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AIP conference proceedings

Transfer matrix spectra of affine–D models

D. Kim, K.-H. Kwon, J.-Y. Choi

Published Jan 1, 1992• 0 citations
Publisher

Abstract

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.

DOI
10.1063/1.41951

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•Jan 1, 1996•0 citations
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 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.
We consider solvable lattice models D L +2 (1) (λ) associated with the Dynkin diagrams of the affine Lie algebras D L +2 (1) and parametrized by the crossing parameter λ (0 < λ < 2π/3). The transfer matrix spectra for the strip widths ⩽ 10 are classified according to the irreducible representations of the symmetry groups under several boundary conditions. All L - and λ-dependences of scaling dimensions are embodied in the thermal scaling dimension X T = 2/{ L 2 (1 - λ/π)}. The operator content of the model is the same as that of the Ashkin-Teller model with the same thermal scaling dimension. This generalizes the results of Pasquier (λ = 0) and of the magnetic hard square model ( L = 3).
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 universal critical point ratio Q is exploited to determine positions of the critical Ising transition lines on the phase diagram of the Ashkin-Teller model on the square lattice. A leading-order expansion of the ratio Q in the presence of a nonvanishing thermal field is found from finite-size scaling and the corresponding expression is fitted by the accurate perturbative transfer-matrix data calculations for the L\ifmmode\times\else\texttimes\fi{}L square clusters with L\ensuremath{\leqslant}9.
The universal critical point ratio of the square of the second moment of the order parameter to its fourth moment, denoted as Q, is exploited to determine the position of the critical Ising transition lines in the phase diagram of the Ashkin-Teller (AT) model on the square lattice. A leading-order expansion of the Q ratio in the presence of a non-vanishing thermal field is found from finite-size scaling and the corresponding expression is fitted to the accurate perturbative transfer-matrix data for the L x L squaxe clusters with L <= 9. The same method is used to determine the critical Ising transition line on the phase diagram of the Blume-Capel (BC) model (L <= 11). We have also calculated the Q ratio for the q=3 Potts model in the critical point. The previously predicted scaling relation for this universality class is confirmed by our results for L <= 12.