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arXiv:cond-mat/0201137 · Physical review. B, Condensed matter

Negative scaling dimensions and conformal invariance at the Nishimori point in the ±J random-bond Ising model

Florian Merz, J. T. Chalker

Published Aug 13, 2002• 18 citations
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Abstract

We reexamine the disorder-dominated multicritical point of the two-dimensional $\ifmmode\pm\else\textpm\fi{}J$-Ising model, known as the Nishimori point (NP). At the NP we investigate numerically and analytically the behavior of the disorder correlator, familiar from the self-dual description of the pure critical point of the two-dimensional Ising model. We consider the logarithmic average and the $q\mathrm{th}$ moments of this correlator in the ensemble average over randomness, for continuous q in the range $0<q<2.5,$ and demonstrate their conformal invariance. At the NP we find, in contrast to the self-dual pure critical point, that the disorder correlators exhibit multiscaling in q which is different from that of spin-spin correlators and that their scaling dimension becomes negative for $q>1$ and $q<0.$ Using properties on the Nishimori line we show that the first moment $(q=1)$ of the disorder correlator is exactly one for all separations. The spectrum of scaling dimensions at the NP is not parabolic in q.

DOI
10.1103/physrevb.66.054413

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We reexamine the disorder-dominated multicritical point of the two-dimensional \ifmmode±\else\textpm\fiJ\ifmmode\pm\else\textpm\fi{}J\ifmmode±\else\textpm\fiJ-Ising model, known as the Nishimori point (NP). At the NP we investigate numerically and analytically the behavior of the disorder correlator, familiar from the self-dual description of the pure critical point of the two-dimensional Ising model. We consider the logarithmic average and the qthq\mathrm{th}qth moments of this correlator in the ensemble average over randomness, for continuous q in the range 0<q<2.5,0<q<2.5,0<q<2.5, and demonstrate their conformal invariance. At the NP we find, in contrast to the self-dual pure critical point, that the disorder correlators exhibit multiscaling in q which is different from that of spin-spin correlators and that their scaling dimension becomes negative for q>1q>1q>1 and q<0.q<0.q<0. Using properties on the Nishimori line we show that the first moment (q=1)(q=1)(q=1) of the disorder correlator is exactly one for all separations. The spectrum of scaling dimensions at the NP is not parabolic in q.
We investigate the critical behavior of the random-bond $\ifmmode\pm\else\textpm\fi{}J$ Ising model on a square lattice at the multicritical Nishimori point in the $T\text{\ensuremath{-}}p$ phase diagram, where TTT is the temperature and ppp is the disorder parameter (p=1p=1p=1 corresponds to the pure Ising model). We perform a finite-size scaling analysis of high-statistics Monte Carlo simulations along the Nishimori line defined by $2p\ensuremath{-}1=\text{tanh}(1/T)$, along which the multicritical point lies. The multicritical Nishimori point is located at ${p}^{\ensuremath{\ast}}=0.890\text{ }81(7)$, ${T}^{\ensuremath{\ast}}=0.9528(4)$, and the renormalization-group dimensions of the operators that control the multicritical behavior are y1=0.655(15){y}_{1}=0.655(15)y1​=0.655(15) and y2=0.250(2){y}_{2}=0.250(2)y2​=0.250(2); they correspond to the thermal exponent $\ensuremath{\nu}\ensuremath{\equiv}1/{y}_{2}=4.00(3)$ and to the crossover exponent $\ensuremath{\phi}\ensuremath{\equiv}{y}_{1}/{y}_{2}=2.62(6)$.
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.
Abstract The Nishimori point of the random bond Ising model is a prototype of renormalization group fixed points with strong disorder. We show that the exact correlation length and crossover critical exponents at this point can be identified in two and three spatial dimensions starting from properties of the Nishimori line. These are the first exact exponents for frustrated random magnets, a circumstance to be also contrasted with the fact that the exact exponents of the Ising model without disorder are not known in three dimensions. Our considerations extend to higher dimensions and models other than Ising.
By considering the quench dynamics of two-dimensional frustrated Ising models through numerical simulations, we investigate the dynamical critical behavior on the multicritical Nishimori point (NP). We calculate several dynamical critical exponents, namely, the relaxation exponent z_{c}, the autocorrelation exponent λ_{c}, and the persistence exponent θ_{c}, after a quench from the high temperature phase to the NP. We confirm their universality with respect to the lattice geometry and bond distribution. For a quench from a power-law correlated initial state to the NP, the aging dynamics are much slower. We also look up the issue of multifractality during the critical dynamics by investigating different moments of the spatial correlation function. We observe a single growth law for all the length scales extracted from different moments, indicating that the equilibrium multifractality at the NP does not affect the dynamics.
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.
The multicritical behavior at the Nishimori point of two-dimensional Ising spin glasses is investigated by using numerical transfer-matrix methods to calculate probability distributions P(C)P(C)P(C) and associated moments of spin-spin correlation-functions C on strips. The angular dependence of the shape of correlation function distributions P(C)P(C)P(C) provides a stringent test of how well they obey predictions of conformal invariance; and an even symmetry of $(1\ensuremath{-}C)P(C)$ reflects the consequences of the Ising spin-glass gauge (Nishimori) symmetry. We show that conformal invariance is obeyed in its strictest form, and the associated scaling of the moments of the distribution is examined, in order to assess the validity of a recent conjecture on the exact localization of the Nishimori point. Power-law divergences of P(C)P(C)P(C) are observed near C=1C=1C=1 and C=0,C=0,C=0, in partial accord with a simple scaling scheme which preserves the gauge symmetry.