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Journal of Physics A Mathematical and General

Scaling dimensions and conformal anomaly in anisotropic lattice spin models

Man-Sun Kim, Paul A. Pearce

Published May 11, 1987• 52 citations
Publisher

Abstract

The effect of anisotropic interactions on the eigenvalue spectrum of the row-to-row transfer matrix of critical lattice spin models is investigated. It is verified that the predictions of conformal theory apply to anisotropic systems if one allows for spatial rescaling by incorporating an anisotropy factor zeta =(a y /a x ) sin theta where a y and a x are lattice spacings and theta is an angle describing the anisotropy. For exactly solvable models these anisotropy angles can be calculated analytically using corner transfer matrices. This is done for the eight-vertex model, hard hexagons and interacting hard squares and it is found that theta = pi u/ lambda , 10u/3 and 5u respectively where u is the spectral parameter and lambda is the crossing parameter. For each of these models, the amplitude of the finite-size corrections to the free energy at criticality is found to be of the form pi zeta c/6N 2 where zeta is the anisotropy factor and the central charge or conformal anomaly is given by c=1, 4/5 and 7/10 respectively. This is an analytic result for the eight-vertex model. For the hard hexagon and square models the largest eigenvalues are found accurately by numerically solving their inversion identities for various anisotropies and strip widths up to N=48. Finally, the authors argue that the anisotropy angle for magnetic hard squares and the q-state Potts models is also given by theta = pi u/ lambda , so this result is quite general.

DOI
10.1088/0305-4470/20/7/006

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Journal of Physics A Mathematical and General•Jun 7, 1989•53 citations
The effect of anisotropic interactions on the eigenvalue spectrum of the row-to-row transfer matrix of critical lattice spin models is investigated. It is verified that the predictions of conformal theory apply to anisotropic systems if one allows for spatial rescaling by incorporating an anisotropy factor zeta =(a y /a x ) sin theta where a y and a x are lattice spacings and theta is an angle describing the anisotropy. For exactly solvable models these anisotropy angles can be calculated analytically using corner transfer matrices. This is done for the eight-vertex model, hard hexagons and interacting hard squares and it is found that theta = pi u/ lambda , 10u/3 and 5u respectively where u is the spectral parameter and lambda is the crossing parameter. For each of these models, the amplitude of the finite-size corrections to the free energy at criticality is found to be of the form pi zeta c/6N 2 where zeta is the anisotropy factor and the central charge or conformal anomaly is given by c=1, 4/5 and 7/10 respectively. This is an analytic result for the eight-vertex model. For the hard hexagon and square models the largest eigenvalues are found accurately by numerically solving their inversion identities for various anisotropies and strip widths up to N=48. Finally, the authors argue that the anisotropy angle for magnetic hard squares and the q-state Potts models is also given by theta = pi u/ lambda , so this result is quite general.
The commuting row-to-row transfer matrices of magnetic hard squares on the multicritical T manifold are shown to satisfy a special functional equation called an inversion identity.For strip widths up to N=32, these equations are solved numerically for the transfer matrix eigenvalues. The central charge or conformal anomaly c=1 is obtained from 1/N 2 corrections to the exact bulk free energy and scaling dimensions are obtained from the eigenvalue gaps. The magnetic scaling dimension is x m =1/8 and the sublattice density difference scaling dimension is x e =1/9(2-y), where y=2 lambda / pi and the interaction-dependent crossing parameter lambda varies between 0 and 2 pi /3. Scaling dimensions for further operators fall in the sequence x n =n 2 x e where n=1,2,3,4. The critical exponents also vary continuously along the multicritical line and are given by alpha =(14-9y)/(16-9y), beta m =9(2-y)/16(16-9y), beta e =1/2(16-9y), nu =9(2-y)/2(16-9y) and delta =15. These exponents are simply related to the exponents of the critical eight-vertex and Ashkin-Teller models, which also exhibit Z 2 *Z 2 symmetry, and the authors argue that these models lie in the same universality class.
Determinantal functional equations satisfied by the row transfer matrix eigenvalues of critical A–D–E lattice spin models are presented. These are obtained for models associated with the Lie algebras [Formula: see text], [Formula: see text], AL, DL and E6,7,8 by exploiting connections with functional equations satisfied by the row transfer matrix eigenvalues of the six-vertex model at rational values of the crossing parameter λ=sπ/h where h is the Coxeter number. In addition, fusion is used to derive special functional equations, called inversion identity hierarchies, which provide the key to the direct calculation of finite-size corrections, central charges and conformal weights for the critical A–D–E lattice models.
A general analytic method for calculating finite-size corrections, central charges, and scaling dimensions of solvable lattice models is presented. The approach is to solve the special functional equations or inversion identities satisfied by the commuting row transfer matrices of these lattice models at criticality. For purposes of illustration, the method is applied to calculate the central charge c=4/5 and leading magnetic scaling dimension x=2/15 of hard hexagons. These numbers are rational due to special values of Rogers dilogarithms.
The eigenspectrum of the critical anisotropic Heisenberg model (or XXZ model), with arbitrary spin S in its disordered-ferroelectric regime (0\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}\ensuremath{\pi}/2S) is solved by the Bethe-Ansatz method. The amplitudes of the leading finite-size corrections are calculated analytically and numerically. Using conformal invariance we give exact results for the conformal anomaly and scaling dimensions. Our results indicate that, for all spin S, the critical behavior is governed by a conformal field theory with central charge c=1.
Transfer matrix spectra can be used to establish a connection between two dimensional critical lattice models and conformal field theories. Affine–D models are a class of spin models with (L+3) state per site (L=3, 4, ...) generalizing the multicritical line of the magnetic hard square model. There are two continuous parameters characterizing the models; crossing parameter λ with which critical exponents vary continuously and spectral parameter u which controls anisotropy of the interactions. Their transfer matrix spectra combined with finite‐size scaling theory show that the models in the positive u regime belong to the universality class of the Ashkin–Teller model and are related to the c=1 conformal field theory by r={2L2(1−λ/π)}−1/2, where r is the orbifold radius. The models in the negative u regime are found to be related to those in the positive u regime by a certain transformation and also belong to the Ashkin–Teller universality class with the correspondence r={2L2λ/π}−1/2.
The authors are concerned with the critical properties of the anisotropic Heisenberg chain, or XXZ model, with arbitrary integer or half-integer spin. The eigenspectra of these Hamiltonians, with periodic boundaries, are calculated for finite chains by solving numerically their associated Bethe ansatz equations. The resulting spectra are found to be in accord with the predictions of conformal invariance and the operator content is identified, for lattices with an even and odd number of sites. The results for spin 1 and spin 3/2 indicate that the conformal anomaly c for these models, in the gapless regime, has the value c=3S/(1+S), independent of the anisotropy, and the exponents vary continuously with the anisotropy as in the eight-vertex model. The operator content of these models indicate that the underlying field theory governing these critical spin-S models is described by composite fields formed by the product of Gaussian and Z(N) fields, with N=2S. Finally some of the irrelevant operators which produce the leading finite-size corrections of the eigenenergies are identified.