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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Mean-field Monte Carlo approach to the Sherrington-Kirkpatrick model with asymmetric couplings

H. Eissfeller, Manfred Opper

Published Aug 1, 1994• 37 citations
Publisher

Abstract

An alternative method is applied to the nonequilibrium zero temperature dynamics of a Sherrington-Kirkpatrick model with nonsymmetric random exchange couplings ${\mathit{J}}_{\mathit{i}\mathit{j}}$\ensuremath{\ne}${\mathit{J}}_{\mathit{j}\mathit{i}}$. Based on exact stochastic single spin equations for the infinite system, this mean-field Monte Carlo approach avoids the strong finite size effects of the usual simulations for this problem. Varying the average symmetry \ensuremath{\eta}=[${\mathit{J}}_{\mathit{i}\mathit{j}}$${\mathit{J}}_{\mathit{j}\mathit{i}}$]/[${\mathit{J}}_{\mathit{i}\mathit{j}}^{2}$], we find a clear transition from ergodic dynamics at \ensuremath{\eta}${\mathrm{\ensuremath{\eta}}}_{\mathit{c}}$=0.825 to a phase (\ensuremath{\eta}>${\mathrm{\ensuremath{\eta}}}_{\mathit{c}}$) where a finite fraction of the spins freezes. Only in the fully symmetric case \ensuremath{\eta}=1 is the entire system frozen.

DOI
10.1103/physreve.50.709

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An alternative method is applied to the nonequilibrium zero temperature dynamics of a Sherrington-Kirkpatrick model with nonsymmetric random exchange couplings Jij{\mathit{J}}_{\mathit{i}\mathit{j}}Jij​\ensuremath{\ne}Jji{\mathit{J}}_{\mathit{j}\mathit{i}}Jji​. Based on exact stochastic single spin equations for the infinite system, this mean-field Monte Carlo approach avoids the strong finite size effects of the usual simulations for this problem. Varying the average symmetry \ensuremath{\eta}=[{\mathit{J}}_{\mathit{i}\mathit{j}}$${\mathit{J}}_{\mathit{j}\mathit{i}}]/[Jij2{\mathit{J}}_{\mathit{i}\mathit{j}}^{2}Jij2​], we find a clear transition from ergodic dynamics at \ensuremath{\eta}\ensuremathηc{\mathrm{\ensuremath{\eta}}}_{\mathit{c}}\ensuremathηc​=0.825 to a phase (\ensuremath{\eta}>\ensuremathηc{\mathrm{\ensuremath{\eta}}}_{\mathit{c}}\ensuremathηc​) where a finite fraction of the spins freezes. Only in the fully symmetric case \ensuremath{\eta}=1 is the entire system frozen.
Sompolinsky and Zippelius obtained Parisi's solution of the Sherrington-Kirkpatrick (SK) model from the relaxational dynamics of soft spins in continuous time, by assuming a hierarchy of widely separated time scales. Here we carry out the same programme for a discrete-time dynamics in which all Ising spins are updated at once. This dynamics, synchronous Monte Carlo (SyncMC), introduces Gaussian auxiliary fields so that the spins become conditionally independent, and it samples the Boltzmann distribution exactly for any system with symmetric pairwise couplings. We show that, for any such system, the correlation and response functions of SyncMC obey an exact fluctuation-dissipation relation in discrete time in equilibrium. The dynamical mean-field equations of SyncMC for the SK model contain two terms absent in continuous-time dynamics, a self-coupling of each spin and a noise correlated through the couplings; both affect only the fast relaxation. As a result, under the same assumptions as in the Sompolinsky-Zippelius construction (widely separated time scales, and on each slow scale the same relation with a reduced ratio x, as expected in relaxation from random initial conditions), the long-time structure of SyncMC reproduces the Parisi equations, including the Parisi partial differential equation and the equation for the distribution of local fields. Equilibrium simulations with replica exchange for up to N=2048 spins exclude the replica-symmetric distribution of the actual local fields and are consistent with the Parisi solution. The function x(q) estimated from the overlap distribution, and from the response in relaxation from random configurations for up to N=4096 spins, is also consistent with Parisi's.
Numerical data on the probability distribution of the equilibrium relaxation time of the Sherrington-Kirkpatrick model are obtained by means of dynamical Monte Carlo simulation, for several values of the system size NNN and temperature TTT. Proper care is taken that the thermal fluctuations on the relaxation time estimates are totally negligible compared to the disorder induced fluctuations. The probability distribution of ln⁡τ−ln⁡τ‾\lnτ-\overline{\lnτ}lnτ−lnτ scales with the scaling variable N1/3(Tc−T)N^{1/3} (T_c-T)N1/3(Tc​−T) strengthening the belief that ln⁡τ‾∝N1/3\overline{\lnτ}\propto N^{1/3}lnτ∝N1/3 in the whole spin glass phase.
A new method is presented that allows the determination of dynamical properties of disordered spin systems avoiding finite-size effects. The method is based on exact dynamical mean-field equations for the infinite large system. The resulting single-spin dynamics is solved by Monte Carlo simulations. We outline the formalism for the parallel dynamics of a fully connected model with random couplings. The decay of remanent magnetization of the model is studied. We find a power-law decay: m(t)-m(\ensuremath{\infty})\ensuremath{\propto}${\mathit{t}}^{\mathrm{\ensuremath{-}}\mathit{a}}$, with a=0.474 and m(\ensuremath{\infty})=0.184 for the infinite system.
The stochastic dynamics of randomly asymmetric fully connected Ising systems is studied. We solve analytically the particularly simple case of fully asymmetric systems. We calculate the relaxation time of the autocorrelation function and show that the system remains paramagnetic even at zero temperature (T=0). The ferromagnetic phase is only slightly affected by the asymmetry, and the paramagnetic-to-ferromagnetic phase transition is characterized by a critical slowing down similar to second-order transition in symmetric (fully connected) systems. Monte Carlo simulations of a fully connected Ising system with random asymmetric interactions, both at finite and zero temperature, are presented. For finite T the autocorrelation function decays completely to zero for all strengths of the asymmetry. The T=0 behavior is more complex. In the fully asymmetric case the system is ergodic, with decaying autocorrelations, in agreement with the theoretical predictions. In the partially asymmetric case all flows terminate at fixed points (i.e., states which are stable to single spin flips). However, the typical time that it takes to converge to a fixed point grows exponentially with the size of the system. This convergence time varies from sample to sample and has a log-normal distribution in large systems. On time scales which are smaller than the convergence time, the system behaves ``ergodically,'' and the autocorrelation function decays to zero, much like the finite-temperature case.
A mean-field spherical model with random couplings between pairs, quartets, and possibly higher multiplets of spins is considered. It has the same critical behavior as the Sherrington-Kirkpatrick model. It thus exhibits replica symmetry breaking. The order parameter function is solved exactly in the whole low temperature phase. The zero-field-cooled susceptibility remains finite at low TTT. Next a quantum version of the system is considered. Whereas the magnetic properties are not altered qualitatively, the thermodynamics is now regular at small temperatures.
We construct a complete numerically exact solution of a mean-field quantum spin glass model---the transverse field Sherrington-Kirkpatrick model---by implementing a continuous-time quantum Monte Carlo method in the presence of full replica symmetry breaking. We extract the full numerically exact phase diagram, displaying a glassy phase with continuous replica symmetry breaking at small transverse fields and low temperatures. A paramagnetic phase emerges once thermal and quantum fluctuations melt the spin glass. We characterize both phases by extracting the order parameter as well as the static and dynamical local spin susceptibilities. The static susceptibility shows a plateau in the glassy phase, but it remains smooth across the phase boundary. For the imaginary part of the dynamical susceptibility, we find an Ohmic, i.e., linear in $\ensuremath{\omega}$, scaling for small frequencies $\ensuremath{\omega}$, with a slope independent of the transverse field. These results compare qualitatively well with ac susceptibility measurements on a dipole-coupled three-dimensional Ising magnet---the ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}$ compound---in a transverse magnetic field. Our work provides a general framework for the exact numerical solution of mean-field quantum glass models, and it constitutes an important step towards understanding glassiness in realistic systems.