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arXiv:cond-mat/9912304 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Multiscaling of energy correlations in the random-bond Potts model

Jesper Lykke Jacobsen

Published Jun 1, 2000• 15 citations
arXivHTMLPDF

Abstract

We numerically calculate the exponents governing the disorder averaged and fixed-sample decay of the energy-energy correlator in the q-state random-bond Potts model. Our results are in good agreement with a two-loop expansion around q=2 recently found from perturbative renormalization group techniques, fulfill the correlation length bound nu>/=2/d, and give further evidence against replica symmetry breaking in this class of models.

DOI
10.1103/physreve.61.r6060

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•Jan 1, 2000•9 citations
We numerically calculate the exponents governing the disorder averaged and fixed-sample decay of the energy-energy correlator in the q-state random-bond Potts model. Our results are in good agreement with a two-loop expansion around q=2 recently found from perturbative renormalization group techniques, fulfill the correlation length bound nu>/=2/d, and give further evidence against replica symmetry breaking in this class of models.
In this paper we provide new analytic results on two-dimensional q q -Potts models ( q ≥ 2 q ≥ 2 ) in the presence of bond disorder correlations which decay algebraically with distance with exponent a a . In particular, our results are valid for the long-range bond disordered Ising model ( q=2 q = 2 ). We implement a renormalization group perturbative approach based on conformal perturbation theory. We extend to the long-range case the RG scheme used in [V. Dotsenko et al., Nucl. Phys. B 455 701-23] for the short-range disorder. Our approach is based on a 2 2 -loop order double expansion in the positive parameters (2-a) ( 2 − a ) and (q-2) ( q − 2 ) . We will show that the Weinrib-Halperin conjecture for the long-range thermal exponent can be violated for a non-Gaussian disorder. We compute the central charges of the long-range fixed points finding a very good agreement with numerical measurements.
In this letter we present supporting evidence for the replica symmetric approach to the random bond q -state Potts models. The evidence is statistically strong enough to reject the applicability of the Parisi replica symmetry breaking scheme to this class of models. The test we use is a generalization of one formerly proposed by Dotsenko et al. ( Nucl. Phys. B , 520 (1998) 633) and consists in measuring scaling laws of disordered-averaged moments of the spin-spin correlation functions. Numerical results, obtained via Monte Carlo simulations for several values of q , are shown to be in fair agreement with the replica symmetric values computed by using perturbative CFT (Dotsenko Vik. S. et al. , Nucl. Phys. B , 520 (1998) 633; Lewis M.-A., Europhys. Lett. , 43 (1998) 189) for the second and third moments of the q = 3 model. RSB effects, which should increase in strength with moment, are unobserved.
We introduce an exact replica method for the study of critical systems with quenched bond randomness in two dimensions. For the q-state Potts model, we show that a line of renormalization group fixed points interpolates from weak to strong randomness as q-2 grows from small to large values. This theory exhibits a q-independent sector, and allows at the same time for a correlation length exponent which keeps the Ising value and continuously varying magnetization exponent and effective central charge. These findings appear to solve long-standing numerical and theoretical puzzles, and to illustrate the peculiarities which may characterize the conformal field theories of random fixed points.
We study two-dimensional q -state random-bond Potts models for both q=8 and q=5 with a linearly varying temperature. By applying a successive Monte Carlo renormalization group procedure, both the static and dynamic critical exponents are obtained for randomness amplitudes (the strong to weak coupling ratio) of r_{0}=3 , 10, 15, and 20. The correlation length exponent nu increases with disorder from less than to larger than unity and this variation is justified by the good collapse of the specific heat near the critical region. The specific heat exponent is obtained by the usual hyperscaling relation alpha=2-dnu and thus indicates no possibility of the activated dynamic scaling. Both r_{0} and q have effects on the critical dynamics of the disordered systems, which can be seen from variations of the rate exponent, the hysteresis exponent, and the dynamic critical exponent. Implications of these results are discussed.
We study the critical behavior of the random q-state Potts quantum chain by density-matrix renormalization techniques. Critical exponents are calculated by scaling analysis of finite lattice data of short chains (L< 16)(L<~16)(L< 16) averaging over all possible realizations of disorder configurations chosen according to a binary distribution. Our numerical results show that the critical properties of the model are independent of q in agreement with a renormalization group analysis of Senthil and Majumdar [Phys. Rev. Lett. 76, 3001 (1996)]. We show how an accurate analysis of moments of the distribution of magnetizations allows a precise determination of critical exponents, circumventing some problems related to binary disorder. Multiscaling properties of the model and dynamical correlation functions are also investigated.
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