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Physical review. B, Condensed matter

Computer simulation of the three-dimensional short-range Heisenberg spin glass

J. A. Olive, A. P. Young, D. Sherrington

Published Nov 1, 1986• 166 citations
Publisher

Abstract

The results of large-scale Monte Carlo simulations on a nearest-neighbor Heisenberg spin-glass model are presented. We find that the transition is at zero temperature and that 3 is below the lower critical dimension. Relaxation times follow conventional dynamical scaling and diverge with a power of the temperature T as T\ensuremath{\rightarrow}0.

DOI
10.1103/physrevb.34.6341

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The results of large-scale Monte Carlo simulations on a nearest-neighbor Heisenberg spin-glass model are presented. We find that the transition is at zero temperature and that 3 is below the lower critical dimension. Relaxation times follow conventional dynamical scaling and diverge with a power of the temperature T as T\ensuremath{\rightarrow}0.
It is shown, by means of Monte Carlo simulation and finite size scaling analysis, that the Heisenberg spin glass undergoes a finite-temperature phase transition in three dimensions. There is a single critical temperature, at which both a spin glass and a chiral glass ordering develop. The Monte Carlo algorithm, adapted from lattice gauge theory simulations, makes it possible to thermalize lattices of size L = 32, larger than in any previous spin-glass simulation in three dimensions. High accuracy is reached thanks to the use of the Marenostrum supercomputer. The large range of system sizes studied allows us to consider scaling corrections.
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The three-dimensional Ising spin-glass in zero field with nearest-neighbor interactions having \ifmmode\pm\else\textpm\fi{}J distribution is studied by Monte Carlo simulations for samples of linear dimension L with 3\ensuremath{\le}L\ensuremath{\le}20. Results for the probability distribution, PL{\mathrm{P}}_{\mathrm{L}}PL​(q), of the spin-glass order parameter are analyzed by finite-size scaling. Data for T\ensuremath{\ge}1.2 are consistent with a conventional phase transition at Tc{\mathrm{T}}_{\mathrm{c}}Tc​${=1.2}_{\mathrm{\ensuremath{-}}0.2}^{+0.1}$, with exponents \ensuremath{\nu}=1.3\ifmmode\pm\else\textpm\fi{}0.3 and \ensuremath{\eta}=-0.3\ifmmode\pm\else\textpm\fi{}0.2. However, results at lower temperature indicate marginal behavior and suggest that the lower critical dimensional is close to three.
A spin-glass transition has been investigated for a long time but we have not reached a conclusion yet due to difficulties in the simulation studies. They are slow dynamics, strong finite-size effects, and sample-to-sample dependencies. We found that a size of the spin-glass order reaches a lattice boundary within a very short Monte Carlo step. A competition between the spin-glass order and a boundary condition causes these difficulties. Once the boundary effect was removed, physical quantities exhibited quite normal behaviors. They became self-averaging in a limit of large replica numbers. These findings suggest that the nonequilibrium relaxation method is a good choice for solving the difficulties if a lattice size and a replica number are set sufficiently large. A dynamic scaling analysis on nonequilibrium relaxation functions gave a result that the spin-glass transition and the chiral-glass transition occurs at the same temperature in the Heisenberg model in three dimensions. The estimated critical exponent ν agrees with the experimental result.
We study the XY spin glass by large-scale Monte Carlo simulations for sizes up to 243{24}^{3}243, down to temperatures below the transition temperature found in earlier work. The data for the larger sizes show more marginal behavior than that for the smaller sizes, indicating that the lower critical dimension is close, and possibly equal to three. We find that the spins and chiralities behave in a similar manner. We also address the optimal ratio of ``over-relaxation'' to ``Metropolis'' sweeps in the simulation.
A short-range \ifmmode\pm\else\textpm\fi{}J Heisenberg spin glass with and without anisotropy on a simple cubic lattice is studied by using a hybrid Monte Carlo spin-dynamics method. The anisotropy leads to a cusplike peak of the susceptibility \ensuremath{\chi} and a very slow relaxation of the spin configuration at low temperatures. From a finite-size-scaling analysis of the spin-glass susceptibility ${\mathrm{\ensuremath{\chi}}}_{\mathrm{sg}}$, we find, for the first time, that the anisotropy induces a phase transition.