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Journal of Physics A Mathematical and General

Perturbation study of the linear renormalization group for spin glasses

J. R. L. de Almeida

Published Jan 21, 1993• 8 citations
Publisher

Abstract

The usual linear renormalization group (LRG) used in spin glasses is studied when a small nonlinear term is introduced in the relationship between block-spin variables. Contrary to what happens in uniform systems, this perturbation gives a singular contribution to the fixed-point Hamiltonian within the approximations used. It is then claimed that the LRG is inappropriate for spin glass systems.

DOI
10.1088/0305-4470/26/2/008

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The usual linear renormalization group (LRG) used in spin glasses is studied when a small nonlinear term is introduced in the relationship between block-spin variables. Contrary to what happens in uniform systems, this perturbation gives a singular contribution to the fixed-point Hamiltonian within the approximations used. It is then claimed that the LRG is inappropriate for spin glass systems.
Most renormalization groups studied heretofore are linear, in that the block-spin variables are linearly related to the old spin variables. These renormalization groups all require an adjustable parameter that must be properly fixed in order that the renormalization-group transformation have a useful fixed point. Niemeijer and van Leeuwen, however, have investigated the critical behavior of the two-dimensional Ising model with a nonlinear renormalization group, and find a fixed point without introducing such a parameter. We introduce here another nonlinear renormalization group and study it in detail for the Gaussian model. We find that (i) within certain limits, no such parameter need be adjusted in order to reach a fixed point; (ii) an eigenvalue of the linearized transformation with no physical significance depends on the nonlinearity of the transformation; and (iii) a physically significant eigenvalue is unchanged (to the order examined).
A renormalization group transformation suitable for spin glasses, and more generally for disordered systems, is presented. The approach is non-standard for two reasons. 1. The coarse graining transformation is performed on the distribution of site overlaps, which clearly is not a Boltzmann-Gibbs distribution. 2. The additional interactions are realized introducing a gauge-field disorder distribution, while the Hamiltonian is kept fixed. Universality classes are thus naturally defined on a large set of models, going from ℤ2 and Gaussian spin glasses to Ising and fully frustrated models, and others. The role of frustration and of the deconfinement transition can also be analysed in terms of fixed points. The proposed analysis is tested numerically on the ℤ2 Edwards-Anderson model in d=4. Good estimates of the critical index ν and of Tc are obtained, and an RG flow diagram is sketched for the first time.
The renormalization group treatment of spin glasses has been a theoretical challenge since shortly after the formulation of the Sherrington–Kirkpatrick model. This chapter brings three perspectives on this topic. Section 4.1, by T. Lubensky, reviews the early replica symmetric description of the problem, which predate and in some ways anticipate the discovery of the instability of that treatment by de Almeida and Thouless. The perturbative renormalization group approach to the ensuing transition is presented in Sec. 4.2, contributed jointly by T. Temesvari and I. Kondor. The findings of this approach motivate the consideration of non-perturbative renormalization group schemes, which are discussed in Sec. 4.3 written by M. C. Angelini. © 2023 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.
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.
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.