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Journal of Physics A Mathematical and General

Critical properties of the Izergin-Korepin and solvable O(n) models and their related quantum spin chains

S. Ole Warnaar, Murray T. Batchelor, Bernard Nienhuis

Published Jun 7, 1992• 85 citations
Publisher

Abstract

The central charge of the Izergin-Korepin model the corresponding quantum spin chain, and the O (n) model is calculated analytically via the Bethe ansatz. The calculation extends a technique recently developed for the Zamolodchikov-Fateev model. In addition critical exponents and the central charge for these models are obtained from numerical solutions of Bethe ansatz equations for finite systems. As a physical application the authors find the exponents v=12/23 and gamma =53/46 for the Theta -transition of polymers in two dimensions.

DOI
10.1088/0305-4470/25/11/016

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The central charge of the Izergin-Korepin model the corresponding quantum spin chain, and the O (n) model is calculated analytically via the Bethe ansatz. The calculation extends a technique recently developed for the Zamolodchikov-Fateev model. In addition critical exponents and the central charge for these models are obtained from numerical solutions of Bethe ansatz equations for finite systems. As a physical application the authors find the exponents v=12/23 and gamma =53/46 for the Theta -transition of polymers in two dimensions.
The sector of zero ZN charge is studied for the ferromagnetic (FM) and antiferromagnetic (AFM) version of the (ZN×Z2)-invariant Fateev-Zamolodchikov quantum spin chain. We conjecture that the relevant Bethe ansatz equations should admit, besides the usual stringlike solutions, exceptional multiplets, and a number of nonphysical solutions. Once the physical ones are identified, we show how completeness and the gapless excitation spectrum can be obtained. The central charge is computed from the specific heat and found to be [Formula: see text] (FM) and c=1 (AFM).
The eigenenergies for the spin SSS generalization of the critical XXZ\mathrm{XXZ}XXZ model are calculated by solving numerically, for finite chains, their associated Bethe-Ansatz equations. The conformal anomaly and scaling dimensions are obtained from the conformal invariance of the infinite chain. Our results reveal that these models are governed by a conformal field theory with central charge c=3S(S+1)c=\frac{3S}{(S+1)}c=(S+1)3S​ and having continuously varying critical exponents, like the eight-vertex model. We also show that they are related to a large class of unitary conformal field theories.
The Bethe-Ansatz approach is used to solve a nineteen-vertex model equivalent to a spin model with O(n) symmetry on the square lattice. The bulk free energy is determined exactly on four branches of critical points. On two of these branches, the central charge is derived from finite-size corrections. On the two other branches, the nature of the Bethe Ansatz solutions prevents us from computing the central charge analytically. In this case, our approach still allows the exact numerical determination of the free energy on strips of hundreds of sites wide.
This paper is concerned with the critical properties of a family of self-dual two-dimensional Z(N) models whose bulk free energy is exactly known at the self-dual point. The author's analysis is performed by studying the finite-size behaviour of the corresponding one-dimensional quantum Hamiltonians which also possess an exact solution at their self-dual point. By exploring finite-size scaling ideas and the conformal invariance of the critical infinite system the author calculates, for N up to 8, the critical temperature and critical exponents as well as the central charge associated with the underlying conformal algebra. The results strongly suggest that the Z(N) quantum field theory of Zamolodchikov and Fateev is the underlying field theory associated with these statistical mechanical systems. The author also tests for the Z(5) case, the conjecture that these models correspond to the bifurcation points in the phase diagram of the General Z(N) spin model, where a massless phase originates.