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Journal of Physics A Mathematical and General

One-dimensional Ising spin-glass model with long-range interactions

M A Moore

Published Mar 11, 1986• 11 citations
Publisher

Abstract

An Ising chain with Hamiltonian H=-J Sigma i<j epsilon ij S i S j / mod i-j mod (1+ sigma )/2 is considered where the epsilon ij are independent random variables. According to Kotliar et al. (1983) this model has a phrase transition with non-mean-field exponents when 1 / 3 < sigma <1 and mean-field exponents beyond the upper critical value of sigma = 1 / 3 . By means of an epsilon expansion about the lower critical value of sigma =1, the model is shown to be replica symmetric as epsilon to 0 and it is speculated that this result holds for 1 / 3 < sigma <1.

DOI
10.1088/0305-4470/19/4/008

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Europhysics Letters (EPL)•Mar 20, 1993•0 citations
An Ising chain with Hamiltonian H=-J Sigma i<j epsilon ij S i S j / mod i-j mod (1+ sigma )/2 is considered where the epsilon ij are independent random variables. According to Kotliar et al. (1983) this model has a phrase transition with non-mean-field exponents when 1 / 3 < sigma <1 and mean-field exponents beyond the upper critical value of sigma = 1 / 3 . By means of an epsilon expansion about the lower critical value of sigma =1, the model is shown to be replica symmetric as epsilon to 0 and it is speculated that this result holds for 1 / 3 < sigma <1.
We consider an Ising chain with Hamiltonian $H=\ensuremath{\Sigma}{i>j}^{}\frac{({\ensuremath{\epsilon}}_{\mathrm{ij}}{S}_{i}{S}_{j})}{{(a|i\ensuremath{-}j|)}^{\ensuremath{\sigma}}}$, where the ${\ensuremath{\epsilon}}_{\mathrm{ij}}$ are independent random variables. We find a phase transition for $\frac{1}{2}<\ensuremath{\sigma}<1$. For $\frac{1}{2}<\ensuremath{\sigma}<\frac{2}{3}$ the critical exponents exhibit mean-field classical behavior. Near $\ensuremath{\sigma}=1$ we find a smoothly varying specific heat. We investigate the critical behavior near the upper and lower critical range by means of an $\ensuremath{\epsilon}$ expansion around $\ensuremath{\sigma}=1 \mathrm{and} \frac{2}{3}$.
In the replica formalism for an Ising spin glass the interaction energy between two sites depends on the number of replicas of the two sites that have parallel or antiparallel spins; therefore, the energy may take n+1 different values even when permutation symmetry is assumed in the space of n replicas. This insight is used to propose a new replica-symmetric solution to the renormalization-group equations for a one-dimensional spin-glass model with long-range interactions. In a simple approximation this solution gives a value of (2\ensuremath{\epsilon}${)}^{\mathrm{\ensuremath{-}}1/2}$ for the correlation-length exponent [\ensuremath{\epsilon}=1-\ensuremath{\sigma} and (1+\ensuremath{\sigma})/2 is the range parameter for the random interactions]. The new solution is argued to be equivalent to the replica-symmetric ansatz for the Sherrington-Kirkpatrick model.
The author considers an Ising chain where the random interaction matrix has the symmetrised form J=PV+VP. The matrix elements of V and P are V ij =exp(- gamma mod i-j mod ) and P ij =P i delta j,i-1 +P i+1 delta j,i+1 , where i, j indicate sites on the chain. The quantities P i are independent random variables with the same probability distribution. Their mean and variance are fixed to be (P i )=J 0 gamma /4; (P i 2 )-(P i ) 2 =J 2 gamma /4, where gamma -1 is the range of the interaction. The author then obtains Sigma j (J ij )=2J 0 , Sigma j (J ij 2 )=J 2 in the limit gamma to 0, as in a model Hamiltonian that describes a spin-glass transition. The author evaluates exactly in this limit the density of eigenvalues of the random matrix PV. An integral equation for the distribution probability of new local random variables is derived by using the continued fraction method and the asymptotic solution in the limit gamma to 0 is obtained analytically. The largest eigenvalue of PV is found to be J 0 independent of J. However the largest eigenvalue of J=PV+VP is found to be mu M =J 0 +(J 0 2 +NJ 2 ) 1 2/, when gamma =0 and N is the total number of sites. An independent evaluation of the ground state energy shows that E 0 <-J square root NN, hence E 0 is not extensive. These results show that the scaling of the variance of the random interactions as J 2 N -1 gives highly unphysical results for the present model.
The one-dimensional long-range Ising spin glass provides useful insights into the properties of finite-dimensional spin glasses with short-range interactions. The defect energy renormalization-group equations derived for it by Kotliar, Anderson, and Stein have been reexamined and a new fixed point has been found. The fixed points which have previously been studied are found to be inappropriate. It is shown that the renormalization-group equations themselves directly imply that that the spin-glass phase is replica symmetric, in agreement with droplet model expectations, when the range exponent $\ensuremath{\sigma}$ lies between the upper and lower critical values.
A mean-field replica-type theory is presented for long-range Ising spin-glasses whose interactions can be written in the form J ij = J ij I J ij SK , where J SK is a quenched Gaussian interaction, as in the SK spin-glass, and J I is a deterministic matrix which defines the structure or inhomogeneity of the model and must have a finite rank in the thermodynamic limit. The spin-glass model proposed by Franz, Parisi and Virasoro ( Europhys. Lett. , 17 (1992) 5) can be written in this form, with J I having rank 2, and is used as a case study to illustrate our theory. In the context of replica theory we also consider the stable states of this model, which appear continuously as the temperature is lowered.