Band spectra of rectangular graph superlattices

TL;DR

This paper investigates the band spectra of rectangular graph superlattices, revealing a fractal structure influenced by the ratio of side lengths and number-theoretic properties, with implications for gap formation and localization.

Contribution

It introduces a detailed analysis of band spectra in rectangular graph superlattices, highlighting fractal structures and number-theoretic effects on spectral gaps and localization.

Findings

Band spectra exhibit a hidden fractal structure related to the ratio of side lengths.

No spectral gaps in weak-coupling for irrational ratios badly approximable by rationals.

Quantization of critical coupling values where spectral gaps open.

Abstract

We consider rectangular graph superlattices of sides l1, l2 with the wavefunction coupling at the junctions either of the delta type, when they are continuous and the sum of their derivatives is proportional to the common value at the junction with a coupling constant alpha, or the "delta-prime-S" type with the roles of functions and derivatives reversed; the latter corresponds to the situations where the junctions are realized by complicated geometric scatterers. We show that the band spectra have a hidden fractal structure with respect to the ratio theta := l1/l2. If the latter is an irrational badly approximable by rationals, delta lattices have no gaps in the weak-coupling case. We show that there is a quantization for the asymptotic critical values of alpha at which new gap series open, and explain it in terms of number-theoretic properties of theta. We also show how the irregularity is manifested in terms of Fermi-surface dependence on energy, and possible localization properties under influence of an external electric field. KEYWORDS: Schroedinger operators, graphs, band spectra, fractals, quasiperiodic systems, number-theoretic properties, contact interactions, delta coupling, delta-prime coupling.