Kagome network with vertex coupling of a preferred orientation

TL;DR

This paper studies the spectral properties of kagome and triangular lattice quantum graphs with a specific vertex coupling that breaks time-reversal symmetry, revealing universal spectral features and band structures.

Contribution

It introduces a novel vertex coupling in quantum graphs that violates time-reversal symmetry and analyzes its spectral implications on kagome and triangular lattices.

Findings

Positive spectrum has infinitely many bands, including flat bands.

Negative spectrum has at most five bands.

Spectral universality holds for incommensurate edges, with a probability of about 0.639 for a number to be in the spectrum.

Abstract

We investigate spectral properties of periodic quantum graphs in the form of a kagome or a triangular lattice in the situation when the condition matching the wave functions at the lattice vertices is chosen of a particular form violating the time-reversal invariance. The positive spectrum consists of infinite number of bands, some of which may be flat; the negative one has at most three and two bands, respectively. The kagome lattice example shows that even in graphs with such an uncommon vertex coupling spectral universality may hold: if its edges are incommensurate, the probability that a randomly chosen positive number is contained in the spectrum is $\approx 0.639$.