Fermionic Mean-Field Theory as a Tool for Studying Spin Hamiltonians

TL;DR

This paper explores various fermionic mappings, including Jordan--Wigner, to transform spin Hamiltonians into fermionic models, enabling mean-field solutions that can accurately approximate energies and correlations in complex quantum systems.

Contribution

It systematically compares different spin-fermion mappings to identify the most effective for applying fermionic mean-field theory to spin Hamiltonians.

Findings

Jordan--Wigner transformation yields exact solutions for certain models.

Fermionic mean-field approaches provide accurate energy estimates.

Different mappings vary in effectiveness depending on the model.

Abstract

The Jordan--Wigner transformation permits one to convert spin $1/2$ operators into spinless fermion ones, or vice versa. In some cases, it transforms an interacting spin Hamiltonian into a noninteracting fermionic one which is exactly solved at the mean-field level. Even when the resulting fermionic Hamiltonian is interacting, its mean-field solution can provide surprisingly accurate energies and correlation functions. Jordan--Wigner is, however, only one possible means of interconverting spin and fermionic degrees of freedom. Here, we apply several such techniques to the XXZ and $J_1\text{--}J_2$ Heisenberg models, as well as to the pairing or reduced BCS Hamiltonian, with the aim of discovering which of these mappings is most useful in applying fermionic mean-field theory to the study of spin Hamiltonians.