A New Perturbative Expansion for Fermionic Functional Integrals

TL;DR

This paper introduces a novel perturbative expansion method for fermionic functional integrals involving Grassmann variables, offering a local and straightforward alternative to traditional expansions, demonstrated through decay of correlations in a lattice model.

Contribution

It develops a new perturbative expansion for fermionic integrals with Grassmann variables, simplifying analysis and avoiding complex traditional methods.

Findings

Expansion exhibits local structure and simplicity.

Proves exponential decay of 2-point correlations in a lattice model.

Provides an alternative to decoupling and cluster expansions.

Abstract

We construct a power series representation of the integrals of form \begin{equation} \text{log} \int d\mu_{S}(\psi, \bar{\psi}) \hspace{0.05 cm} e^{f(\psi, \bar{\psi}, \eta, \bar{\eta})} \nonumber \end{equation} where $\psi, \bar{\psi}$ and $\eta, \bar{\eta}$ are Grassmann variables on a finite lattice in $d \geqslant 2$. Our expansion has a local structure, is clean and provides an easy alternative to decoupling expansion and Mayer-type cluster expansions in any analysis. As an example, we show exponential decay of 2-point truncated correlation function (uniform in volume) in massive Gross-Neveu model on a unit lattice.