Quantum harmonic oscillator systems with disorder

TL;DR

This paper investigates localization phenomena in disordered quantum harmonic oscillator lattices, establishing conditions for dynamical localization and decay of correlations using eigenfunction correlators, applicable to various models and temperatures.

Contribution

It introduces a generalized framework for eigenfunction correlators, including singular functions, and applies Anderson localization techniques to prove localization and correlation decay in disordered oscillator systems.

Findings

Dynamical localization characterized by zero-velocity Lieb-Robinson bounds.

Exponential decay of static correlations at low energies.

Applicability to both finite and infinite oscillator systems.

Abstract

We study many-body properties of quantum harmonic oscillator lattices with disorder. A sufficient condition for dynamical localization, expressed as a zero-velocity Lieb-Robinson bound, is formulated in terms of the decay of the eigenfunction correlators for an effective one-particle Hamiltonian. We show how state-of-the-art techniques for proving Anderson localization can be used to prove that these properties hold in a number of standard models. We also derive bounds on the static and dynamic correlation functions at both zero and positive temperature in terms of one-particle eigenfunction correlators. In particular, we show that static correlations decay exponentially fast if the corresponding effective one-particle Hamiltonian exhibits localization at low energies, regardless of whether there is a gap in the spectrum above the ground state or not. Our results apply to finite as well as to infinite oscillator systems. The eigenfunction correlators that appear are more general than those previously studied in the literature. In particular, we must allow for functions of the Hamiltonian that have a singularity at the bottom of the spectrum. We prove exponential bounds for such correlators for some of the standard models.