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Progress of Theoretical Physics Supplement

Monte Carlo Simulation of the ±JHeisenberg Model in Parisi States

Fumitaka Matsubara, Shinsuke Takahashi, Takayuki Shirakura

Published Jan 1, 2005• 0 citations
Publisher

Abstract

We performed Monte Carlo simulation of the three dimensional ±J Heisenberg model in a Parisi state. That is, having started with one of Parisi states A, we calculated the magnetization MA(t) projected on A. We found that at low temperatures (T ≪ TSG) MA(t) for small t exhibits a relaxation similar to that of the ferromagnetic phase, where TSG is the spin glass transition temperature estimated before. From this behavior, we estimated temperature dependence of the order parameter of that Parisi state.

DOI
10.1143/ptps.157.50

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We performed Monte Carlo simulation of the three dimensional ±J Heisenberg model in a Parisi state. That is, having started with one of Parisi states A, we calculated the magnetization MA(t) projected on A. We found that at low temperatures (T ≪ TSG) MA(t) for small t exhibits a relaxation similar to that of the ferromagnetic phase, where TSG is the spin glass transition temperature estimated before. From this behavior, we estimated temperature dependence of the order parameter of that Parisi state.
The three-dimensional Heisenberg spin glass model with ±J interactions is studied using an over-relaxed Monte Carlo algorithm. We have measured the correlation length ξ, and using the crossing of ξ∕L for different L find a finite temperature spin glass transition, with TSG=0.220(5). In addition, we have varied the number of Monte Carlo steps used prior to, and during, thermal averaging to control the effects of finite time averaging. We find that the over-relaxation algorithm allows for an accurate measurements using as few as 300 Monte Carlo steps.
We describe results of Monte Carlo simulation studies on the Ising spin glass in four dimensions on a hypercubic lattice with nearest neighbor bonds. Studies of the equilibrium static properties show that the system undergoes a genuine phase transition to an ordered spin glass phase. Critical dynamical behavior is analyzed to obtain the dynamic exponent. Finally, we describe results on the spin glass phase, in particular the finite size scaling of the order parameter distribution function, and compare it with existing models of the spin glass phase, namely the droplet model and the Parisi solution for the low temperature phase of the infinite range spin glass.
We study the phase transition of the ±J\pm J±J Heisenberg model in three dimensions. Using a dynamical simulation method that removes a drift of the system, the existence of the spin-glass (SG) phase at low temperatures is suggested. The transition temperature is estimated to be TSG∼0.18JT_{\rm SG} \sim 0.18JTSG​∼0.18J from both equilibrium and off-equilibrium Monte-Carlo simulations. Our result contradicts the chirality mechanism of the phase transition reported recently by Kawamura which claims that it is not the spins but the chiralities of the spins that are ordered in Heisenberg SG systems.
The recent interpretation of Parisi's order-parameter function q(x)q(x)q(x) in terms of a probability distribution for the overlap between magnetizations in different phases is investigated by Monte Carlo computer simulation for the infinite-range Ising spin-glass model. The main features of the solution for q(x)q(x)q(x) are reproduced, in particular $q(x)\ensuremath{\propto}x$ as $x\ensuremath{\rightarrow}0$ and ${q}^{\ensuremath{'}}(x)=0$ at q=qmaxq={q}_{max}q=qmax​, the largest value. Finite-size effects prevent one from establishing with certainty whether there is a "plateau," i.e., ${q}^{\ensuremath{'}}(x)=0$ for a range of xxx.
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.