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Physical Review Letters

Ising spin-glass order in vector spin glasses with random anisotropy

Michel J. P. Gingras

Published Sep 6, 1993• 19 citations
Publisher

Abstract

Vector spin glasses with random short-range anisotropic couplings are studied using the defect-wall renormalization-group method. By calculating the correlation length exponent \ensuremath{\nu} and the fractal dimension of the defect wall, ${\mathit{d}}_{\mathit{s}}$, we obtain the first direct evidence that in two dimensions these systems have a zero-temperature transition in the same universality class as the Ising spin glass. We also present results which strongly suggest that these systems have a transition at nonzero temperature in three dimensions.

DOI
10.1103/physrevlett.71.1637

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Vector spin glasses with random short-range anisotropic couplings are studied using the defect-wall renormalization-group method. By calculating the correlation length exponent \ensuremath{\nu} and the fractal dimension of the defect wall, ds{\mathit{d}}_{\mathit{s}}ds​, we obtain the first direct evidence that in two dimensions these systems have a zero-temperature transition in the same universality class as the Ising spin glass. We also present results which strongly suggest that these systems have a transition at nonzero temperature in three dimensions.
The dependence of the effective coupling J'(L,m) on length scale L and spin dimension m is studied numerically for vector spin glasses using the 'defect energy' method for space dimension d=2,3 (m=2,3) and 1<or=d<or=4 (m= infinity ). Scaling towards weak coupling, J'(L,m) approximately L -1 nu /, L>>L c (m), is found in all cases studied implying a power-law divergence of the correlation length, xi approximately T - nu , for T to 0 but no finite-temperature transition. The characteristic length L c (m), which increases with m, separates a preasymptotic (small L) regime where m= infinity behaviour is observed, J'(L,m) approximately=J'(L, infinity ), from the asymptotic (large-L) regime which gives the true critical behaviour for finite m. Values of nu (m,d) obtained are nu ( infinity ,2) approximately=0.65, nu ( infinity ,3) approximately=1.0, nu ( infinity ,4) approximately=1.5 nu (2,2) approximately=1,1, nu (2.3) approximately=2.2. The m=2 exponents are in good agreement with a Migdal-Kadanoff-like renormalisation group calculation. For d=1 the unexpected results nu approximately=O for m= infinity is obtained if frustration is included. For d=3, anisotropy is expected to induce an Ising-like phase transition at a finite temperature, and a crossover expression for the transition temperature is derived.
The random anisotropy axis model (RAM) in two dimensions (d=2) is studied in the limit of infinite anisotropy strength using the large-cell renormalisation group method. It is found that the model exhibits a 'zero-temperature phase transition', to a nonferromagnetic ground state, with critical exponents consistent with those of the d=2 Ising spin glass model. It is concluded that the RAM belongs to the universality class of the Ising spin glass, at least for d=2 and probably for all d<or=4.
We present the results of numerical simulations on Ising spin glasses in zero magnetic field with nearest-neighbor interactions on hyper-cubic lattices in two, three, and four dimensions with both Gaussian and \ifmmode\pm\else\textpm\fi{}J bond distributions. Finite-size scaling is used to analyze the results. In two dimensions (d=2) we agree with earlier work that the transition temperature is at Tc{T}_{c}Tc​=0, and obtain the correlation-length exponent \ensuremath{\nu}, and the exponent \ensuremath{\eta}, at the zero-temperature transition for the \ifmmode\pm\else\textpm\fi{}J model. In d=3 dimensions we concentrate on results for the Gaussian distribution, since our results for the \ifmmode\pm\else\textpm\fi{}J distribution have been presented earlier. As expected, we find similar results for the two distributions, namely a nonzero Tc{T}_{c}Tc​ but evidence that d=3 is close to the lower critical dimension. In a four-dimensional spin glass with Gaussian bonds we find that only a modest amount of computer time is required to show that Tc{T}_{c}Tc​ is nonzero with a long-range-ordered phase below Tc{T}_{c}Tc​. Our estimates for critical exponents in d=4 dimensions agree well with results from recent high-temperature-series expansions.
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 $\ensuremath{\alpha}=\ensuremath{-}7\ifmmode\pm\else\textpm\fi{}3$.