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arXiv:cond-mat/9307065

Chaos in short-range spin glasses

Félix Ritort

Published Jul 29, 1993• 1 citation
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Abstract

The nature of static chaos in Ising spin glasses is studied. For the problem of chaos with magnetic field, scaling relations in the case of the SK model and short-range models are presented. Our results also suggest that if there is de Almeida-Thouless line then it is similar to that of mean-field theory for d=4 and close to the h=0 axis for d=3. We estimate the lower critical dimension to be in the range 2.7-2.9. Numerical studies at d=4 show that there is chaos against temperature changes dT and the correlation length diverges like 1/dT.

DOI
10.48550/arxiv.cond-mat/9307065

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The nature of static chaos in Ising spin glasses is studied. For the problem of chaos with magnetic field, scaling relations in the case of the SK model and short-range models are presented. Our results also suggest that if there is de Almeida-Thouless line then it is similar to that of mean-field theory for d=4 and close to the h=0 axis for d=3. We estimate the lower critical dimension to be in the range 2.7-2.9. Numerical studies at d=4 show that there is chaos against temperature changes dT and the correlation length diverges like 1/dT.
We discuss the problem of static chaos in spin glasses. In the case of magnetic-field perturbations, we propose a scaling theory for the spin-glass phase. Using the mean-field approach we argue that some pure states are suppressed by the magnetic field and their free-energy cost is determined by the finite-temperature fixed-point exponents. In this framework, numerical results suggest that mean-field chaos exponents are probably exact in finite dimensions. If we use the droplet approach, numerical results suggest that the zero-temperature fixed-point exponent \ensuremath{\theta} is very close to (d-3)/2. In both approaches d=3 is the lower critical dimension in agreement with recent numerical simulations.
We use high-temperature series expansions to study the ±J Ising spin glass in a magnetic field in d-dimensional hypercubic lattices for d=5-8 and in the infinite-range Sherrington-Kirkpatrick (SK) model. The expansions are obtained in the variable w=tanh^{2}J/T for arbitrary values of u=tanh^{2}h/T complete to order w^{10}. We find that the scaling dimension Δ associated with the ordering-field h^{2} equals 2 in the SK model and for d≥6. However, in agreement with the work of Fisher and Sompolinsky [Phys. Rev. Lett. 54, 1063 (1985)PRLTAO0031-900710.1103/PhysRevLett.54.1063], there is a violation of scaling in a finite field, leading to an anomalous h-T dependence of the de Almeida-Thouless (AT) [J. Phys. A 11, 983 (1978)JPHAC50305-447010.1088/0305-4470/11/5/028] line in high dimensions, whereas scaling is restored as d→6. Within the convergence of our series analysis, we present evidence supporting an AT line in d≥6. In d=5, the exponents γ and Δ are substantially larger than mean-field values, but we do not see clear evidence for the AT line in d=5.
We investigate the chaotic temperature dependence of the renormalized couplings in a short-range Ising spin glass. Some of our analytic and numerical results, obtained by the Migdal-Kadanoff renormalization scheme but argued to have general validity, differ from those of the scaling and droplet theory. First, chaos is present also at and above the critical temperature. Second, between dl{\mathit{d}}_{\mathit{l}}dl​=2.46 (the lower critical dimension) and d+{\mathit{d}}_{+}d+​=3.4, the chaos in the critical region is characterized by a new critical exponent ${\mathrm{\ensuremath{\zeta}}}_{\mathit{c}}$ that has hitherto escaped attention.
We propose a new picture of the Ising-spin-glass phase, based on an Ansatz for the scaling of low-lying large-scale-droplet excitations. We find behavior very different from the infinite-range model. The truncated spatial correlations decay as a power of distance, the ac nonlinear susceptibility diverges as a power of $\mathrm{ln}\ensuremath{\omega}$, and the magnetization noise power diverges as $\frac{1}{\ensuremath{\omega}}$ with logarithmic corrections. A magnetic field destroys the spin-glass phase so that there is no de Almeida-Thouless transition. Defect excitations should yield similar dynamic phenomena in vector spin-glasses.
Abstract We present results of numerical simulations on a one-dimensional Ising spin glass with long-range interactions. Parameters of the model are chosen such that it is a proxy for a short-range spin glass above the upper critical dimension (i.e. in the mean-field regime). The system is quenched to a temperature well below the transition temperature and the growth of correlations is observed. The spatial decay of the correlations at distances less than the dynamic correlation length agrees quantitatively with the predictions of a static theory, the ‘metastate’, evaluated according to the replica symmetry breaking (RSB) theory. We also compute the dynamic exponent z ( T ) defined by and find that it is compatible with the mean-field value of the critical dynamical exponent for short range spin glasses.
We study in Ising spin glasses the finite-size effects near the spin-glass transition in zero field and at the de Almeida-Thouless transition in a field by Monte Carlo methods and by analytical approximations. In zero field, the finite-size scaling function associated with the spin-glass susceptibility of the Sherrington-Kirkpatrick mean-field spin-glass model is of the same form as that of one-dimensional spin-glass models with power-law long-range interactions in the regime where they can be a proxy for the Edwards-Anderson short-range spin-glass model above the upper critical dimension. We also calculate a simple analytical approximation for the spin-glass susceptibility crossover function. The behavior of the spin-glass susceptibility near the de Almeida-Thouless transition line has also been studied, but here we have only been able to obtain analytically its behavior in the asymptotic limit above and below the transition. We have also simulated the one-dimensional system in a field in the non-mean-field regime to illustrate that when the Imry-Ma droplet length scale exceeds the system size one can then be erroneously lead to conclude that there is a de Almeida-Thouless transition even though it is absent.