Universality of the momentum band density of periodic networks

TL;DR

This paper demonstrates that the probability of a momentum belonging to the spectrum of a periodic quantum graph exhibits universal properties, independent of edge lengths and certain graph topologies.

Contribution

It reveals universal invariance properties of the spectral band density in periodic networks, regardless of specific edge lengths or certain topological classes.

Findings

Probability is independent of edge lengths for generic cases.

Probability remains invariant within certain graph topologies.

Spectral band density exhibits universal behavior.

Abstract

The momentum spectrum of a periodic network (quantum graph) has a band-gap structure. We investigate the relative density of the bands or, equivalently, the probability that a randomly chosen momentum belongs to the spectrum of the periodic network. We show that this probability exhibits universal properties. More precisely, the probability to be in the spectrum does not depend on the edge lengths (as long as they are generic) and is also invariant within some classes of graph topologies.