Band touching from real space topology in frustrated hopping models

TL;DR

This paper investigates the connection between real-space topology and band touching phenomena in frustrated hopping models with flat bands, revealing topological states that protect band degeneracies.

Contribution

It demonstrates that band touchings in frustrated hopping models are linked to topologically non-trivial states supported on winding loops, with a counting method to predict their occurrence.

Findings

Band touchings are associated with topological states on winding loops.

A counting argument predicts band touching occurrences.

Topological protection prevents removal of band touchings without splitting flat bands.

Abstract

We study ``frustrated'' hopping models, in which at least one energy band, at the maximum or minimum of the spectrum, is dispersionless. The states of the flat band(s) can be represented in a basis which is fully localized, having support on a vanishing fraction of the system in the thermodynamic limit. In the majority of examples, a dispersive band touches the flat band(s) at a number of discrete points in momentum space. We demonstrate that this band touching is related to states which exhibit non-trivial topology in real space. Specifically, these states have support on one-dimensional loops which wind around the entire system (with periodic boundary conditions). A counting argument is given that determines, in each case, whether there is band touching or not, in precise correspondence to the result of straightforward diagonalization. When they are present, the topological structure protects the band touchings in the sense that they can only be removed by perturbations which {\sl also} split the degeneracy of the flat band.