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arXiv:cond-mat/9910390 · Journal of Physics A Mathematical and General

Absence of ageing in the remanent magnetization in Migdal-Kadanoff spin glasses

Federico Ricci‐Tersenghi, Félix Ritort

Published May 11, 2000• 10 citations
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Abstract

We study the non-equilibrium behaviour of three-dimensional spin glasses in the Migdal-Kadanoff approximation. This approximation is exact for disordered hierarchical lattices which have a unique ground state and equilibrium properties correctly described by the droplet model. Extensive numerical simulations show that this model lacks ageing in the remanent magnetization as well as a maximum in the magnetic viscosity in disagreement with experiments as well as with numerical studies of the Edwards-Anderson model. This result strongly limits the validity of the droplet model (at least in its simplest form) as a good model for real spin glasses.

DOI
10.1088/0305-4470/33/20/301

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We study the non-equilibrium behaviour of three-dimensional spin glasses in the Migdal-Kadanoff approximation. This approximation is exact for disordered hierarchical lattices which have a unique ground state and equilibrium properties correctly described by the droplet model. Extensive numerical simulations show that this model lacks ageing in the remanent magnetization as well as a maximum in the magnetic viscosity in disagreement with experiments as well as with numerical studies of the Edwards-Anderson model. This result strongly limits the validity of the droplet model (at least in its simplest form) as a good model for real spin glasses.
The aging phenomena in Ising spin glasses in a magnetic field are investigated both by the numerical simulation on three-dimensional Ising Edwards-Anderson model and by the experiment on the Fe0.55Mn0.45TiO3 Ising spin glass. After reviewing our recent numerical work on the field-shift aging protocol, by which we argue that the spin-glass (SG) phase is unstable in a static magnetic field of any strength, we present the nonequilibrium aspects of the field-cooled magnetization in a relatively small field observed numerically and experimentally. The results are compatible with the absence of the equilibrium SG phase in a field.
We have studied the Parisi overlap distribution for the three-dimensional Ising spin glass in the Migdal-Kadanoff approximation. For temperatures $T\ensuremath{\simeq}{0.7T}_{c}$ and system sizes up to L=32L\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}32L=32, we found a P(q)P(q)P(q) as expected for full Parisi replica-symmetry breaking, just as was also observed in recent Monte Carlo simulations on a cubic lattice. However, for lower temperatures our data agree with predictions from the droplet or scaling picture. The failure to see droplet model behavior in Monte Carlo simulations is due to the fact that all existing simulations are done at temperatures too close to the transition temperature so that system sizes larger than the correlation length have not been achieved.
We have argued in recent papers that the Monte Carlo results for the equilibrium properties of the Edwards-Anderson spin glass in three dimensions, which had been interpreted earlier as providing evidence for replica symmetry breaking, can be explained quite simply within the droplet model once finite-size effects and proximity to the critical point are taken into account. In this paper we show that similar considerations are sufficient to explain the Monte Carlo data in four dimensions. In particular, we study the Parisi overlap and the link overlap for the four-dimensional Ising spin glass in the Migdal-Kadanoff approximation. Similar to what is seen in three dimensions, we find that temperatures well below those studied in the Monte Carlo simulations have to be reached before the droplet model predictions become apparent. We also show that the double-peak structure of the link overlap distribution function is related to the difference between domain-wall excitations that cross the entire system and droplet excitations that are confined to a smaller region.
The low temperature dynamics of the three dimensional Ising spin glass in zero field with a discrete bond distribution is investigated via MC simulations. The thermoremanent magnetization is found to decay algebraically and the temperature dependent exponents agree very well with the experimentally determined values. The nonequilibrium autocorrelation function C(t,tw) shows a crossover at the waiting (or aging) time tw from algebraic quasi-equilibrium decay for times t≪tw to another, faster algebraic decay for t≫tw with an exponent similar to one for the remanent magnetization.
Some recent experiments on the non‐equilibrium dynamics in two‐ and three‐dimensional spin glasses are reviewed. The experimental results are discussed within new domain (droplet) theories of the spin glass phase. It is found that the experiments convincingly verify some fundamental concepts of these theories. Especially, it is shown that if a spin glass has been ‘‘aged’’ at constant temperature and afterwards subjected to a persistent temperature shift or a temperature cycling, the dynamics are governed by an interplay between the growth of spin glass domains and the overlap length, which is the maximum length scale where equilibrium spin correlations at two nearby temperatures are indistinguishable.