### 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 language | English (US) |
---|---|

Pages (from-to) | 969-1012 |

Number of pages | 44 |

Journal | Journal of Statistical Physics |

Volume | 149 |

Issue number | 6 |

DOIs | |

State | Published - Dec 2012 |

### Fingerprint

### Keywords

- Correlation decay
- Dynamical localization
- Harmonic oscillator systems

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*149*(6), 969-1012. https://doi.org/10.1007/s10955-012-0652-1

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

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 149, no. 6, pp. 969-1012. https://doi.org/10.1007/s10955-012-0652-1

}

TY - JOUR

T1 - Quantum Harmonic Oscillator Systems with Disorder

AU - Nachtergaele, Bruno

AU - Sims, Robert J

AU - Stolz, Günter

PY - 2012/12

Y1 - 2012/12

N2 - 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.

AB - 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.

KW - Correlation decay

KW - Dynamical localization

KW - Harmonic oscillator systems

UR - http://www.scopus.com/inward/record.url?scp=84871267355&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84871267355&partnerID=8YFLogxK

U2 - 10.1007/s10955-012-0652-1

DO - 10.1007/s10955-012-0652-1

M3 - Article

AN - SCOPUS:84871267355

VL - 149

SP - 969

EP - 1012

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 6

ER -