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International Journal of Modern Physics B

Superuniversal Critical Behavior in Two-Dimensional Random Anisotropic Models with Discrete Symmetries

Giancarlo Jug, B. N. Shalaev

Published May 30, 1998• 0 citations
Publisher

Abstract

We study the critical behavior of two-dimensional (2D) anisotropic systems with weak quenched disorder described by the Ising model (IM) with random bonds, the dilute N-color Ashkin–Teller model (ATM) and some its generalizations. It is shown that all these models exhibit the same critical behavior as that of the 2D-IM apart from some logarithmic corrections. The minimal conformal field theory (CFT) models with randomness are found to be described by critical exponents which are numerically very close to those of the pure 2D-IM.

DOI
10.1142/s0217979298000727

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We study the critical behavior of two-dimensional (2D) anisotropic systems with weak quenched disorder described by the Ising model (IM) with random bonds, the dilute N-color Ashkin–Teller model (ATM) and some its generalizations. It is shown that all these models exhibit the same critical behavior as that of the 2D-IM apart from some logarithmic corrections. The minimal conformal field theory (CFT) models with randomness are found to be described by critical exponents which are numerically very close to those of the pure 2D-IM.
The critical behavior of 2D anisotropic systems with weak quenched disorder described by the so{called generalized Ashkin{Teller model (GATM), including the Ising model with random bonds, the dilute Baxter model, the impure N{color Ashkin{Teller model, and minimal conformal eld theory models (MCFTM) with c < 1 (c is the central charge) perturbed by randomness is discussed.All these models except MCFTM were found to belong to the Ising model universality class.Critical exponents of disordered MCFTM are calculated within perturbative expansions in the powers of = c 1=2. RG ows exhibit the rounding of uctuation{driven rst{order phase transitions by disorder.
The critical behavior of two-dimensional (2D) anisotropic systems with weak quenched disorder described by the so-called generalized Ashkin-Teller model (GATM) is studied. In the critical region this model is shown to be described by a multifermion field theory similar to the Gross-Neveu model with a few independent quartic coupling constants. Renormalization group calculations are used to obtain the temperature dependence near the critical point of some thermodynamic quantities and the large-distance behavior of the two-spin correlation function. The equation of state at criticality is also obtained in this framework. We find that random models described by the GATM belong to the same universality class as that of the two-dimensional Ising model. The critical exponent \ensuremath{\nu} of the correlation length for the three- and four-state random-bond Potts models is also calculated in a three-loop approximation. We show that this exponent is given by an apparently convergent series in \ensuremath{\epsilon}=c-1/2 (with c the central charge of the Potts model) and that the numerical values of \ensuremath{\nu} are very close to that of the 2D Ising model. This work therefore supports the conjecture (valid only approximately for the three- and four-state Potts models) of a superuniversality for the 2D disordered models with discrete symmetries. \textcopyright{} 1996 The American Physical Society.
Abstract The N -color Ashkin–Teller model corresponds to N Ising models coupled by four-spin interactions. We consider the two-dimensional case in presence of quenched disorder and use scale invariant scattering theory to determine all the solutions of the exact renormalization group fixed points equations. The weak disorder sector is characterized by a solution that, for any fixed N larger than 1, is a line of fixed points with Ising thermal exponents and continuously varying magnetic exponents. The number of fixed point solutions allowed by the symmetries of the model increases at strong disorder illustrating the growing dependence on the distributions of the two random couplings. The presence of some critical exponents which do not depend on the symmetry parameter N confirms this type of superuniversality as a peculiar feature of random criticality.
We study the critical behavior of the 2D N-color Ashkin-Teller model in the presence of random bond disorder whose correlations decays with the distance r as a power-law r -a .We consider the case when the spins of different colors sitting at the same site are coupled by the same bond and map this problem onto the 2D system of N/2 flavors of interacting Dirac fermions in the presence of correlated disorder.Using renormalization group we show that for N = 2, a "weakly universal" scaling behavior at the continuous transition becomes universal with new critical exponents.For N > 2, the first-order phase transition is rounded by the correlated disorder and turns into a continuous one.
We consider the two-dimensional randomly site diluted Ising model and the random-bond +/-J Ising model (also called the Edwards-Anderson model), and study their critical behavior at the paramagnetic-ferromagnetic transition. The critical behavior of thermodynamic quantities can be derived from a set of renormalization-group equations, in which disorder is a marginally irrelevant perturbation at the two-dimensional Ising fixed point. We discuss their solutions, focusing in particular on the universality of the logarithmic corrections arising from the presence of disorder. Then, we present a finite-size scaling analysis of high-statistics Monte Carlo simulations. The numerical results confirm the renormalization-group predictions, and in particular the universality of the logarithmic corrections to the Ising behavior due to quenched dilution.
We investigate and contrast, via entropic sampling based on the Wang–Landau algorithm, the effects of quenched bond randomness on the critical behavior of two Ising spin models in 2D. The random bond version of the superantiferromagnetic (SAF) square model with nearest- and next-nearest-neighbor competing interactions and the corresponding version of the simple Ising model are studied, and their general universality aspects are inspected by means of a detailed finite size scaling (FSS) analysis. We find that the random bond SAF model obeys weak universality, hyperscaling, and exhibits a strong saturating behavior of the specific heat due to the competing nature of interactions. On the other hand, for the random Ising model we encounter some difficulties as regards a definite discrimination between the two well-known scenarios of the logarithmic corrections versus the weak universality. However, a careful FSS analysis of our data favors the field theoretically predicted logarithmic corrections.