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Physical review. B, Condensed matter

Magnetic correlations in two-dimensional spin-glasses

I. Morgenstern, Kurt Binder

Published Jul 1, 1980• 304 citations
Publisher

Abstract

By a recursive method numerically exact free energies are calculated for square $L\ifmmode\times\else\texttimes\fi{}L$ Ising lattices, with $6<~L<~18$, for several kinds of frozen-in bond disorder: (i) bonds $\ifmmode\pm\else\textpm\fi{}J$ with various concentrations of negative bonds; (ii) bonds distributed according to a Gaussian distribution. Ground states of these systems are identified, the response to "ordering fields" is studied, and the correlation function ${〈{S}_{0}{S}_{R}〉}_{T}^{2}$ is calculated as a function of temperature for various distances $R$ in the lattice. This correlation is found to decay strongly (presumably exponentially) with increasing $R$ even at temperatures distinctly below the apparent freezing temperature ${T}_{f}$ of previous Monte Carlo simulations; this "freezing transition" is hence unambiguously identified as a nonequilibrium effect. However, the correlation length is found to become long ranged at low temperatures, and it is suggested that a phase transition still occurs at $T=0$; while in the Gaussian model the spin-glass order parameter $q(T=0)=1$, it is found that $q\ensuremath{\equiv}0$ in the $\ifmmode\pm\else\textpm\fi{}J$ model where rather a power-law decay of correlations ${〈{S}_{0}{S}_{R}〉}_{T=0}^{2}$ occurs. Performing Monte Carlo simulations for precisely the same systems, the cooling times necessary to reach the true ground states of the system are identified, as well as the simulation times necessary to reach thermal equilibrium for the correlation functions. These times are found to increase so strongly with $L$ that for systems of macroscopic size the correct thermal equilibrium is probably irrelevant for experimental purposes. Rather a statistical mechanics based on the many long-lived metastable states would be required.

DOI
10.1103/physrevb.22.288

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By a recursive method numerically exact free energies are calculated for square L\ifmmode×\else\texttimes\fiLL\ifmmode\times\else\texttimes\fi{}LL\ifmmode×\else\texttimes\fiL Ising lattices, with 6< L< 186<~L<~186< L< 18, for several kinds of frozen-in bond disorder: (i) bonds \ifmmode±\else\textpm\fiJ\ifmmode\pm\else\textpm\fi{}J\ifmmode±\else\textpm\fiJ with various concentrations of negative bonds; (ii) bonds distributed according to a Gaussian distribution. Ground states of these systems are identified, the response to "ordering fields" is studied, and the correlation function 〈S0SR〉T2{〈{S}_{0}{S}_{R}〉}_{T}^{2}〈S0​SR​〉T2​ is calculated as a function of temperature for various distances RRR in the lattice. This correlation is found to decay strongly (presumably exponentially) with increasing RRR even at temperatures distinctly below the apparent freezing temperature Tf{T}_{f}Tf​ of previous Monte Carlo simulations; this "freezing transition" is hence unambiguously identified as a nonequilibrium effect. However, the correlation length is found to become long ranged at low temperatures, and it is suggested that a phase transition still occurs at T=0T=0T=0; while in the Gaussian model the spin-glass order parameter q(T=0)=1q(T=0)=1q(T=0)=1, it is found that q\ensuremath≡0q\ensuremath{\equiv}0q\ensuremath≡0 in the \ifmmode±\else\textpm\fiJ\ifmmode\pm\else\textpm\fi{}J\ifmmode±\else\textpm\fiJ model where rather a power-law decay of correlations 〈S0SR〉T=02{〈{S}_{0}{S}_{R}〉}_{T=0}^{2}〈S0​SR​〉T=02​ occurs. Performing Monte Carlo simulations for precisely the same systems, the cooling times necessary to reach the true ground states of the system are identified, as well as the simulation times necessary to reach thermal equilibrium for the correlation functions. These times are found to increase so strongly with LLL that for systems of macroscopic size the correct thermal equilibrium is probably irrelevant for experimental purposes. Rather a statistical mechanics based on the many long-lived metastable states would be required.
By recursive methods, numerically exact free energies are calculated for $L\ifmmode\times\else\texttimes\fi{}L$ Ising lattices with bonds of randomly chosen sign, with 6< L< 186<~L<~186< L< 18. Ground states of these systems are identified, and the response to ordering fields is studied. By performing Monte Carlo simulations for precisely the same systems we are able to unambiguously distinguish nonequilibrium phenomena from equilibrium properties. The LLL dependence of our results suggests that there is no nonzero spin-glass order parameter for $L\ensuremath{\rightarrow}\ensuremath{\infty}$.
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.
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.
Understanding the low-temperature pure state structure of spin glasses remains an open problem in the field of statistical mechanics of disordered systems. Here we study Monte Carlo dynamics, performing simulations of the growth of correlations following a quench from infinite temperature to a temperature well below the spin-glass transition temperature T_{c} for a one-dimensional Ising spin-glass model with diluted long-range interactions. In this model, the probability P_{ij} that an edge {i,j} has nonvanishing interaction falls as a power law with chord distance, P_{ij}∝1/R_{ij}^{2σ}, and we study a range of values of σ with 1/2<σ<1. We consider a correlation function C_{4}(r,t). A dynamic correlation length that shows power-law growth with time ξ(t)∝t^{1/z} can be identified in the data and, for large time t, C_{4}(r,t) decays as a power law r^{-α_{d}} with distance r when r≪ξ(t). The calculation can be interpreted in terms of the maturation metastate averaged Gibbs state, or MMAS, and the decay exponent α_{d} differentiates between a trivial MMAS (α_{d}=0), as expected in the droplet picture of spin glasses, and a nontrivial MMAS (α_{d}≠0), as in the replica-symmetry-breaking (RSB) or chaotic pairs pictures. We find nonzero α_{d} even in the regime σ>2/3 which corresponds to short-range systems below six dimensions. For σ<2/3, the decay exponent α_{d} follows the RSB prediction for the decay exponent α_{s}=3-4σ of the static metastate, consistent with a conjectured statics-dynamics relation, while it approaches α_{d}=1-σ in the regime 2/3<σ<1; however, it deviates from both lines in the vicinity of σ=2/3.
We present the results of numerical simulations on Ising spin glasses in zero magnetic field with nearest-neighbor interactions on hyper-cubic lattices in two, three, and four dimensions with both Gaussian and \ifmmode\pm\else\textpm\fi{}J bond distributions. Finite-size scaling is used to analyze the results. In two dimensions (d=2) we agree with earlier work that the transition temperature is at Tc{T}_{c}Tc​=0, and obtain the correlation-length exponent \ensuremath{\nu}, and the exponent \ensuremath{\eta}, at the zero-temperature transition for the \ifmmode\pm\else\textpm\fi{}J model. In d=3 dimensions we concentrate on results for the Gaussian distribution, since our results for the \ifmmode\pm\else\textpm\fi{}J distribution have been presented earlier. As expected, we find similar results for the two distributions, namely a nonzero Tc{T}_{c}Tc​ but evidence that d=3 is close to the lower critical dimension. In a four-dimensional spin glass with Gaussian bonds we find that only a modest amount of computer time is required to show that Tc{T}_{c}Tc​ is nonzero with a long-range-ordered phase below Tc{T}_{c}Tc​. Our estimates for critical exponents in d=4 dimensions agree well with results from recent high-temperature-series expansions.
First results of massive Monte Carlo simulations of the d=3 Ising spin glass with ± J bond distribution, performed on a fast special purpose computer, are discussed. Equilibrium static and dynamic correlation functions have been recorded for temperatures T/J=1.10 to 2.00. A qualitative change in the behavior of the system, and the best fits for the spin glass correlation length and relaxation time favor a continuous phase transition at Tc/J≊1.20±0.05. We find no evidence for the Tc=0 freezing picture.