Local models and global constraints for degeneracies and band crossings

TL;DR

This paper investigates the topological properties of Hamiltonian families with degeneracies and band crossings, using local models and global constraints to understand invariants like Chern classes and Berry phases, especially under symmetries.

Contribution

It introduces a novel analysis of local models for degeneracies and establishes new global topological constraints, including symmetry considerations, for Hamiltonian families.

Findings

Derived constraints for Chern classes and Berry phases in degenerate Hamiltonians.

Analyzed examples including Gyroid and honeycomb geometries with Weyl and Dirac points.

Enhanced understanding of topological invariants in systems with degeneracies and symmetries.

Abstract

We study topological properties of families of Hamiltonians which may contain degenerate energy levels aka. band crossings. The primary tool are Chern classes, Berry phases and slicing by surfaces. To analyse the degenerate locus, we study local models. These give information about the Chern classes and Berry phases. We then give global constraints for the topological invariants. This is an hitherto relatively unexplored subject. The global constraints are more strict when incorporating symmetries such as time reversal symmetries. The results can also be used in the study of deformations. We furthermore use these constraints to analyse examples which include the Gyroid geometry, which exhibits Weyl points and triple crossings and the honeycomb geometry with its two Dirac points.