Quantum Intermittency in Almost-Periodic Lattice Systems Derived from their Spectral Properties

TL;DR

This paper constructs almost-periodic Hamiltonians with multi-fractal spectral measures, analyzes their quantum dynamics, and establishes a relation between long-time behavior of moments and spectral dimensions, revealing exact and approximate cases.

Contribution

It introduces a recursive method to construct almost-periodic Hamiltonians with multi-fractal spectra and links their spectral properties to quantum dynamical behavior.

Findings

Derived a relation between moment scaling and spectral dimensions.

Identified cases of exact and approximate relations, explaining discrepancies.

Numerically and theoretically analyzed quantum dynamics of constructed systems.

Abstract

Hamiltonian tridiagonal matrices characterized by multi-fractal spectral measures in the family of Iterated Function Systems can be constructed by a recursive technique here described. We prove that these Hamiltonians are almost-periodic. They are suited to describe quantum lattice systems with nearest neighbours coupling, as well as chains of linear classical oscillators, and electrical transmission lines. We investigate numerically and theoretically the time dynamics of the systems so constructed. We derive a relation linking the long-time, power-law behaviour of the moments of the position operator, expressed by a scaling function $\beta$ of the moment order $\alpha$, and spectral multi-fractal dimensions, D_q, via $\beta(\alpha) = D_{1-\alpha}$. We show cases in which this relation is exact, and cases where it is only approximate, unveiling the reasons for the discrepancies.