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arXiv:hep-th/9501017 · Nuclear Physics B

Renormalisation-group calculation of correlation functions for the 2D random bond Ising and Potts models

Vladimir S. Dotsenko, Marco Picco, Pierre Pujol

Published Sep 1, 1995• 115 citations
arXivHTMLPDF
DOI
10.1016/0550-3213(95)00534-y

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We compute the combined two and three loop order correction to the spin-spin correlation functions for the 2D Ising and q-states Potts model with random bonds at the critical point. The procedure employed is the renormalisation group approach for the perturbation series around the conformal field theories representing the pure models. We obtain corrections for the correlations functions which produce crossover in the amplitude but don't change the critical exponent in the case of the Ising model and which produce a shift in the critical exponent, due to randomness, in the case of the Potts model. Comparison with numerical data is discussed briefly.
The two-dimensional random-bond Q-state Potts model is studied for Q near 2 via the perturbative renormalisation group to one loop. It is shown that weak disorder induces cross-correlations between the quenched-averages of moments of the two-point spin/spin and energy/energy correlation functions, which should be observable numerically in specific linear combinations of various quenched correlation functions. The random-bond Ising model in (2 + ǫ) dimensions is similarly treated. As a byproduct, a simple method for deriving the scaling dimensions of all moments of the local energy operator is presented. 1
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.