Caustics in the Grassmann Integral

TL;DR

This paper investigates caustics in a Grassmann integral model with four-body interactions, demonstrating the effectiveness of the auxiliary field method when higher order effects are included, even in small systems.

Contribution

It reveals how higher order corrections improve the auxiliary field method's accuracy in models with caustics, especially for small N.

Findings

AFM works better with higher order effects included

Caustics appear in Grassmann integrals with four-body interactions

Higher order series expansions improve approximation accuracy

Abstract

It is shown that a simple model of 2N-Grassmann variables with a four-body coupling involves caustics when the integral has been converted to a bosonic form with the aid of the auxiliary field. Approximation is then performed to assure validity of the auxiliary field method(AFM). It turns out that even in N=2, the smallest case in which a four-body interaction exists, AFM does work more excellently if higher order effects, given by a series in terms of $1/N^{1/3}$ around a caustic and of 1/N around a saddle point, would be taken into account.