Spectral properties of hexagonal lattices with the -R coupling

TL;DR

This paper investigates the spectral properties of hexagonal lattice graphs with a specific vertex coupling that breaks time reversal symmetry, revealing how high-energy behavior and lattice dilation affect the spectrum and flat bands.

Contribution

It introduces a novel analysis of spectral behavior in hexagonal lattices with non-reversible vertex couplings, including effects of lattice dilation and edge length variations.

Findings

High-energy decoupling of edges at even vertices

Spectral invariance under lattice dilation with commensurate edges

Absence of flat bands with incommensurate edge lengths

Abstract

We analyze the spectrum of the hexagonal lattice graph with a vertex coupling which manifestly violates the time reversal invariance and at high energies it asymptotically decouples edges at even degree vertices; a comparison is made to the case when such a decoupling occurs at odd degree vertices. We also show that the spectral character does not change if the equilateral elementary cell of the lattice is dilated to have three different edge lengths, except that flat bands are absent if those are incommensurate.