Geometry of the momentum space: From wire networks to quivers and monopoles

TL;DR

This paper develops a geometric and algebraic framework for analyzing the band structure of wire networks, especially gyroids, using non-commutative geometry, K-theory, and topological invariants like Chern classes.

Contribution

It introduces a unified approach combining quiver representations, $C^*$-algebra techniques, and topological methods to study band structures in complex materials.

Findings

K-theory provides insights into band topology and stability.

Geometric methods reveal band intersection properties.

Berry connection relates to monopole charges in momentum space.

Abstract

A new nano--material in the form of a double gyroid has motivated us to study (non-commutative $C^*$ geometry of periodic wire networks and the associated graph Hamiltonians. Here we present the general abstract framework, which is given by certain quiver representations, with special attention to the original case of the gyroid as well as related cases, such as graphene. In these geometric situations, the non- commutativity is introduced by a constant magnetic field and the theory splits into two pieces: commutative and non-commutative, both of which are governed by a $C^*$ geometry. In the non-commutative case, we can use tools such as K-theory to make statements about the band structure. In the commutative case, we give geometric and algebraic methods to study band intersections; these methods come from singularity theory and representation theory. We also provide new tools in the study, using $K$-theory and Chern classes. The latter can be computed using Berry connection in the momentum space. This brings monopole charges and issues of topological stability into the picture.