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arXiv:0709.1552

Some aspects of infinite range models of spin glasses: theory and numerical simulations

Alain H. Billoire

Published Sep 11, 2007• 3 citations
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Abstract

These notes give an introduction to the physics of the infinite range version of the Edwards--Anderson model, the so-called Sherrington--Kirkpatrick model. In a first part, I motivate and introduce the Edwards--Anderson and Sherrington--Kirkpatrick models. In the second part, I explain the analytical solution of the Sherrington--Kirkpatrick model, following Giorgio Parisi. I next give the physical interpretation of this solution. This is a vast subject, and I concentrate on the major points and give references for more details. The third part presents the simulation approach and compare its results to theoretical expectations: thermodynamics, finite size scaling, determination of the critical temperature using "parameters" like the "Binder parameter", and fluctuations. The last part gives a summary of our current understanding of finite size effects for the free energy and internal energy of the Sherrington--Kirkpatrick model

DOI
10.48550/arxiv.0709.1552

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These notes give an introduction to the physics of the infinite range version of the Edwards--Anderson model, the so-called Sherrington--Kirkpatrick model. In a first part, I motivate and introduce the Edwards--Anderson and Sherrington--Kirkpatrick models. In the second part, I explain the analytical solution of the Sherrington--Kirkpatrick model, following Giorgio Parisi. I next give the physical interpretation of this solution. This is a vast subject, and I concentrate on the major points and give references for more details. The third part presents the simulation approach and compare its results to theoretical expectations: thermodynamics, finite size scaling, determination of the critical temperature using "parameters" like the "Binder parameter", and fluctuations. The last part gives a summary of our current understanding of finite size effects for the free energy and internal energy of the Sherrington--Kirkpatrick model
We discuss the results of numerical simulations of two spin-glass models, the infinite range Sherrington–Kirkpatrick (SK) model, and a nearest–neighbor model in three dimensions. The probability distribution PN(q) for the spin glass order parameter q has been evaluated for the SK model below Tc with N, the number of spins, up to 192. It shows the same basic features as the Parisi function dx/dq as expected, but with sizeable finite size corrections. Results for the short-range model show evidence for a finite temperature transition but further work is needed to confirm this.
Spin glasses have been the subject of intense study and considerable controversy for decades, and the low-temperature phase of short-range spin glasses is still poorly understood. Our main goal is to improve our understanding in this area and find an answer to the following question: Are there only a single pair or a countable infinity of pure states in the low temperature phase of the EA spin glass? To that aim we first start by introducing spin glasses and provide a brief history of their research, then proceed to describe our method of simulation, the parallel tempering Monte Carlo algorithm. Next, we present the results of a large-scale numerical study of the equilibrium three-dimensional Edwards-Anderson Ising spin glass with Gaussian disorder. In order to understand how the parallel tempering algorithm works, we measure various static, as well as dynamical quantities, such as the autocorrelation times and round-trip times for the parallel tempering Monte Carlo method. We examine the correlation between static and dynamic observables for _ 5000 disorder realizations and up to 1000 spins down to temperatures at 20% of the critical temperature, and our results show that autocorrelation times are directly correlated with the roughness of the free-energy landscape. In the following chapters, the three- and four-dimensional Edwards-Anderson and mean-field Sherrington-Kirkpatrick Ising spin glasses are studied again via large-scale Monte Carlo simulations at low temperatures, deep within the spinglass phase. Performing a careful statistical analysis of several thousand independent disorder realizations and using an observable that detects peaks in the overlap distribution, we show that the Sherrington-Kirkpatrick and Edwards-Anderson models have a distinctly different low-temperature behavior. We arrive to the following conclusion: The structure of the spin-glass overlap distribution for the Edwards-Anderson model suggests that its low-temperature phase has only a single pair of pure states. Finally we present results for several new observables, along with a few preliminary studies and suggestions for future research.
The Sherrington-Kirkpatrick and the nearest-neighbour X-Y spin-glass models are studied.The effect of a small uniform symmetry breaking field on the infinite-ranged Sherrington-Kirkpatrick Ising spin-glass model is discussed.It is shown that the off-diagonal contributions to the fluctuation-dissipation formula vanish for a symmetric distribution of interactions and the limit of zero field.This contradicts the result of Young and Kirkpatrick based on numerical evidence for small samples.The dynamics of the Sherrington-Kirkpatrick spin-glass are investigated.Below the critical temperature this model exhibits non-ergodic behaviour.Consequently, in order to obtain timedependent correlation functions it is necessary to average over the initial states with a Boltzmann weight.This necessitates the use of "replicas".It is shown that in Parisi's scheme the "dynamical" Edwards-Anderson order parameter is given by q(x =1), whereas the statistical mechanics order parameter is given by r 1 q(x)dx.The result for the statistical mechanics order parameter j 0 implies that there is no violation of the fluctuation-dissipation theorem once an average over samples is performed.It is also shown that the system in general keeps its memory of the initial state for all time.However, this memory is "lost" if one makes a replica symmetric ansatz for the order parameters.This illus trates the connection between replica symmetry breaking and nonergodicity.The nearest neighbour X-Y spin-glass in both two-and threedimensions is investigated by Monte Carlo simulation.In two--11ln[ln Xcn + 1-42] versus In T SG } X2 (Tq ) for T (3D) 4.11 4.12a 4.12b 4.12c 4.13a ' 4.13b > Plots of the data for T (3D) 4.13c 4.14a 4.14b 4.15a 4.15b Plots of £ ^ for 3D SG A plot of the data for £ assuming that LCD = 3 SG A plot of the data for £ assuming that LCD = 3 SG and p__ = 1.2 SG A5.1a A5.
The relaxation of the Sherrington-Kirkpatrick model for spin glasses is studied at low temperatures. A recently developed numerical method is used by which an infinitely large system is simulated, hence finite size effects are avoided. For parallel dynamics the decay of the energy as well as of the magnetization is investigated. For low temperatures we find evidence for a relaxation into a state which is characterized by non-equilibrium values of the energy and a non-vanishing remanent magnetization.
The Sherrington-Kirkpatrick (SK) spin glass model (1975) is studied by Monte Carlo simulation. Below T c there is a spectrum of relaxation times which diverge like exp(cN 1 4/) as N, the number of spins, tends to infinity. In zero field, h, there is one more timescale, tau eg , which is the time to turn over all the spins. Averaging over samples (ln tau eg ) j (=ln tau eg ) varies as N 1 2/. However, ln tau eg is not a self-averaging quantity and there are large sample-to-sample variations even for N to infinity . For h>>h*=T/N 1 2/, where T is the temperature, fluctuations on timescale tau eg may not occur. Above the Almeida-Thouless line (1978) relaxation times are finite and the SK solution is correct. Below this line, the results agree well with Parisi's theory (1983) if the latter is correctly interpreted.