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EXACT SOLUTION OF A VERTEX MODEL WITH UNLIMITED NUMBER OF STATES PER BOND

F. C. Alcaraz, R. Z. Bariev

Published Jan 1, 1999• 2 citations
OpenAlex

Abstract

The exact solution is obtained for the eigenvalues and eigenvectors of the row-to-row transfer matrix of a two-dimensional vertex model with unlimited number of states per bond. This model is a classical counterpart of a quantum spin chain with an unlimited value of spin. This quantum chain is studied using general predictions of conformal field theory. The long-distance behaviour of some ground-state correlation functions is derived from a finite-size analysis of the gapless excitations.

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Exact solution of a vertex model with unlimited number of states per bond

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The exact solution is obtained for the eigenvalues and eigenvectors of the row-to-row transfer matrix of a two-dimensional vertex model with an unlimited number of states per bond. This model is a classical counterpart of a quantum spin chain with an unlimited value of spin. This quantum chain is studied using general predictions of conformal field theory. The long-distance behaviour of some ground state correlation functions is derived from a finite-size analysis of the gapless excitations.

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The exact solution is obtained for the eigenvalues and eigenvectors of the row-to-row transfer matrix of a two-dimensional vertex model with unlimited number of states per bond. This model is a classical counterpart of a quantum spin chain with an unlimited value of spin. This quantum chain is studied using general predictions of conformal field theory. The long-distance behaviour of some ground-state correlation functions is derived from a finite-size analysis of the gapless excitations.
We introduce and solve a special family of integrable interacting vertex models that generalizes the well known six-vertex model. In addition to the usual nearest neighbour interactions among the vertices, there exist extra hard-core interactions among pairs of vertices at larger distances. The associated row-to-row transfer matrices are diagonalized by using the recently introduced matrix product ansatz. Like for the relation of the six-vertex model with the XXZ quantum chain, the row-to-row transfer matrices of these new models are also the generating functions of an infinite set of commuting conserved charges. Among these charges we identify the integrable generalization of the XXZ chain that contains hard-core exclusion interactions among the spins. These quantum chains have already appeared in the literature. The present paper explains their integrability.
The exact solution of the A (2) n integrable lattice models is presented. The free energy and excitation energies of the vertex models are found as functions of the spectral and anisotropy parameters as well as the finite-size corrections yielding the central charges and conformal dimensions. These lattice models yield a new massive QFT where the mass spectrum and the S-matrix are obtained through the light-cone approach. The ground state and excitation energies of a new anisotropic spin-one integrable chain associated to the A (2) 2 model are computed.
Abstract This chapter considers the special case of the six-vertex model on a square lattice using a trigonometric parameterization of the vertex weights. It demonstrates how, by exploiting the Yang-Baxter relations, the six-vertex model is diagonalized and the Bethe ansatz equations are derived. The Hamiltonian of the Heisenberg quantum spin chain is obtained from the transfer matrix for a special value of the spectral parameter together with an infinite set of further conserved quantum operators. By the diagonalization of the transfer matrix the exact solution of the one-dimensional quantum spin chain Hamiltonian has automatically also been obtained, which is given by the same Bethe ansatz equations.
We generalize the matrix product states method using the chiral vertex operators of conformal field theory and apply it to study the ground states of the XXZ spin chain, the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model and random Heisenberg models. We compute the overlap with the exact wave functions, spin-spin correlators, and the Renyi entropy, showing that critical systems can be described by this method. For rotational invariant ansatzs we construct an inhomogenous extension of the Haldane-Shastry model with long-range exchange interactions.