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Physical review. B, Condensed matter

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

W. L. McMillan

Published Jan 1, 1985• 109 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 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 ${T}_{g}$=(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.

DOI
10.1103/physrevb.31.340

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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 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 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 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.
An Ising spin glass of random bonds with the Gaussian and two-peaked distributions in two- and three-dimensions is investigated with the cumulant expansion and two-step decimation transformations. It is found that the spin glass state exists in two- and three-dimensions and that the ferromagnetic phase shrinks to that of the pure system in two-dimension for the two-peaked distribution. The specific heat exponent at the spin glass transition point is found to be -1.08 and -0.471 for the square and simple cubic lattices, respectively. Also the effect of bond correlations on the spin glass state is considered.