Nature of perturbation theory in spin glasses

TL;DR

This paper investigates the high-order behavior of perturbation series in spin glass field theories, revealing alternating signs and complex subdominant terms, and corrects previous epsilon expansion errors.

Contribution

It provides a detailed analysis of the high-order perturbation behavior in spin glass theories and corrects prior epsilon expansion inaccuracies.

Findings

High-order coefficients exhibit alternating signs.

Subdominant terms have complex sign dependencies.

Errors in third-order epsilon expansions are corrected.

Abstract

The high-order behavior of the perturbation expansion in the cubic replica field theory of spin glasses in the paramagnetic phase has been investigated. The study starts with the zero-dimensional version of the replica field theory and this is shown to be equivalent to the problem of finding finite size corrections in a modified spherical spin glass near the critical temperature. We find that the high-order behavior of the perturbation series is described, to leading order, by coefficients of alternating signs (suggesting that the cubic field theory is well-defined) but that there are also subdominant terms with a complicated dependence of their sign on the order. Our results are then extended to the d-dimensional field theory and in particular used to determine the high-order behavior of the terms in the expansion of the critical exponents in a power series in epsilon=6-d. We have also corrected errors in the existing epsilon expansions at third order.