Spin Glass Field Theory with Replica Fourier Transforms

TL;DR

This paper introduces a novel field theory framework for spin glasses using Replica Fourier Transforms, enabling simplified analysis of replica symmetry breaking and Gaussian fluctuations around the Parisi solution.

Contribution

It develops a comprehensive RFT-based formalism for spin glasses, including block-diagonalization of the mass matrix and calculation of fluctuation contributions.

Findings

Block-diagonalization of the four-replica mass matrix into modes

Explicit eigenvalues in terms of RFT mass and propagators

Calculation of Gaussian fluctuation contributions to free energy

Abstract

We develop a field theory for spin glasses using Replica Fourier Transforms (RFT). We present the formalism for the case of replica symmetry and the case of replica symmetry breaking on an ultrametric tree, with the number of replicas $n$ and the number of replica symmetry breaking steps $R$ generic integers. We show how the RFT applied to the two-replica fields allows to construct a new basis which block-diagonalizes the four-replica mass-matrix, into the replicon, anomalous and longitudinal modes. The eigenvalues are given in terms of the mass RFT and the propagators in the RFT space are obtained by inversion of the block-diagonal matrix. The formalism allows to express any $i$-replica vertex in the new RFT basis and hence enables to perform a standard perturbation expansion. We apply the formalism to calculate the contribution of the Gaussian fluctuations around the Parisi solution for the free-energy of an Ising spin glass.