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Physical review. B, Condensed matter

Two-dimensional Ising model in random magnetic fields

I. Morgenstern, Kurt Binder, Richard M. Hornreich

Published Jan 1, 1981• 97 citations
Publisher

Abstract

The statistical mechanics of finite $L\ifmmode\times\else\texttimes\fi{}L$ Ising square lattices in a field $\ifmmode\pm\else\textpm\fi{}h$ of random sign is investigated numerically by a modified recursive transfer matrix method for $6<~L<~16$. Our results are consistent with the absence of a spontaneous magnetization for $h\ensuremath{\ne}0$ even in the ground state. The singularities occurring at $T=0$ in the range $0<~h<~4J$, $J$ being the exchange constant, are discussed in terms of a cluster expansion. For nonzero $T$, less than the critical temperature of the pure two-dimensional Ising model, and $h$ sufficiently small, the system exhibits a nonzero spin-glass order parameter of the Mattis type although there is no magnetization. The ferromagnetic correlation function becomes long ranged for $h\ensuremath{\rightarrow}0$ and is calculated from the domain-wall density.

DOI
10.1103/physrevb.23.287

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The statistical mechanics of finite L\ifmmode×\else\texttimes\fiLL\ifmmode\times\else\texttimes\fi{}LL\ifmmode×\else\texttimes\fiL Ising square lattices in a field \ifmmode±\else\textpm\fih\ifmmode\pm\else\textpm\fi{}h\ifmmode±\else\textpm\fih of random sign is investigated numerically by a modified recursive transfer matrix method for 6< L< 166<~L<~166< L< 16. Our results are consistent with the absence of a spontaneous magnetization for h\ensuremath≠0h\ensuremath{\ne}0h\ensuremath=0 even in the ground state. The singularities occurring at T=0T=0T=0 in the range 0< h< 4J0<~h<~4J0< h< 4J, JJJ being the exchange constant, are discussed in terms of a cluster expansion. For nonzero TTT, less than the critical temperature of the pure two-dimensional Ising model, and hhh sufficiently small, the system exhibits a nonzero spin-glass order parameter of the Mattis type although there is no magnetization. The ferromagnetic correlation function becomes long ranged for h\ensuremath→0h\ensuremath{\rightarrow}0h\ensuremath→0 and is calculated from the domain-wall density.
We study ground-state properties of the two-dimensional random-bond Ising model with couplings having a concentration p∊[0,1]p∊[0,1]p∊[0,1] of antiferromagnetic and $(1\ensuremath{-}p)$ of ferromagnetic bonds. We apply an exact matching algorithm which enables us the study of systems with linear dimension LLL up to 700700700. We study the behavior of the domain-wall energies and of the magnetization. We find that the paramagnet-ferromagnet transition occurs at ${p}_{c}\ensuremath{\sim}0.103$ compared to the concentration ${p}_{n}\ensuremath{\sim}0.109$ at the Nishimori point, which means that the phase diagram of the model exhibits a re-entrance. Furthermore, we find no indications for an (intermediate) spin-glass ordering at finite temperature.
Transfer-matrix methods are used to calculate spin-spin correlation functions (G), Helmholtz free energies (f) and magnetizations (m) in the two-dimensional random-field Ising model close to the zero-field bulk critical temperature T(c 0), on long strips of width L=3-18 sites, for binary field distributions. Analysis of the probability distributions of G for varying spin-spin distances R shows that describing the decay of their averaged values by effective correlation lengths is a valid procedure only for not very large R. Connections between field and correlation function distributions at high temperatures are established, yielding approximate analytical expressions for the latter, which are used for computation of the corresponding structure factor. It is shown that, for fixed R/L, the fractional widths of correlation-function distributions saturate asymptotically with L-2.2. Considering an added uniform applied field h, a connection between f(h), m(h), the Gibbs free energy g(m) and the distribution function for the uniform magnetization in a zero uniform field, P0(m), is derived and first illustrated for pure systems, and then applied for nonzero random field. From finite-size scaling and crossover arguments, coupled with numerical data, it is found that the width of P0(m) varies against (nonvanishing, but small) random-field intensity H0 as H(-3/7)(0).
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 $L\ensuremath{\rightarrow}\ensuremath{\infty}$.
The statistics of the ground-state and domain-wall energies for the two-dimensional random-bond Ising model on square lattices with independent, identically distributed bonds having probability ppp of ${J}_{ij}=\ensuremath{-}1$ and $(1\ensuremath{-}p)$ of Jij=+1{J}_{ij}=+1Jij​=+1 are studied. We are able to consider large samples of up to 3202{320}^{2}3202 spins by using sophisticated matching algorithms. We study $L\ifmmode\times\else\texttimes\fi{}L$ systems, but we also consider $L\ifmmode\times\else\texttimes\fi{}M$ samples, for different aspect ratios R=L∕MR=L∕MR=L∕M. We find that the scaling behavior of the ground-state energy and its sample-to-sample fluctuations inside the spin-glass region $({p}_{c}\ensuremath{\leqslant}p\ensuremath{\leqslant}1\ensuremath{-}{p}_{c})$ are characterized by simple scaling functions. In particular, the fluctuations exhibit a cusplike singularity at pc{p}_{c}pc​. Inside the spin-glass region, the average domain-wall energy converges to a finite nonzero value as the sample size becomes infinite, holding RRR fixed. Here, large finite-size effects are visible, which can be explained for all ppp by a single exponent $\ensuremath{\omega}\ensuremath{\approx}2∕3$, provided higher-order corrections to scaling are included. Finally, we confirm the validity of aspect-ratio scaling for $R\ensuremath{\rightarrow}0$: the distribution of the domain-wall energies converges to a Gaussian for $R\ensuremath{\rightarrow}0$, although the domain walls of neighboring subsystems of size $L\ifmmode\times\else\texttimes\fi{}L$ are not independent.
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.
We use the transfer-matrix method to calculate density-of-states (DOS) functions for fully finite Ising-model systems. In this way, we obtain expressions for the corresponding partition functions for systems of sizes up to 12\ifmmode\times\else\texttimes\fi{}12 in zero field and up to 8\ifmmode\times\else\texttimes\fi{}8 in nonzero external magnetic field. We also establish a way of representing the DOS functions, in terms of the suitably scaled variables, that reveals a remarkably rapid convergence of these functions towards their thermodynamic limiting values as the system size increases. This rapid convergence implies that results obtained for systems of small sizes contain a surprisingly large amount of information about the behavior of infinite systems. In particular, we demonstrate that the finite-size-scaling-theory predictions are satisfied for small systems (treated exactly via this method) comparably well as for much larger systems (treated approximately via the Monte Carlo method). In addition, applying the method to the square Ising model in a nonzero field, we discover that the DOS functions for small systems display a regularity that one would expect to appear only for very large systems.