On the exact continuous mapping of fermions

TL;DR

This paper presents an exact quantum mechanical mapping of fermionic operators to continuous variables, enabling new ways to analyze many-fermion systems and their complex interactions.

Contribution

It introduces a rigorous, exact mapping of fermionic operators to continuous variables, facilitating analysis of complex many-fermion Hamiltonians.

Findings

Efficient mappings for Anderson impurity and Hubbard models.

Provides an exact alternative method for static and dynamical properties of fermionic systems.

Lays groundwork for semiclassical and quantum-classical methods in fermionic problems.

Abstract

We derive a rigorous, quantum mechanical map of fermionic creation and annihilation operators to continuous Cartesian variables that exactly reproduces the matrix structure of the many-fermion problem. We show how our scheme can be used to map a general many-fermion Hamiltonian and then consider two specific models that encode the fundamental physics of many fermionic systems, the Anderson impurity and Hubbard models. We use these models to demonstrate how efficient mappings of these Hamiltonians can be constructed using a judicious choice of index ordering of the fermions. This development provides an alternative exact route to calculate the static and dynamical properties of fermionic systems and sets the stage to exploit the quantum-classical and semiclassical hierarchies to systematically derive methods offering a range of accuracies, thus enabling the study of problems where the fermionic degrees of freedom are coupled to complex anharmonic nuclear motion and spins which lie beyond the reach of most currently available methods.