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Renormalization Group in Spin Glasses

T. C. Lubensky, T. Temesvári, Imre Kondor, Maria Chiara Angelini

Published Aug 1, 2023• 3 citations

Abstract

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.

DOI: 10.1142/9789811273926_0004

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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.

This thesis is concerned with the mean field theory of spin glasses within the replica method approach. We have studied the Sherrington-Kirkpatrick model [33,4δ] both with Ising and m-component spins. It is shown that the replica symmetric solution gives an instability in the condensed phase which requires a breaking of replica symmetry in the spin glass phase. The various broken replica symmetry solutions proposed thus far (Bray and Moore [52], Blandin et al [53], Parisi [54]) are reviewed and particular attention is given to the one proposed by Parisi. The order parameter susceptibility and the stability of the solution are studied within this breaking of replica symmetry scheme which give results in agreement with the predictions of TAP [47] and Bray and Moore [52]. The breakdown of linear response theory in the spin glass phase is also discussed.