Canonical representation for electrons and its application to the Hubbard model

TL;DR

This paper introduces a canonical, constraint-free representation of electrons using spinless fermions and Pauli operators, simplifying the Hubbard model and providing new insights into superexchange and ferromagnetism.

Contribution

It presents a novel, invertible electron representation that simplifies the Hubbard model and elucidates the origins of superexchange and ferromagnetism.

Findings

Reproduces large U superexchange accurately

Simplifies the analysis of Nagaoka ferromagnetism

Links ferromagnetism to gauge invariance of spinless fermions

Abstract

A new representation for electrons is introduced, in which the electron operators are written in terms of a spinless fermion and the Pauli operators. This representation is canonical, invertible and constraint-free. Importantly, it simplifies the Hubbard interaction. On a bipartite lattice, the Hubbard model is reduced to a form in which the exchange interaction emerges simply by decoupling the Pauli subsystem from the spinless fermion bath. This exchange correctly reproduces the large $U$ superexchange. Also derived, for $U=\pm\infty$, is the Hamiltonian to study Nagaoka ferromagnetism. In this representation, the infinite-$U$ Hubbard problem becomes elegant and easier to handle. Interestingly, the ferromagnetism in Hubbard model is found to be related to the gauge invariance of the spinless fermions. Generalization of this representation for the multicomponent fermions, a new representation for bosons, the notion of a `soft-core' fermion, and some interesting unitary transformations are introduced and discussed in the appendices.