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arXiv:cond-mat/0007113

Spin Glasses at Low Temperatures

Helmut G. Katzgraber, Matteo Palassini, A. P. Young

Published Jul 6, 2000• 2 citations
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Abstract

We report the results of Monte Carlo simulations on several spin glass models at low temperatures. By using the parallel tempering (Exchange Monte Carlo) technique we are able to equilibrate down to low temperatures, for moderate sizes, and hence the data should not be affected by critical fluctuations. Our results for short range models are consistent with a picture proposed earlier that there are large scale excitations which cost only a finite energy in the thermodynamic limit, and these excitations have a surface whose fractal dimension is less than the space dimension. For the infinite range Viana-Bray model, our results obtained for a similar number of spins are consistent with standard replica symmetry breaking.

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•Jan 1, 2011•0 citations
We report the results of Monte Carlo simulations on several spin glass models at low temperatures. By using the parallel tempering (Exchange Monte Carlo) technique we are able to equilibrate down to low temperatures, for moderate sizes, and hence the data should not be affected by critical fluctuations. Our results for short range models are consistent with a picture proposed earlier that there are large scale excitations which cost only a finite energy in the thermodynamic limit, and these excitations have a surface whose fractal dimension is less than the space dimension. For the infinite range Viana-Bray model, our results obtained for a similar number of spins are consistent with standard replica symmetry breaking.
The nature of the spin glass state is investigated by studying changes to the ground state when a weak perturbation is applied to the bulk of the system. We consider short range models in three and four dimensions and the infinite range Sherrington-Kirkpatrick (SK) and Viana-Bray models. Our results for the SK and Viana-Bray models agree with the replica symmetry breaking picture. The data for the short range models fit naturally a picture in which there are large scale excitations which cost a finite energy but whose surface has a fractal dimension, d(s), less than the space dimension d. We also discuss the possible crossover to other behavior at larger length scales than the sizes studied.
Results are presented for the geometry of low-energy excitations in the one-dimensional Ising spin chain with power-law interactions, in which the model parameters are chosen to yield a finite spin-glass transition temperature. Both finite-temperature and ground-state studies are carried out. For the range of sizes studied the data cannot be fitted to any of the standard spin-glass scenarios without including corrections to scaling. Incorporating such corrections we find that the fractal dimension of the surface of the excitations, is either equal to the space dimension, consistent with replica symmetry breaking, or very slightly less than it. The latter case is consistent with the droplet and ``trivial-nontrivial'' pictures.
Some recent progress in Monte Carlo simulations of spin glasses will be presented. The problem of slow dynamics at low temperatures is partially alleviated by use of the parallel tempering (replica exchange) method. A useful technique to check for equilibration (applicable only for a Gaussian distribution) will be discussed. It will be argued that a finite size scaling analysis of the scaled correlation length of the system is a good approach with which to investigate phase transitions in spin glasses. This method will be used to study two questions: (i) whether there is a phase transition in zero field in the Heisenberg spin glass in three dimensions, and (ii) whether there is phase transition in a magnetic field in an Ising spin glass, also in three dimensions.
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 study the low-energy excitations of the three dimensional spin glass through a large-scale Monte Carlo simulation on lattices up to L=18L=18L=18. We find smooth extrapolations down to zero temperature, which, in the case of the energy and of the link overlap, can be directly -- and favourably -- compared with previous investigations featuring ground states (i.e., at zero temperature). The best fit for the fractal dimension of the excitations is provided by Replica-Symmetry Breaking theory, but we also consider the alternative TNT description. The P(q)P(q)P(q) is found to verify the Parisi-Toulouse temperature scaling. Our data provides a spectacular confirmation of the overlap-equivalence hypothesis.
We investigate the low temperature phase of a vector spin glass in the limit of large component numbers mmm of the spins, where this model is expected to be replica symmetric. On a Bethe lattice, a random graph with fixed and finite connectivity, we employ a self-consistent effective-field approach to the cavity method to describe equilibrium states of the system in terms of the configuration of cavity fields, which are reduced mean fields onto each spin. We make use of an iterative algorithm to find such equilibrium states for finite system sizes NNN and observe a phase transition, which is indicated by an order parameter describing the preferred direction each spin points into in terms of the configuration of cavity, resp. mean local fields. On single realizations of Bethe lattices the transition is sharp and accompanied by the Generalized-Bose-Einstein condensation, in which the cavity, resp. local fields of the spins condense into a subspace of the available mmm-dimensional space for large mmm. The dimension of this subspace, meff(T)m_{\rm eff}(T)meff​(T) depends on the temperature. Is either one or two right at the critical temperature and increases below. Furthermore, for fixed temperature it is expected to scale with the system size NNN with an exponent μm\mu_mμm​ which depends on the connectivity. Averaging over many realizations of disorder we find meff‾(Tc)=2\overline{m_{\rm eff}}(T_c)=2meff​​(Tc​)=2 and meff‾(T)∼Nμm\overline{m_{\rm eff}}(T)\sim N^{\mu_m}meff​​(T)∼Nμm​. Even more, the cavity fields are subject to rotations in the low temperature phase, i.e. there is no freezing in time as generally expected. In case of meff(Tc)=2m_{\rm eff}(T_c)=2meff​(Tc​)=2 the whole configuration rotates at the critical temperature about the same angle and the same axis of rotation corresponding to a single droplet. In addition, we find replica symmetry breaking below the large-mmm limit, i.e. for meff(T)>mm_{\rm eff}(T) > mmeff​(T)>m. These results are supported by results of parallel tempering Monte Carlo simulations. Replica symmetry breaking is further investigated by an analytical method in the large connectivity limit, where we compare the free energy in terms of spin-spin-correlations of different replicas calculated with a replica symmetric and a one-step broken replica symmetry ansatz. In the large-mmm limit, we find that both solutions coexist for finite connectivities and that replica symmetry holds for full connectivity up to corrections in second order around the saddle point for m→∞m\to\inftym→∞. In the last chapter we calculate the sample-to-sample fluctuations of the free energy of the SK model of the large-mmm limit. We derive an exact connection between these fluctuations and bond chaos. Making use of a replica calculation for the {\it large deviations} and a scaling ansatz for the {\it small deviations}, we quantify bond chaos, i.e. calculate the probability of finding a given link overlap for two systems with possibly different sets of bonds. With this, we find the sample-to-sample fluctuations to scale with the number of spins, NNN, with an exponent μ\muμ in between 1/51/51/5 and 3/103/103/10.