Quantum Harmonic Oscillator Systems with Disorder

Bruno Nachtergaele, Robert J Sims, Günter Stolz

Research output: Contribution to journalArticle

11 Citations (Scopus)

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.

Original languageEnglish (US)
Pages (from-to)969-1012
Number of pages44
JournalJournal of Statistical Physics
Volume149
Issue number6
DOIs
StatePublished - Dec 2012

Fingerprint

Correlator
correlators
Harmonic Oscillator
harmonic oscillators
Disorder
disorders
Eigenfunctions
eigenvectors
Standard Model
Decay
Anderson Localization
Exponential Bound
Zero
decay
Ground State
Correlation Function
oscillators
Singularity
ground state
Sufficient Conditions

Keywords

  • Correlation decay
  • Dynamical localization
  • Harmonic oscillator systems

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Quantum Harmonic Oscillator Systems with Disorder. / Nachtergaele, Bruno; Sims, Robert J; Stolz, Günter.

In: Journal of Statistical Physics, Vol. 149, No. 6, 12.2012, p. 969-1012.

Research output: Contribution to journalArticle

Nachtergaele, Bruno ; Sims, Robert J ; Stolz, Günter. / Quantum Harmonic Oscillator Systems with Disorder. In: Journal of Statistical Physics. 2012 ; Vol. 149, No. 6. pp. 969-1012.
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