Flat bands with band crossings enforced by symmetry representation

TL;DR

This paper demonstrates that certain flat band crossings are enforced by symmetry representation, and introduces a new class of protected degeneracies called SR-enforced band crossings, with implications for flat band topology.

Contribution

It introduces the concept of SR-enforced band crossings, showing they are protected by symmetry representation and flatness, and provides a no-go theorem for isolated flat bands.

Findings

SR-enforced band crossings are unavoidable due to symmetry representation.

Lifting degeneracy can lead to nearly flat Chern bands.

The theory explains band crossings in various flat band models.

Abstract

Flat bands have band crossing points with other dispersive bands in many systems including the canonical flat band models in the Lieb and kagome lattices. Here we show that some of such band degeneracy points are unavoidable because of the symmetry representation (SR) of the flat band under unitary symmetry. We refer to such a band degeneracy point of flat bands as a SR-enforced band crossing. SR-enforced band crossing is distinct from the conventional band degeneracy protected by symmetry eigenvalues or topological charges in that its protection requires both specific symmetry representation and band flatness of the flat band, simultaneously. Even $n$-fold rotation $C_n$ ($n=2,3,4,6$) symmetry, which cannot protect band degeneracy without additional symmetries due to its abelian nature, can protect SR-enforced band crossings in flat band systems. In two-dimensional flat band systems with $C_n$ symmetry, when the degeneracy of a SR-enforced band crossing is lifted by a $C_n$ symmetry-preserving perturbation, we obtain a nearly flat Chern band. Our theory not only explains the origin of the band crossing points of FBs existing in various models, but also gives a strict no-go theorem for isolated FBs in a given lattice arising from the SR.