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arXiv:1005.0561 · Physical Review B

Ordered phase of the one-dimensional Ising spin glass with long-range interactions

M A Moore

Published Jul 16, 2010• 20 citations
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Abstract

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.

DOI
10.1103/physrevb.82.014417

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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 \ensuremathσ\ensuremath{\sigma}\ensuremathσ lies between the upper and lower critical values.
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 one-dimensional Ising spin-glass model with power-law long-range interactions is a useful proxy model for studying spin glasses in higher space dimensions and for finding the dimension at which the spin-glass state changes from having broken replica symmetry to that of droplet behavior. To this end we have calculated the exponent that describes the difference in free energy between periodic and antiperiodic boundary conditions. Numerical work is done to support some of the assumptions made in the calculations and to determine the behavior of the interface free-energy exponent of the power law of the interactions. Our numerical results for the interface free-energy exponent are badly affected by finite-size problems.
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 investigate zero and finite-temperature properties of the one-dimensional spin-glass model for vector spins in the limit of an infinite number mmm of spin components where the interactions decay with a power, $\ensuremath{\sigma}$, of the distance. A diluted version of this model is also studied, but found to deviate significantly from the fully connected model. At zero temperature, defect energies are determined from the difference in ground-state energies between systems with periodic and antiperiodic boundary conditions to determine the dependence of the defect-energy exponent $\ensuremath{\theta}$ on $\ensuremath{\sigma}$. A good fit to this dependence is $\ensuremath{\theta}=3/4\ensuremath{-}\ensuremath{\sigma}$. This implies that the upper critical value of $\ensuremath{\sigma}$ is 3/43/43/4, corresponding to the lower critical dimension in the ddd-dimensional short-range version of the model. For finite temperatures, the large mmm saddle-point equations are solved self-consistently, which gives access to the correlation function, the order parameter, and the spin-glass susceptibility. Special attention is paid to the different forms of finite-size scaling effects below and above the lower critical value, $\ensuremath{\sigma}=5/8$, which corresponds to the upper critical dimension 8 of the hypercubic short-range model.
A model of the quantum (transverse) Ising spin glass with a long-range random interaction is proposed and studied here. In this model the random interaction between two spins at a distance rij{r}_{\mathrm{ij}}rij​ apart falls algebraically as ${1/r}_{\mathrm{ij}}^{(d+\ensuremath{\sigma})/2}.$ Here, apart from the strength of the quantum fluctuations, the interaction range $\ensuremath{\sigma}$ is also tunable. We have studied the ground-state properties of the model, extending the ``Droplet model'' of the short-range quantum spin glass. The model is also studied using a modified form of the effective Landau action describing the transition of the short-range quantum spin glass. The important features which are due to the long-range interaction are clearly mentioned. Field theoretical renormalization group calculations fail to locate any stable weak-coupling fixed point in the non-mean-field region. The simplest conjecture is that beyond the mean-field region, the critical behavior is governed by the infinite randomness fixed point. We extend the phenomenological scaling relation for the short-range quantum spin glass (based on the assumption of the existence of a dangerously irrelevant operator) to the present long-range interacting case. Most of the possible interesting aspects associated with the quantum transition in the present model are elaborately discussed.
We apply the real space renormalization group (RG) to the replica Hamiltonian of the two dimensional Ising Edwards–Anderson model to discuss the existence of the spin glass (SG) phase. We derive the RG equations under the replica symmetric (RS) ansatz and the resulting flow diagram indicates the existence of the SG phase. The critical exponent for the SG transition has a plausible value while that of the multicritical point (MCP) is a complex number. We consider that this failure at the MCP is due to not taking the RS breaking into account. Indeed we find that the RS breaking parameter is relevant both at the SG critical point and the MCP, indicating the nontriviality of the SG phase of this model.