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arXiv:cond-mat/0606312 · Journal of Statistical Mechanics Theory and Experiment

Strong disorder fixed points in the two-dimensional random-bond Ising model

Marco Picco, A. Honecker, Pierre Pujol

Published Sep 13, 2006• 50 citations
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Abstract

The random-bond Ising model on the square lattice has several disordered critical points, depending on the probability distribution of the bonds. There are a finite-temperature multicritical point, called the Nishimori point, and a zero-temperature fixed point, for both a binary distribution where the coupling constants take the values ± J and a Gaussian disorder distribution. Inclusion of dilution in the ± J distribution ( J = 0 for some bonds) gives rise to another zero-temperature fixed point which can be identified with percolation in the non-frustrated case ( J ≥ 0). We study these fixed points using numerical (transfer matrix) methods. We determine the location, critical exponents, and central charge of the different fixed points and study the spin–spin correlation functions. Our main findings are the following. (1) We confirm that the Nishimori point is universal with respect to the type of disorder, i.e. we obtain the same central charge and critical exponents for the ± J and Gaussian distributions of disorder. (2) The Nishimori point, the zero-temperature fixed point for the ± J and Gaussian distributions of disorder, and the percolation point in the diluted case all belong to mutually distinct universality classes. (3) The paramagnetic phase is re-entrant below the Nishimori point, i.e. the zero-temperature fixed points are not located exactly below the Nishimori point, either for the ± J distribution, or for the Gaussian distribution.

DOI
10.1088/1742-5468/2006/09/p09006

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The random-bond Ising model on the square lattice has several disordered critical points, depending on the probability distribution of the bonds. There are a finite-temperature multicritical point, called the Nishimori point, and a zero-temperature fixed point, for both a binary distribution where the coupling constants take the values ± J and a Gaussian disorder distribution. Inclusion of dilution in the ± J distribution ( J = 0 for some bonds) gives rise to another zero-temperature fixed point which can be identified with percolation in the non-frustrated case ( J ≥ 0). We study these fixed points using numerical (transfer matrix) methods. We determine the location, critical exponents, and central charge of the different fixed points and study the spin–spin correlation functions. Our main findings are the following. (1) We confirm that the Nishimori point is universal with respect to the type of disorder, i.e. we obtain the same central charge and critical exponents for the ± J and Gaussian distributions of disorder. (2) The Nishimori point, the zero-temperature fixed point for the ± J and Gaussian distributions of disorder, and the percolation point in the diluted case all belong to mutually distinct universality classes. (3) The paramagnetic phase is re-entrant below the Nishimori point, i.e. the zero-temperature fixed points are not located exactly below the Nishimori point, either for the ± J distribution, or for the Gaussian distribution.
We study the universality class of the Nishimori point in the 2D $\ifmmode\pm\else\textpm\fi{}J$ random-bond Ising model by means of the numerical transfer-matrix method. Using the domain-wall free energy, we locate the position of the fixed point along the Nishimori line at the critical concentration value ${p}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.1094\ifmmode\pm\else\textpm\fi{}0.0002$ and estimate $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.33\ifmmode\pm\else\textpm\fi{}0.03$. Then, we obtain the exponents for the moments of the spin-spin correlation functions as well as the value for the central charge $c\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.464\ifmmode\pm\else\textpm\fi{}0.004$. The main qualitative result is the fact that percolation is now excluded as a candidate for describing the universality class of this fixed point.
We consider the random-bond ±J Ising model on a square lattice as a function of the temperature T and of the disorder parameter p (p=1 corresponds to the pure Ising model). We investigate the critical behavior along the paramagnetic-ferromagnetic transition line at low temperatures, below the temperature of the multicritical Nishimori point at T *=0.9527(1), p *=0.89083(3). We present finite-size scaling analyses of Monte Carlo results at two temperature values, T≈0.645 and T=0.5. The results show that the paramagnetic-ferromagnetic transition line is reentrant for T<T *, that the transitions are continuous and controlled by a strong-disorder fixed point with critical exponents ν=1.50(4), η=0.128(8), and β=0.095(5). This fixed point is definitely different from the Ising fixed point controlling the paramagnetic-ferromagnetic transitions for T>T *. Our results for the critical exponents are consistent with the hyperscaling relation 2β/ν−η=d−2=0.
The two-dimensional q-state Potts model is subjected to a Z(q) symmetric disorder that allows for the existence of a Nishimori line. At q=2, this model coincides with the +/- J random-bond Ising model. For q>2, apart from the usual pure- and zero-temperature fixed points, the ferro/paramagnetic phase boundary is controlled by two critical fixed points: a weak disorder point, whose universality class is that of the ferromagnetic bond-disordered Potts model, and a strong disorder point which generalizes the usual Nishimori point. We numerically study the case q=3, tracing out the phase diagram and precisely determining the critical exponents. The universality class of the Nishimori point is inconsistent with percolation on Potts clusters.
We numerically study the phase diagram and critical properties of the two-dimensional disordered O(n) loop model by using the transfer matrix and the worm Monte Carlo methods. The renormalization group flow is extracted from the landscape of the effective central charge obtained by the transfer matrix method based on the Zamolodchikov's C-theorem. We find a line of random fixed points (FPs) for nc<n<1n_c < n <1nc​<n<1, with nc∼0.5n_c \sim 0.5nc​∼0.5, for which the central charge and critical exponents agree well with the results of the 1−n1-n1−n perturbative expansion. Furthermore, for n>ncn> n_cn>nc​, we find a line of multicritical FPs at strong randomness. The FP at n=1n=1n=1 has c=0.4612(4)c=0.4612(4)c=0.4612(4), which suggests that it belongs to the universality class of the Nishimori point in the ±J\pm J±J random-bond Ising model. For n>2n>2n>2, we find another critical line that connects the hard-hexagon FP in the pure model to a finite-randomness zero-temperature FP.
We consider a classical random-bond Ising model (RBIM) with binary distribution of $\ifmmode\pm\else\textpm\fi{}K$ bonds on the square lattice at finite temperature. In the phase diagram of this model there is the so-called Nishimori line which intersects the phase boundary at a multicritical point. It is known that the correlation functions obey many exact identities on this line. We use a supersymmetry method to treat the disorder. In this approach the transfer matrices of the model on the Nishimori line have an enhanced supersymmetry osp(2n+1∣2n),\mathrm{osp}(2n+1|2n),osp(2n+1∣2n), in contrast to the rest of the phase diagram, where the symmetry is osp(2n∣2n)\mathrm{osp}(2n|2n)osp(2n∣2n) (where n is an arbitrary positive integer). An anisotropic limit of the model leads to a one-dimensional quantum Hamiltonian describing a chain of interacting superspins, which are irreducible representations of the osp(2n+1∣2n)\mathrm{osp}(2n+1|2n)osp(2n+1∣2n) superalgebra. By generalizing this superspin chain, we embed it into a wider class of models. These include other models that have been studied previously in one and two dimensions. We suggest that the multicritical behavior in two dimensions of a class of these generalized models (possibly not including the multicritical point in the RBIM itself) may be governed by a single fixed point, at which the supersymmetry is enhanced still further to osp(2n+2∣2n).\mathrm{osp}(2n+2|2n).osp(2n+2∣2n). This suggestion is supported by a calculation of the renormalization-group flows for the corresponding nonlinear sigma models at weak coupling.
We report our Monte Carlo results on the critical and multicritical behavior of the ± J Ising model [with a random-exchange probability P ( J xy ) = p δ( J xy - J )+(1- p )δ( J xy + J )], in two and three dimensions. We study the transition line between the paramagnetic and ferromagnetic phase, which extends from p = 1 to a multicritical (Nishimori) point. By a finite-size scaling analysis, we provide strong numerical evidence that in three dimensions the critical behavior along this line belongs to the same universality class as that of the critical transition in the randomly dilute Ising model. In two dimensions we confirm that the critical behavior is controlled by the pure Ising fixed point and that disorder is marginally irrelevant, giving rise to universal logarithmic corrections. In both two and three dimensions, we also determine the location of the multicritical Nishimori point, as well as the renormalization-group dimensions of the operators that control the renormalization-group flow close to it.