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arXiv:cond-mat/9607201 · Physical review. B, Condensed matter

Critical behavior of weakly disordered anisotropic systems in two dimensions

Giancarlo Jug, B. N. Shalaev

Published Aug 1, 1996• 28 citations

Abstract

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.

DOI: 10.1103/physrevb.54.3442

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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.

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.

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.

The Ashkin-Teller model can be formulated as a pair of 2D Ising models, interacting via a four-spin interaction. I consider the case of weak anisotropy (slight a-symmetry between the two Ising layers) and weak coupling. I show that the system admits two critical temperatures whose difference varies continuously with the strength of the coupling, scaling with an anomalous exponent as one let the a-symmetry parameter go to zero. The specific heat diverges logarithmically at the critical points (as for Ising) but the constant in front of the logarithm is renormalized by an anomalous critical exponent. The logarithmic divergence of the specific heat dominates only in an exponentially small interval around the critical temperatures and outside it is modified into an anomalous power law behaviour. The proof is based on an exact mapping of Ashkin-Teller into a model of (1+1)D interacting fermions and on the implementation of constructive Renormalization group methods on the fermionic system. This PhD thesis includes: (1) a review of the exact solution of 2D Ising in terms of Grassmann functional integrals; (2) a review of the Grassmann representation for a class of interacting Ising models, including Ashkin-Teller and the 8 vertex model; (3) a detailed discussion of the multiscale analysis of the Ashkin-Teller model, based on fermionic Renormalization Group methods, including the study of the flow of the effective coupling constants (in particular a proof of vanishing of the beta function for this and similar Luttinger-like models, based on modified approximate Ward identities, is included). This thesis is based on joint work with V. Mastropietro.

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 study the effects of quenched disorder on the first-order phase transition in the two-dimensional three-color Ashkin-Teller model by means of large-scale Monte Carlo simulations. We demonstrate that the first-order phase transition is rounded by the disorder and turns into a continuous one. Using a careful finite-size-scaling analysis, we provide strong evidence for the emerging critical behavior of the disordered Ashkin-Teller model to be in the clean two-dimensional Ising universality class, accompanied by universal logarithmic corrections. This agrees with perturbative renormalization-group predictions by Cardy. As a byproduct, we also provide support for the strong-universality scenario for the critical behavior of the two-dimensional disordered Ising model. We discuss consequences of our results for the classification of disordered phase transitions as well as generalizations to other systems.