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Physical review. B, Condensed matter

Replica symmetry in a one-dimensional spin-glass model with long-range interactions

Anil Khurana

Published Aug 1, 1989• 4 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.40.2602

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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})\ensuremath−1/2{)}^{\mathrm{\ensuremath{-}}1/2})\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 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.
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 have studied the diluted Heisenberg spin glass model in a three-component random field for the commonly used one-dimensional long-range model where the probability that two spins separated by a distance r interact with one another falls as 1/r^{2σ}, for two values of σ, 0.75 and 0.85. No de Almeida-Thouless line is expected at these σ values. The spin glass correlation length ξ_{SG} varies with the random field as expected from the Imry-Ma argument and the droplet scaling picture of spin glasses. However, when ξ_{SG} becomes comparable to the system size L, there are departures which we attribute to the features deriving from the replica symmetry breaking picture of spin glasses. For the case σ=0.85 these features go away for system sizes with L>L^{*}, where L^{*} is large (≈4000-8000 lattice spacings). In the case of σ=0.75 we have been unable to study large enough systems to determine its value of L^{*}. We sketch a renormalization group scenario to explain how these features could arise. In this scenario finite size effects on the droplet scaling picture in low-dimensional spin glasses produce some aspects of Parisi's replica symmetry breaking theory of the Sherrington-Kirkpatrick model.
We consider a system of linear chains of Ising spins with short-range nearest-neighbor ferromagnetic interactions of strength K along the chains, and with infinite-range random interchain interactions of a Sherrington-Kirkpatrick type. Also included in the model is a local Gaussian random field with variance \ensuremath{\Delta}. In the replica symmetric case, the system is mapped exactly on the one-dimensional Ising model in a random field and a recursion method for calculating the spin-correlation functions is developed. For \ensuremath{\Delta}=0, an analytic relation for freezing temperature Tf{\mathit{T}}_{\mathit{f}}Tf​(K) is obtained. The replica-symmetric spin-glass phase is shown to be stable above the freezing temperature Tf{\mathit{T}}_{\mathit{f}}Tf​(K,\ensuremath{\Delta}), which is determined numerically.
We solve the fermionic version of the Ising spin glass for arbitrary filling µ and temperature T taking into account replica symmetry breaking. Using a simple exact mapping from µ to the anisotropy parameter D , we also obtain the solution of the S = 1 Sherrington-Kirkpatrick model. An analytic expression for T = 0 gives an improved critical value for the first-order phase transition. We revisit the question of stability against replica-diagonal fluctuations and find that the appearance of complex eigenvalues of the Almeida-Thouless matrix is not an artefact of the replica-symmetric approximation.
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.