Spectrum and diffusion for a class of tight-binding models on hypercubes

TL;DR

This paper introduces exactly solvable tight-binding models on hypercubes, revealing how spectral properties influence wave diffusion, with results showing diffusion exponents vary widely despite spectral continuity.

Contribution

It provides an analytical framework for understanding spectral and diffusion properties in high-dimensional hypercube models, highlighting the nuanced relationship between spectrum type and diffusion behavior.

Findings

Spectral spectrum can be fractal or absolutely continuous.

Diffusion exponent varies between 0 and 1 regardless of spectrum type.

Analytical expressions for spectral and diffusion exponents are derived.

Abstract

We propose a class of exactly solvable anisotropic tight-binding models on an infinite-dimensional hypercube. The energy spectrum is analytically computed and is shown to be fractal and/or absolutely continuous according to the value hopping parameters. In both cases, the spectral and diffusion exponents are derived. The main result is that, even if the spectrum is absolutely continuous, the diffusion exponent for the wave packet may be anything between 0 and 1 depending upon the class of models.