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Model factorized S-matrix and an integrable spin-1 Heisenberg chain

Al.B. Zamolodchikov, V.A. Fateev

Published Aug 1, 1980• 21 citations
OpenAlex

Abstract

A solution of the factorization equations in dimension 3 is obtained. The solution is expressed in terms of trigonometric functions and contains one free parameter. It corresponds to a factorized S-matrix with U(1) symmetry. The solution also allows one to construct a commuting family of transfer matrices for a 19-vertex lattice statistical system and the quantum Hamiltonian of an integrable anisotropic spin-1 Heisenberg chain.

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Integrable quantum spin chains

Anastasia Doikou

Integrable quantum spin chains are exactly solvable quantum mechanical models of N quantum spins, of which the Heisenberg model solved by Bethe is the prototype. In the antiferromagnetic regime, such spin chains can be regarded as lattice versions of corresponding integrable relativistic quantum field theories. For integrable spin chains, quantities of physical interest (spectrum, S matrix, etc.) can be calculated exactly by direct means, starting from the microscopic Hamiltonian, while for the corresponding field theories, such exact information has been primarily obtained by indirect means, such as the “ bootstrap ” approach and semiclassical approximations. We have been focused basically on the Heisenberg XXX and XXZ chains and their generalizations. Basic aims of this work were the study of the symmetries of these models and the computation, by means of Bethe ansatz equations, of the bulk and boundary S matrices for the closed and open chains respectively...

•Jan 1, 1999•0 citations
A solution of the factorization equations in dimension 3 is obtained. The solution is expressed in terms of trigonometric functions and contains one free parameter. It corresponds to a factorized S-matrix with U(1) symmetry. The solution also allows one to construct a commuting family of transfer matrices for a 19-vertex lattice statistical system and the quantum Hamiltonian of an integrable anisotropic spin-1 Heisenberg chain.
An elliptic generalization of the D2(2)D^{(2)}_{2}D2(2)​ Spin chain
An integrable elliptic generalization of the closed D2(2)D^{(2)}_2D2(2)​ quantum spin chain is constructed. Taking the trigonometric limit, we have shown that the corresponding R-matrix reduces to that of the D2(2)D^{(2)}_2D2(2)​ one, up to a similarity transformation. Basing on the very fact of the factorization for the R-matrix and the corresponding transfer matrix which indeed ensures the integrability of the model, we give the exact solution of the model which can be expressed in terms of a homogeneous T-Q relation.
Integrable quantum spin chains are exactly solvable quantum mechanical models of N quantum spins, of which the Heisenberg model solved by Bethe is the prototype. In the antiferromagnetic regime, such spin chains can be regarded as lattice versions of corresponding integrable relativistic quantum field theories. For integrable spin chains, quantities of physical interest (spectrum, S matrix, etc.) can be calculated exactly by direct means, starting from the microscopic Hamiltonian, while for the corresponding field theories, such exact information has been primarily obtained by indirect means, such as the “ bootstrap ” approach and semiclassical approximations. We have been focused basically on the Heisenberg XXX and XXZ chains and their generalizations. Basic aims of this work were the study of the symmetries of these models and the computation, by means of Bethe ansatz equations, of the bulk and boundary S matrices for the closed and open chains respectively...