The noncommutative geometry of wire networks from triply periodic surfaces

Abstract

We study wire networks that are the complements of triply periodic minimal surfaces. Here we consider the P, D, G surfaces which are exactly the cases in which the corresponding graphs are symmetric and self-dual. Our approach is using the Harper Hamiltonian in a constant magnetic field. We treat this system with the methods of noncommutative geometry and obtain a classification for all the $C^{\ast }$ geometries that appear.

15 pages, 5 figures

References in corpus (1)

• Effective dynamics for Bloch electrons: Peierls substitution and beyond

Cited by in corpus (2)

• Singularities, swallowtails and Dirac points. An analysis for families of Hamiltonians and applications to wire networks, especially the Gyroid
• Geometry of the momentum space: From wire networks to quivers and monopoles