Re-gauging groupoid, symmetries and degeneracies for Graph Hamiltonians and applications to the Gyroid wire network

TL;DR

This paper explores how gauge transformations and symmetries in graph Hamiltonians influence degeneracies in spectra, with applications to complex wire networks like the Gyroid and honeycomb structures, revealing symmetry-driven degeneracies such as Dirac points.

Contribution

It introduces a framework for understanding gauge and symmetry actions on graph Hamiltonians via groupoids and noncommutative geometry, with applications to physical wire networks.

Findings

Degeneracies are explained by extended symmetries and projective representations.

The framework applies to Gyroid and honeycomb wire networks, elucidating Dirac points.

Symmetry actions lead to super-selection rules and spectral decompositions.

Abstract

We study a class of graph Hamiltonians given by a type of quiver representation to which we can associate (non)--commutative geometries. By selecting gauging data these geometries are realized by matrices through an explicit construction or a Kan-extension. We describe the changes in gauge via the action of a regauging groupoid. It acts via matrices that give rise to a noncommutative 2--cocycle and hence to a groupoid extension (gerbe). We furthermore show that automorphisms of the underlying graph of the quiver can be lifted to extended symmetry groups of regaugings. In the commutative case, we deduce that the extended symmetries act via a projective representation. This yields isotypical decompositions and super--selection rules. We apply these results to the PDG and honeycomb wire--networks using representation theory for projective groups and show that all the degeneracies in the spectra are consequences of these enhanced symmetries. This includes the Dirac points of the G(yroid) and the honeycomb systems.