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Physical review. B, Condensed matter

Domain-wall renormalization-group study of the three-dimensional random Ising model

W. L. McMillan

Published Jul 1, 1984• 185 citations
Publisher

Abstract

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 $\ensuremath{\alpha}=\ensuremath{-}7\ifmmode\pm\else\textpm\fi{}3$.

DOI
10.1103/physrevb.30.476

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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 \ensuremathν=3.0\ifmmode±\else\textpm\fi1.0\ensuremath{\nu}=3.0\ifmmode\pm\else\textpm\fi{}1.0\ensuremathν=3.0\ifmmode±\else\textpm\fi1.0 and the heat capacity exponent is \ensuremathα=\ensuremath−7\ifmmode±\else\textpm\fi3\ensuremath{\alpha}=\ensuremath{-}7\ifmmode\pm\else\textpm\fi{}3\ensuremathα=\ensuremath−7\ifmmode±\else\textpm\fi3.
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 free energies at finite temperature is calculated with the use of the Metropolis Monte Carlo algorithm for finite lattices. A renormalization-group transformation is set up which preserves the domain-wall free-energy distribution when the lattice parameter is changed. The spin-glass transition temperature is found to be Tg{T}_{g}Tg​=(1.0\ifmmode\pm\else\textpm\fi{}0.2)J\ifmmode \tilde{}\else \~{}\fi{} where J\ifmmode \tilde{}\else \~{}\fi{} is the variance of the Gaussian interaction distribution. The thermal exponent is \ensuremath{\nu}=1.8\ifmmode\pm\else\textpm\fi{}0.5 and the heat-capacity exponent is \ensuremath{\alpha}=-3.4\ifmmode\pm\else\textpm\fi{}1.5. The heat capacity exhibits a rounded peak at higher temperatures.
The random Heisenberg 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 spin-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 weak coupling and therefore exhibits a ``phase transition at zero temperature'' in both two and three dimensions. The correlation-length exponent is \ensuremath{\nu}=0.714\ifmmode\pm\else\textpm\fi{}0.015 in two dimensions and \ensuremath{\nu}=1.54\ifmmode\pm\else\textpm\fi{}0.19 in three dimensions. The lower critical dimension is four.
For the Ising model with Gaussian random coupling of average J0{J}_{0}J0​ and unit variance, the zero-temperature spin-glass--ferromagnetic transition as a function of the control parameter J0{J}_{0}J0​ can be studied via the size-LLL-dependent renormalized coupling defined as the domain-wall energy ${J}^{R}(L)\ensuremath{\equiv}{E}_{\text{GS}}^{(\text{AF})}(L)\ensuremath{-}{E}_{\text{GS}}^{(\text{F})}(L)$ (i.e., the difference between the ground-state energies corresponding to antiferromagnetic and ferromagnetic boundary conditions in one direction). We study numerically the critical exponents of this zero-temperature transition within the Migdal-Kadanoff approximation as a function of the dimension d=2,3,4,5,6d=2,3,4,5,6d=2,3,4,5,6. We then compare with the mean-field spherical model. Our main conclusion is that in low dimensions, the critical stiffness exponent ${\ensuremath{\theta}}^{c}$ is clearly bigger than the spin-glass stiffness exponent ${\ensuremath{\theta}}^{\text{SG}}$, but that they turn out to coincide in high enough dimension and in the mean-field spherical model. We also discuss the finite-size scaling properties of the averaged value and of the width of the distribution of the renormalized couplings.
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.