Dimensional Reduction of a Generalized Flux Problem

TL;DR

This paper demonstrates how a generalized flux problem in lattice models can be reduced in dimension for Abelian fluxes, revealing spectral equivalences, while non-Abelian cases require specific conditions for reduction.

Contribution

It introduces a method to reduce the dimensionality of the flux problem in lattice models and identifies conditions for spectral equivalence classes.

Findings

Abelian flux problems reduce to lower-dimensional hopping models.

Spectral equivalence classes of Hamiltonians are identified.

Non-Abelian flux reduction requires flux tensor factorization.

Abstract

A generalized flux problem with Abelian and non-Abelian fluxes is considered. In the Abelian case we shall show that the generalized flux problem for tight-binding models of noninteracting electrons on either $2n$ or $2n+1$ dimensional lattice can always be reduced to a $n$ dimensional hopping problem. A residual freedom in this reduction enables to identify equivalence classes of hopping Hamiltonians which have the same spectrum. In the non-Abelian case the reduction is not possible in general unless the flux tensor factorizes into an Abelian one times an element of the corresponding algebra.