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arXiv:cond-mat/9707272 · Physical review. B, Condensed matter

Scaling and chaos in periodic approximations to the two-dimensional Ising spin glass

David A. Huse, Lee-Fen Ko

Published Dec 1, 1997• 11 citations
arXivHTMLPDF

Abstract

We approximate a two-dimensional spin glass by tiling an infinite lattice with large identical unit cells. The interactions within the unit cell are chosen at random, just as when one studies finite-size systems with periodic boundary conditions. But here the unit cells are instead connected to form an infinite lattice, so one may examine correlations on all length scales, and the system can have true phase transitions. For such approximations to the Ising spin glass on the square lattice, we apply the free-fermion method of Onsager and the anticommuting operator approach of Kaufman to obtain numerically exact results for each realization of the quenched disorder. Each such sample shows one or more critical points, with the distribution of critical temperatures scaling with the unit cell size, consistent with what is expected from the scaling theory of low-dimensional spin glasses. Due to ``chaos,'' the correlations between unit cells can change sign with changing temperature. We examine the scaling of this chaos with unit cell size. Many samples have multiple critical points due to the interactions between unit cells changing sign at temperatures within the ordered phases.

DOI
10.1103/physrevb.56.14597

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We approximate a two-dimensional spin glass by tiling an infinite lattice with large identical unit cells. The interactions within the unit cell are chosen at random, just as when one studies finite-size systems with periodic boundary conditions. But here the unit cells are instead connected to form an infinite lattice, so one may examine correlations on all length scales, and the system can have true phase transitions. For such approximations to the Ising spin glass on the square lattice, we apply the free-fermion method of Onsager and the anticommuting operator approach of Kaufman to obtain numerically exact results for each realization of the quenched disorder. Each such sample shows one or more critical points, with the distribution of critical temperatures scaling with the unit cell size, consistent with what is expected from the scaling theory of low-dimensional spin glasses. Due to ``chaos,'' the correlations between unit cells can change sign with changing temperature. We examine the scaling of this chaos with unit cell size. Many samples have multiple critical points due to the interactions between unit cells changing sign at temperatures within the ordered phases.
We have calculated finite-size correlation lengths for strips of the square-lattice Ising spin glass using the transfer-matrix method. In order to minimize sampling errors we study strip lengths up to 5\ifmmode\times\else\texttimes\fi{}106{10}^{6}106 lattice spacings. A phenomenological renormalization-group analysis indicates that there are strong corrections to simple power-law scaling near the zero-temperature critical point, as is to be expected near a lower critical dimension. We examine models with Gaussian, exp(-J2{J}^{2}J2/2), and exponential, exp(-\ensuremath{\Vert}J\ensuremath{\Vert}), distributions of couplings; the Gaussian distribution shows stronger finite-size corrections. The correlation-length exponent is estimated to be \ensuremath{\nu}=4.2\ifmmode\pm\else\textpm\fi{}0.5, although we do not want to rule out the possibility that \ensuremath{\nu} is significantly larger than this.
We analyze changes in the thermodynamic properties of a spin system when it passes from the classical two-dimensional Ising model to the spin glass model, where spin-spin interactions are random in their values and signs. Formally, the transition reduces to a gradual change in the amplitude of the multiplicative noise (distributed uniformly with a mean equal to one) superimposed over the initial Ising matrix of interacting spins. Considering the noise, we obtain analytical expressions that are valid for lattices of finite sizes. We compare our results with the results of computer simulations performed for square N = L × L lattices with linear dimensions L = 50 ÷ 1000. We find experimentally the dependencies of the critical values (the critical temperature, the internal energy, entropy and the specific heat) as well as the dependencies of the energy of the ground state and its magnetization on the amplitude of the noise. We show that when the variance of the noise reaches one, there is a jump of the ground state from the fully correlated state to an uncorrelated state and its magnetization jumps from 1 to 0. In the same time, a phase transition that is present at a lower level of the noise disappears.
By a recursive method numerically exact free energies and correlation functions are calculated for finite two- and three-dimensional Ising lattices with bonds chosen randomly either ±J or from a gaussian distribution. While (dynamic) Monte Carlo simulations of precisely the same systems show the onset of Edwards-Anderson ordering at rather sharply defined nonzero freezing temperature Tf, the exact static calculations imply that there is no static order parameter qEA = [<Si≳2T]av at nonzero temperature. The correlation length, which would become infinite when qEA would start to become nonzero, is found at Tf to be of the order of a few lattice spacings only. As a result, the freezing-in of these models is identified as a dynamic phenomenon, and not a phase transition associated with static Edwards-Anderson order. The specific heat of the gaussian spin glass is found to vary nearly linear with temperature. On the other hand, in two-dimensional Ising systems with random fields a nonzero (Mattis-like) spin glass order is found.
Monte Carlo data of the two-dimensional Ising spin glass with bimodal interactions are presented with the aim of understanding the low-temperature physics of the model. An analysis of the specific heat, spin-glass susceptibility, finite-size correlation length, and the Binder ratio is performed to try to verify a recent proposal in which for large system sizes and finite but low temperatures the effective critical exponents are identical to the critical exponents of the two-dimensional Ising spin glass with Gaussian interactions. Our results show that with present system sizes the recently proposed scenario, in which the two-dimensional Ising spin glass with bimodally distributed interactions is in the same universality class as the model with Gaussian-distributed disorder at low but finite temperatures, cannot be reliably proven.
Renyi mutual information, computed from second Renyi entropies, can identify classical phase transitions from their finite-size scaling at critical points. We apply this technique to examine the presence or absence of finite temperature phase transitions in various two-dimensional models on a square lattice, which are extensions of the conventional Ising model by adding a quenched disorder. When the quenched disorder causes the nearest neighbor bonds to be both ferromagnetic and antiferromagnetic, (a) a spin glass phase exists only at zero temperature, and (b) a ferromagnetic phase exists at a finite temperature when the antiferromagnetic bond distributions are sufficiently dilute. Furthermore, finite temperature paramagnetic-ferromagnetic transitions can also occur when the disordered bonds involve only ferromagnetic couplings of random strengths. In our numerical simulations, the 'zero temperature only' phase transitions are identified when there is no consistent finite-size scaling of the Renyi mutual information curves, while for finite temperature critical points, the curves can identify the critical temperature T-c by their crossings at T-c and 2 T-c.
The standard short-range two-dimensional Ising spin glass is numerically well accessible, in particular, because there are polynomial-time ground-state algorithms. On the other hand, in contrast to higher dimensional spin glasses, it does not exhibit a rich behavior, i.e., no ordered phase at finite temperature. Here, we investigate whether long-range correlated bonds change this behavior. This would still keep the model numerically well accessible while exhibiting a more interesting behavior. The bonds are drawn from a Gaussian distribution with a two-point correlation for bonds at distance r that decays as (1+r^{2})^{-a/2}, a≥0. We study numerically with exact algorithms the ground-state and domain-wall excitations. Our results indicate that the inclusion of bond correlations still does not lead to a spin-glass order at any finite temperature. A further analysis reveals that bond correlations have a strong effect at local length scales, inducing ferro- and antiferromagnetic domains into the system. The length scale of ferro- and antiferromagnetic order diverges exponentially as the correlation exponent approaches a critical value, a→a_{crit}=0. Thus, our results suggest that the system becomes a ferro- or antiferromagnet only in the limit a→0.