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Physical review. B, Condensed matter

Monte Carlo renormalization-group study of Ising spin glasses

Jian‐Sheng Wang, Robert H. Swendsen

Published May 1, 1988• 34 citations
Publisher

Abstract

We present a new Monte Carlo renormalization-group approach to spin glasses. Application of this approach to the \ifmmode\pm\else\textpm\fi{}J Ising spin glass shows clear differences between two-, three-, and four-dimensional models. The data are consistent with a zero-temperature transition in two dimensions, and finite-temperature transitions in three and four dimensions.

DOI
10.1103/physrevb.37.7745

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We present a new Monte Carlo renormalization-group approach to spin glasses. Application of this approach to the \ifmmode\pm\else\textpm\fi{}J Ising spin glass shows clear differences between two-, three-, and four-dimensional models. The data are consistent with a zero-temperature transition in two dimensions, and finite-temperature transitions in three and four dimensions.
The three-dimensional random Ising model with a Gaussian distribution of nearest-neighbor interactions is studied for the pure spin-glass case where the average interaction vanishes. The distribution of domain-wall energies at zero temperature is calculated using a Monte Carlo-quench algorithm to find the ground-state energy for finite lattices. A renormalization-group transformation is set up which preserves the domain-wall energy distribution when the lattice parameter is changed. In the strong-coupling regime (zero temperature) the model iterates toward strong coupling and therefore exhibits a spin-glass phase transition at nonzero temperature. The thermal exponent is $\ensuremath{\nu}=3.0\ifmmode\pm\else\textpm\fi{}1.0$ and the heat capacity exponent is $\ensuremath{\alpha}=\ensuremath{-}7\ifmmode\pm\else\textpm\fi{}3$.
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.
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.
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.