Spectrum of periodic chain graphs with time-reversal non-invariant vertex coupling

TL;DR

This paper analyzes the spectral properties of periodic chain graphs with non-time-reversal-invariant vertex couplings, revealing how topology, geometry, and symmetry influence their energy spectra and band structures.

Contribution

It introduces a detailed spectral analysis of quantum chain graphs with non-invariant vertex conditions, highlighting effects of topology, geometry, and symmetry on spectral properties.

Findings

Spectral properties depend on graph topology and geometry.

Probability of energy belonging to spectrum varies with vertex parity and symmetry.

Band patterns are affected by the commensurability of edge lengths.

Abstract

We investigate spectral properties of quantum graphs in the form of a periodic chain of rings with a connecting link between each adjacent pair, assuming that wave functions at the vertices are matched through conditions manifestly non-invariant with respect to time reversal. We discuss, in particular, the high-energy behavior of such systems and the limiting situations when one of the edges in the elementary cell of such a graph shrinks to zero. The spectrum depends on the topology and geometry of the graph. The probability that an energy belongs to the spectrum takes three different values reflecting the vertex parities and mirror symmetry, and the band patterns are influenced by commensurability of graph edge lengths.