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Physical review. B, Solid state

Mean field and ε -expansion study of spin glasses

Jing-Huei Chen, T. C. Lubensky

Published Sep 1, 1977• 155 citations
Publisher

Abstract

A Landau-Ginzburg phenomenological free energy for the Edwards and Anderson spin-glass model when there is competition between spin-glass and ferromagnetic ordering is developed. This free energy obtained with the use of the $n\ensuremath{\rightarrow}0$ replication procedure is analyzed using mean-field theory and the $\ensuremath{\epsilon}$ expansion. Critical exponents for the ferromagnetic-spin-glass multicritical point are calculated in $6\ensuremath{-}\ensuremath{\epsilon}$ dimensions. For Ising systems, $\ensuremath{\nu}=\frac{1}{2}+(\frac{2}{3})\ensuremath{\epsilon}$ and $\ensuremath{\varphi}=1+(\frac{1}{2})\ensuremath{\epsilon}$. For $\mathrm{XY}$ and Heisenberg systems, these exponents are complex. This result is not fully understood. The Harris-Plischke-Zuckermann model for amorphous magnetism is shown to have an Ising-like spin-glass fixed point in high enough dimension.

DOI
10.1103/physrevb.16.2106

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A Landau-Ginzburg phenomenological free energy for the Edwards and Anderson spin-glass model when there is competition between spin-glass and ferromagnetic ordering is developed. This free energy obtained with the use of the n\ensuremath→0n\ensuremath{\rightarrow}0n\ensuremath→0 replication procedure is analyzed using mean-field theory and the \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ expansion. Critical exponents for the ferromagnetic-spin-glass multicritical point are calculated in 6\ensuremath−\ensuremathϵ6\ensuremath{-}\ensuremath{\epsilon}6\ensuremath−\ensuremathϵ dimensions. For Ising systems, \ensuremathν=12+(23)\ensuremathϵ\ensuremath{\nu}=\frac{1}{2}+(\frac{2}{3})\ensuremath{\epsilon}\ensuremathν=21​+(32​)\ensuremathϵ and \ensuremathφ=1+(12)\ensuremathϵ\ensuremath{\varphi}=1+(\frac{1}{2})\ensuremath{\epsilon}\ensuremathφ=1+(21​)\ensuremathϵ. For XY\mathrm{XY}XY and Heisenberg systems, these exponents are complex. This result is not fully understood. The Harris-Plischke-Zuckermann model for amorphous magnetism is shown to have an Ising-like spin-glass fixed point in high enough dimension.
We have developed 15th-order high-temperature series expansions for the study of the critical behavior of the Ising spin glass with nearest-neighbor exchange interactions each of which assumes the values \ifmmode\pm\else\textpm\fi{}J randomly. Series for the Edwards-Anderson spin-glass susceptibility (${\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}$) and two of its derivatives with respect to the ordering field have been evaluated for hypercubic lattices in general dimension, d. These extend previous general-dimension series by five terms. Certain measurable universal amplitude ratios have been estimated from the new series. Accurate critical data for d=5 and the first reliable estimates of the exponent \ensuremath{\beta} for d=4 and 5, are given. We quote \ensuremath{\gamma}=1.73\ifmmode\pm\else\textpm\fi{}0.03, 2.00\ifmmode\pm\else\textpm\fi{}0.25, and 2.${7}_{\mathrm{\ensuremath{-}}0.6}^{+1.0}$ and \ensuremath{\beta}=0.95\ifmmode\pm\else\textpm\fi{}0.04, 0.9\ifmmode\pm\else\textpm\fi{}0.1, and 0.7\ifmmode\pm\else\textpm\fi{}0.2 in 5, 4, and 3 dimensions, respectively. Our results provide a smooth extrapolation between the mean-field results above six dimensions and experiments and simulations in physical dimensions. We relate our calculated derivatives of ${\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}$ to measurements of derivatives of the magnetization with respect to a uniform magnetic field.
The large scale behavior of the simplest non-mean-field spin-glass system is analyzed, and the critical exponent related to the divergence of the correlation length is computed at two loops within the ε-expansion technique with two independent methods. The techniques presented show how the underlying ideas of the renormalization group apply also in this disordered model, in such a way that an ε-expansion can be consistently set up. By pushing such calculation to high orders in ε, a consistent non-mean-field theory for such disordered system could be established, giving a substantial contribution the development of a predictive theory for real spin glasses.
In a recent work [M. Castellana and G. Parisi, Phys. Rev. E 82, 040105(R) (2010)], the large-scale behavior of the simplest non-mean-field spin-glass system has been analyzed, and the critical exponent related to the divergence of the correlation length has been computed at two loops within the ε-expansion technique by two independent methods. By performing the explicit calculation of the critical exponents at two loops, one obtains that the two methods yield the same result. This shows that the underlying renormalization group ideas apply consistently in this disordered model, in such a way that an ε-expansion can be set up. The question of the extension to high orders of this ε-expansion is particularly interesting from the physical point of view. Indeed, once high orders of the series in ε for the critical exponents are known, one could check the convergence properties of the series, and find out if the ordinary series resummation techniques, yielding very accurate predictions for the Ising model, work also for this model. If this is the case, a consistent and predictive non-mean-field theory for such a disordered system could be established. In that regard, in this work we expose the underlying techniques of such a two-loop computation. We show with an explicit example that such a computation could be quite easily automatized, i.e., performed by a computer program, in order to compute high orders of the ε-expansion, and so eventually make this theory physically predictive. Moreover, all the underlying renormalization group ideas implemented in such a computation are widely discussed and exposed.
The critical properties of the model of a spin-glass proposed by Edwards and Anderson are studied using the renormalization group. The critical exponents are calculated in $6\ensuremath{-}\ensuremath{\epsilon}$ spatial dimensions. It is argued that a tricritical point can exist where the nonordering field is the skewness of the distribution of JJJ.
The replica-symmetric cluster expansion for the Ising spin glass is reviewed and explained. The expansion allows computation of various quantities and correlations directly in finite dimensions, without recourse to the mean-field theory or the saddle point. The expansion for the free energy is recovered with the use of external fields that couple to the replica-symmetric Edwards-Anderson order parameter. The expansion scheme is applied to the Edwards-Anderson susceptibility. The expansion diverges in the mean-field [infinite dimensionality] limit. The divergent expansion, when naively summed, yields the negative replicon mass of the mean-field theory. This so-called Almeida-Thouless instability of the replica-symmetric mean-field theory is argued to be absent in finite dimensions. The argument, similar to the one already reported for the entropy, depends on the low-temperature behavior of terms in the expansion at finite dimensions. \textcopyright{} 1996 The American Physical Society.
By mapping the Edwards-Anderson model onto a nonrandom Ising model we argue that the lower critical dimension d∗{d}^{*}d∗ for the Edwards-Anderson spin-glass is close to 3 and may depend on the probability distribution of the bond strengths. For d>d∗d>{d}^{*}d>d∗, the approach to spin-glass order from the paramagnetic phase is characterized by Ising exponents. At d=d∗d={d}^{*}d=d∗, however, the critical exponents defined for T>TcT>{T}_{c}T>Tc​ are twice the corresponding Ising ones. These results are consistent with recent computer simulations.