Flux Hamiltonians, Lie Algebras and Root Lattices With Minuscule Decorations

TL;DR

This paper explores Hamiltonians of fermions on decorated root lattices linked to Lie algebras, revealing spectral properties and connections to known algebraic structures, with implications for models like kagome and pyrochlore lattices.

Contribution

It introduces a novel class of flux Hamiltonians based on Lie algebraic root lattice decorations and analyzes their spectral characteristics and physical realizations.

Findings

Spectra resemble Dirac spectra and relate to Lie algebra properties.

Lattices like kagome and pyrochlore can be interpreted through Lie algebraic frameworks.

Flux Hamiltonians correspond to mean-field models of spin-1/2 Heisenberg systems.

Abstract

We study a family of Hamiltonians of fermions hopping on a set of lattices in the presence of a background gauge field. The lattices are constructed by decorating the root lattices of various Lie algebras with their minuscule representations. The Hamiltonians are, in momentum space, themselves elements of the Lie algebras in these same representations. We describe various interesting aspects of the spectra--which exhibit a family resemblance to the Dirac spectrum, and in many cases are able to relate them to known facts about the relevant Lie algebras. Interestingly, various realizable lattices such as the kagom\'{e} and pyrochlore can be given this Lie algebraic interpretation and the particular flux Hamiltonians arise as mean-field Hamiltonians for spin-1/2 Heisenberg models on these lattices.