Phase space methods for Majorana fermions

TL;DR

This paper develops phase space methods using Majorana fermions and antisymmetric matrices to facilitate stochastic simulations of fermionic systems with Majorana excitations, relevant for topological quantum systems.

Contribution

It introduces a formalism that expresses fermionic Gaussian operators in terms of Majorana operators and antisymmetric matrices, enabling new stochastic simulation techniques.

Findings

Derivation of differential identities for Majorana operators

Formulation of Fokker-Planck equations for fermionic systems

Application to topological Hamiltonians with dissipation

Abstract

Fermionic phase space representations are a promising method for studying correlated fermion systems. The fermionic Q-function and P-function have been defined using Gaussian operators of fermion annihilation and creation operators. The resulting phase-space of covariance matrices belongs to the symmetry class D, one of the non-standard symmetry classes. This was originally proposed to study mesoscopic normal-metal-superconducting hybrid structures, which is the type of structure that has led to recent experimental observations of Majorana fermions. Under a unitary transformation, it is possible to express these Gaussian operators using real anti-symmetric matrices and Majorana operators, which are much simpler mathematical objects. We derive differential identities involving Majorana fermion operators and an antisymmetric matrix which are relevant to the derivation of the corresponding Fokker-Planck equations on symmetric space. These enable stochastic simulations either in real or imaginary time. This formalism has direct relevance to the study of fermionic systems in which there are Majorana type excitations, and is an alternative to using expansions involving conventional Fermi operators. The approach is illustrated by showing how a linear coupled Hamiltonian as used to study topological excitations can be transformed to Fokker-Planck and stochastic equation form, including dissipation through particle losses.