Schwinger-Dyson equations and disorder

TL;DR

This paper investigates the accuracy of Schwinger-Dyson equations in simple disordered models, showing they require dual variable transformation to match exact results when multiple saddle points dominate.

Contribution

It demonstrates the necessity of transforming actions into dual variables for Schwinger-Dyson equations to correctly capture physics in models with multiple saddle points.

Findings

Schwinger-Dyson equations fail without dual transformation in multi-saddle scenarios

Exact partition functions and correlations are computed in low-dimensional models

Comparison reveals limitations of saddle-point approximations in disordered systems

Abstract

Using simple models in D=0+0 and D=0+1 dimensions we construct partition functions and compute two-point correlations. The exact result is compared with saddle-point approximation and solutions of Schwinger-Dyson equations. When integrals are dominated by more than one saddle-point we find Schwinger-Dyson equations do not reproduce the correct results unless the action is first transformed into dual variables.