The semiclassical propagator in fermionic Fock space

TL;DR

This paper derives a rigorous semiclassical propagator for fermionic systems in Fock space, enabling better understanding of quantum interference effects in many-body fermionic systems without relying on auxiliary fields.

Contribution

It introduces a novel exact path integral formulation for fermionic propagators using commuting variables, bypassing auxiliary field methods and fermionic determinants.

Findings

Provides a rigorous definition of the classical limit for interacting fermions.

Constructs a semiclassical van Vleck-Gutzwiller propagator for fermionic Fock space.

Analyzes how universality classes influence many-body interference effects.

Abstract

We present a rigorous derivation of a semiclassical propagator for anticommuting (fermionic) degrees of freedom, starting from an exact representation in terms of Grassmann variables. As a key feature of our approach the anticommuting variables are integrated out exactly, and an exact path integral representation of the fermionic propagator in terms of commuting variables is constructed. Since our approach is not based on auxiliary (Hubbard-Stratonovich) fields, it surpasses the calculation of fermionic determinants yielding a standard form $\int {\cal D}[\psi,\psi^{*}] {\rm e}^{i R[\psi,\psi^{*}]}$ with real actions for the propagator. These two features allow us to provide a rigorous definition of the classical limit of interacting fermionic fields and therefore to achieve the long-standing goal of a theoretically sound construction of a semiclassical van Vleck-Gutzwiller propagator in fermionic Fock space. As an application, we use our propagator to investigate how the different universality classes (orthogonal, unitary and symplectic) affect generic many-body interference effects in the transition probabilities between Fock states of interacting fermionic systems.