Gaussian phase-space representations for fermions

TL;DR

This paper introduces a positive phase-space representation for fermions using Gaussian operators, enabling quantum dynamical and equilibrium calculations in many-body Fermi systems, including strongly correlated gases and nanostructures.

Contribution

It generalizes bosonic phase-space methods to fermions with a Gaussian operator basis, establishing operator correspondences and stochastic mappings for fermionic quantum systems.

Findings

Derived equivalences between quantum and stochastic moments.

Mapped quantum operator evolution onto stochastic processes.

Demonstrated applications to ideal gases, Hubbard model, and open systems.

Abstract

We introduce a positive phase-space representation for fermions, using the most general possible multi-mode Gaussian operator basis. The representation generalizes previous bosonic quantum phase-space methods to Fermi systems. We derive equivalences between quantum and stochastic moments, as well as operator correspondences that map quantum operator evolution onto stochastic processes in phase space. The representation thus enables first-principles quantum dynamical or equilibrium calculations in many-body Fermi systems. Potential applications are to strongly interacting and correlated Fermi gases, including coherent behaviour in open systems and nanostructures described by master equations. Examples of an ideal gas and the Hubbard model are given, as well as a generic open system, in order to illustrate these ideas.