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arXiv:cond-mat/0007509 · Physical Review Letters

Nature of Ergodicity Breaking in Ising Spin Glasses as Revealed by Correlation Function Spectral Properties

Jairo Sinova, Geoffrey Canright, Allan H. MacDonald

Published Sep 18, 2000• 13 citations
arXivHTMLPDF

Abstract

In this Letter we address the nature of broken ergodicity in the low temperature phase of Ising spin glasses by examining spectral properties of spin correlation functions C(ij) identical with<S(i)S(j)>. We argue that more than one extensive [i.e., O(N)] eigenvalue in this matrix signals replica symmetry breaking. Monte Carlo simulations of the infinite-range Ising spin-glass model, above and below the Almeida-Thouless line, support this conclusion. Exchange Monte Carlo simulations for the short-range model in four dimensions find a single extensive eigenvalue and a large subdominant eigenvalue consistent with droplet model expectations.

DOI
10.1103/physrevlett.85.2609

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•Apr 23, 2019•0 citations
In this Letter we address the nature of broken ergodicity in the low temperature phase of Ising spin glasses by examining spectral properties of spin correlation functions C(ij) identical with<S(i)S(j)>. We argue that more than one extensive [i.e., O(N)] eigenvalue in this matrix signals replica symmetry breaking. Monte Carlo simulations of the infinite-range Ising spin-glass model, above and below the Almeida-Thouless line, support this conclusion. Exchange Monte Carlo simulations for the short-range model in four dimensions find a single extensive eigenvalue and a large subdominant eigenvalue consistent with droplet model expectations.
We study the nature of the broken ergodicity in the low temperature phase of Ising spin glass systems, using as a diagnostic tool the spectrum of eigenvalues of the spin-spin correlation function. We show that multiple extensive eigenvalues of the correlation matrix ${C}_{\mathrm{ij}}\ensuremath{\equiv}〈{S}_{i}{S}_{j}〉$ occur if and only if there is replica symmetry breaking. We support our arguments with Exchange Monte Carlo results for the infinite-range problem. Here we find multiple extensive eigenvalues in the replica symmetry breaking (RSB) phase for $N\ensuremath{\gtrsim}200,$ but only a single extensive eigenvalue for phases with long-range order but no RSB. Numerical results for the short-range model in four spatial dimensions, for N< 1296,N<~1296,N< 1296, are consistent with the presence of a single extensive eigenvalue, with the subdominant eigenvalue behaving in agreement with expectations derived from the droplet model. Because of the small system sizes we cannot exclude the possibility of replica symmetry breaking with finite size corrections in this regime.
Using Monte Carlo simulations, we study in detail the overlap distribution for individual samples for several spin-glass models including the infinite-range Sherrington-Kirkpatrick model, short-range Edwards-Anderson models in three and four space dimensions, and one-dimensional long-range models with diluted power-law interactions. We study three long-range models with different powers as follows: The first is approximately equivalent to a short-range model in three dimensions, the second to a short-range model in four dimensions, and the third to a short-range model in the mean-field regime. We study an observable proposed earlier by some of us which aims to distinguish the ``replica symmetry breaking'' picture of the spin-glass phase from the ``droplet picture,'' finding that larger system sizes would be needed to unambiguously determine which of these pictures describes the low-temperature state of spin glasses best, except for the Sherrington-Kirkpatrick model, which is unambiguously described by replica symmetry breaking. Finally, we also study the median integrated overlap probability distribution and a typical overlap distribution, finding that these observables are not particularly helpful in distinguishing the replica symmetry breaking and the droplet pictures.
We determine the low-energy spectrum and Parisi replica-symmetry-breaking function for the spin-glass phase of the quantum Ising model with infinite-range random exchange interactions and transverse and longitudinal ( h ) fields. We show that, for all h , the spin-glass state has full replica symmetry breaking and the local spin spectrum is gapless, with a spectral density that vanishes linearly with frequency. These results are obtained using an action functional—argued to yield exact results at low frequencies—that expands in powers of a spin-glass order parameter, which is bilocal in time, and a matrix in replica space. We also present the exact solution of the infinite-range spherical quantum p -rotor model at nonzero h : here, the spin-glass state has one-step replica symmetry breaking and gaplessness only appears after imposition of an additional marginal stability condition. Possible connections to experiments on random arrays of trapped Rydberg atoms are noted. Published by the American Physical Society 2024
We have studied the diluted Heisenberg spin glass model in a three-component random field for the commonly used one-dimensional long-range model where the probability that two spins separated by a distance r interact with one another falls as 1/r^{2σ}, for two values of σ, 0.75 and 0.85. No de Almeida-Thouless line is expected at these σ values. The spin glass correlation length ξ_{SG} varies with the random field as expected from the Imry-Ma argument and the droplet scaling picture of spin glasses. However, when ξ_{SG} becomes comparable to the system size L, there are departures which we attribute to the features deriving from the replica symmetry breaking picture of spin glasses. For the case σ=0.85 these features go away for system sizes with L>L^{*}, where L^{*} is large (≈4000-8000 lattice spacings). In the case of σ=0.75 we have been unable to study large enough systems to determine its value of L^{*}. We sketch a renormalization group scenario to explain how these features could arise. In this scenario finite size effects on the droplet scaling picture in low-dimensional spin glasses produce some aspects of Parisi's replica symmetry breaking theory of the Sherrington-Kirkpatrick model.
The study of the low temperature phase of spin glass models by means of Monte Carlo simulations is a challenging task, because of the very slow dynamics and the severe finite-size effects they show. By exploiting at the best the capabilities of standard modern CPUs (especially the streaming single instruction, multiple data extensions), we have been able to simulate the four-dimensional Edwards-Anderson model with Gaussian couplings up to sizes L=70 and for times long enough to accurately measure the asymptotic behavior. By quenching systems of different sizes to the critical temperature and to temperatures in the whole low temperature phase, we have been able to identify the regime where finite-size effects are negligible: ξ(t)≲L/7. Our estimates for the dynamical exponent (z≃1/T) and for the replicon exponent (α≃1.0 and T independent), that controls the decay of the spatial correlation in the zero overlap sector, are consistent with the replica symmetry breaking theory, but the latter differs from the theoretically conjectured value.
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.