WKB structure in a scalar model of flat bands

TL;DR

This paper investigates the WKB structure of solutions in a scalar model with flat bands, providing a theoretical explanation for quantisation conditions and confirming results through numerical experiments.

Contribution

It presents a general theorem on solution structure and a heuristic WKB explanation for flat band quantisation in scalar periodic operators.

Findings

WKB structure explains flat band quantisation conditions

Numerical experiments confirm the theoretical predictions

Simplified model allows complex WKB analysis

Abstract

We consider a family of periodic scalar operators for which one can define flat bands in the sense of Floquet-Bloch theory. One puzzling question originating in recent physics literature is a quantisation rule for the values of parameters at which these flat bands occur. We present a general theorem about the structure of solutions to the corresponding equation and a heuristic argument explaining their WKB structure in a specific case. That structure also explains the quantisation condition - both the WKB structure and that rule are confirmed by numerical experiments. Finally, we consider a simplified model in which separation of variables allows the use of complex WKB methods.