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arXiv:cond-mat/9312020

Replica Real Space Renormalization Group for Spin Glasses

J. R. L. de Almeida, S. Coutinho

Published Dec 5, 1993• 0 citations
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Abstract

We construct a real space renormalization group (RG) approach for Ising spin glasses on hypercubic lattices within the scheme of the Migdal-Kadanoff approximation using replicas. Our replica symmetric solution yields results consistent with simple decimation previously obtained and the introduction of breaking of replica symmetry within the RG is discussed, which inserts in a natural fashion non-linear RG into the problem.

DOI
10.48550/arxiv.cond-mat/9312020

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We construct a real space renormalization group (RG) approach for Ising spin glasses on hypercubic lattices within the scheme of the Migdal-Kadanoff approximation using replicas. Our replica symmetric solution yields results consistent with simple decimation previously obtained and the introduction of breaking of replica symmetry within the RG is discussed, which inserts in a natural fashion non-linear RG into the problem.
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.
Here is the first part of the summary of my work on random Ising model using real-space renormalization group (RSRG), also known as a Migdal-Kadanoff one. This approximate renormalization scheme was applied to the analysis thermodynamic properties of the model, and of probabilistic properties of a pair correlator, which is a fluctuating object in disordered systems. PACS numbers: 02.50.-r, 05.20.-y, 05.70.Fh, 64.60.ae, 75.10.-b, 75.10.Hk, 87.10.+e Keywords: statistical physics, magnetism, spin glass, neural networks
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.
A zero-temperature Migdal-Kadanoff renormalization-group scheme is used to investigate two classical spin-glass lattice models of a random superconductor in a magnetic field. We show that the two models considered here become the same under renormalization and belong to a different universality class from the Ising and XY spin glasses. We find that, in contrast with results from recent Monte Carlo simulations, the lower critical dimension for a nonzero-temperature vortex-glass transition in a random superconductor is greater than 3.
A real-space Renormalization Group approach is presented for a non-mean-field spin-glass. This approach has been conceived in the effort to develop an alternative method to the Renormalization Group approaches based on the replica method. Indeed, non-perturbative effects in the latter are quite generally out of control, in such a way that such approaches are non-predictive. On the contrary, we show that the real-space method presented here yields a precise prediction of the critical behavior and exponents of the model.