
The author considers one-dimensional n-state quantum chains having as symmetry an Abelian group A of order n (generalised Ising chain). These Hamiltonians depend on 2(n-1) coupling constants. For special values of the coupling constants the symmetry of the quantum chains is larger than A corresponding to a (in general non-Abelian) group G with m elements (m)n). For each element of G he finds explicitly a boundary condition which leaves the Hamiltonian translationally invariant. This problem is relevant in finite-size scaling studies at the critical point of a second-order phase transition.
Abstract Sutherland showed that the XYZ quantum spin-chain Hamiltonian commutes with the eight-vertex model transfer matrix, so that Baxter’s subsequent tour de force proves the integrability of both. The proof requires parametrising the Boltzmann weights using elliptic theta functions and showing they satisfy the Yang-Baxter equation. We here give a simpler derivation of the integrability of the XYZ chain by explicitly constructing an extensive sequence of conserved charges from a matrix-product operator. We show that they commute with the XYZ Hamiltonian with periodic boundary conditions or an arbitrary boundary magnetic field. A straightforward generalisation yields impurity interactions that preserve the integrability. Placing such an impurity at the edge gives an integrable generalisation of the Kondo problem with a gapped bulk. We make contact with the traditional approach by relating our matrix-product operator to products of the eight-vertex model transfer matrix.