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Spin--spin critical point correlation functions for the 2D random bond Ising and Potts models

Vladimir S. Dotsenko, Marco Picco, Pierre Pujol

Published May 2, 1994• 27 citations
OpenAlex

Abstract

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.

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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
Using conformal field theory techniques, we compute the disorder-averaged p -th power of the spin-spin correlation function (overline⟨σ(0)σ( R )⟩ p , p ∊ Z ) for the ferromagnetic random bond Potts model. We thus generalize the calculations of Dotsenko, Dotsenko and Picco ( hep-th/9709136 , 1997), where the case p = 2 was considered, and of Ludwig ( Nucl. Phys. B , 85 (1987) 97, 330 (1990) 639), where first-order computations where made for general p . Perturbative calculations are made up to second order in ( being proportional to the central charge deviation of the pure model from the Ising model value). The explicit dependence of the correlation function on p gives an upper bound for the validity of the -expansion, which seems to be valid, in the three-states case, only if p ⩽ 4.
We consider the critical behavior of two-dimensional Potts models in the presence of a bond disorder in which the correlation decays as a power law. In some recent work the thermal sector of this theory was investigated by a renormalization group computation based on perturbed conformal field theory. Here we apply the same approach to study instead the magnetic sector. In particular, we compute the leading corrections to the Potts spin scaling dimension. Our results include as a special case the long-range disorder Ising model. We compare our prediction to Monte Carlo simulations. Finally, by studying the magnetization scaling function, we show a clear numerical evidence of a cross-over between the long-range and the short-range class of universality.
We present results of a numerical simulation of the qqq-state random bond Potts model in two dimensions and for large qqq. In particular, care is taken to study the crossover from the pure model to the random model, as well as the crossover from the percolation to the random model. We show how to determine precisely the random fixed point and measure critical exponents at this point.