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Journal of Physics C Solid State Physics

Real space rescaling study of spin glass behaviour in three dimensions

B. W. Southern, A. P. Young

Published Jun 28, 1977• 177 citations
Publisher

Abstract

The Edwards and Anderson model (1975) for spin glasses is investigated by a real space rescaling transformation on an S= 1 / 2 Ising Hamiltonian with nearest-neighbour-only interactions in three dimensions. A transition to a spin glass ordered phase is found at about half the mean transition temperature, in contrast to earlier work on a two-dimensional model where no transition was found. The results suggest that the lower critical dimensionality, d c , for spin glass order is slightly greater than 2, although possibly d c =2 exactly. The specific heat is calculated and found to exhibit a broad maximum above T c , with no discernable singularity at the transition since the exponent alpha is large and negative.

DOI
10.1088/0022-3719/10/12/023

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The Edwards and Anderson model (1975) for spin glasses is investigated by a real space rescaling transformation on an S= 1 / 2 Ising Hamiltonian with nearest-neighbour-only interactions in three dimensions. A transition to a spin glass ordered phase is found at about half the mean transition temperature, in contrast to earlier work on a two-dimensional model where no transition was found. The results suggest that the lower critical dimensionality, d c , for spin glass order is slightly greater than 2, although possibly d c =2 exactly. The specific heat is calculated and found to exhibit a broad maximum above T c , with no discernable singularity at the transition since the exponent alpha is large and negative.
The Edwards and Anderson model for spin glasses is investigated by a decimation rescaling transformation applied to a spin- 1 / 2 Ising model. The technique reproduces the exactly known results for a linear chain and, in particular, predicts ordering to a spin-glass type ground state at T=0. For a two-dimensional square lattice it is shown that approximations which give qualitatively correct results in other situations predict no spin-glass transition at any temperature. A comparison is made with results for spin glasses on a Bethe lattice and with an infinite-range spin-glass model, which has been found to exhibit a transition. It is argued that the difference between infinite-range and short-range models is that the former has entirely extended eigenvectors of the random J ij matrix whereas the latter has at least some localized eigenvectors.
The real space block spin renormalisation method is applied to an Ising model spin glass with various symmetric distributions of nearest-neighbour interactions. For different blocks in two and three dimensions, the first-order cumulant approximation yields a phase transition with a critical exponent alpha <or approximately=-2. However, two and three block-spin cluster approximations do not yield a phase transition. The authors conclude that previous results obtained by them and by other authors on this level of approximation are inconclusive.
The Edwards-Anderson model of a spin glass is studied by position-space renormalization-group techniques, using an inhomogeneous generalization of Migdal's approximate recursion relation. We treat the spin-1/2 Ising model with independently random nearest-neighbor interactions in dimensionalities d=2,3,and4d=2, 3, \mathrm{and} 4d=2,3,and4. The phase diagram, which is in qualitative agreement with mean-field results, exhibits paramagnetic, ferromagnetic, antiferromagnetic, and spin-glass phases. The spin-glass and paramagnetic phases meet along an extended second-order phase boundary, which terminates in two tricritical points. Critical and tricritical exponents are calculated. The spin-glass specific-heat exponent turns out to be large and negative, compatibly with recent experiments which show a rounded specific-heat anomaly.
We study spin glass behavior in a random Ising Coulomb antiferromagnet in two and three dimensions using Monte Carlo simulations. In two dimensions, we find a transition at zero temperature with critical exponents consistent with those of the Edwards-Anderson model, though with large uncertainties. In three dimensions, evidence for a finite-temperature transition, as occurs in the Edwards-Anderson model, is rather weak. This may indicate that the sizes are too small to probe the asymptotic critical behavior, or possibly that the universality class is different from that of the Edwards-Anderson model and has a lower critical dimension equal to three.
We report a high-precision finite-size scaling study of the critical behavior of the three-dimensional Ising Edwards-Anderson model (the Ising spin glass). We have thermalized lattices up to L=40L=40L=40 using the Janus dedicated computer. Our analysis takes into account leading-order corrections to scaling. We obtain Tc=1.1019(29){T}_{\mathrm{c}}=1.1019(29)Tc​=1.1019(29) for the critical temperature, $\ensuremath{\nu}=2.562(42)$ for the thermal exponent, $\ensuremath{\eta}=\ensuremath{-}0.3900(36)$ for the anomalous dimension, and $\ensuremath{\omega}=1.12(10)$ for the exponent of the leading corrections to scaling. Standard (hyper)scaling relations yield $\ensuremath{\alpha}=\ensuremath{-}5.69(13)$, $\ensuremath{\beta}=0.782(10)$, and $\ensuremath{\gamma}=6.13(11)$. We also compute several universal quantities at Tc{T}_{\mathrm{c}}Tc​.