Quantum harmonic oscillator systems with disorder

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.

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• Localization Properties of the XY Spin Chain. A review of mathematical results with an eye toward Many-Body Localization
• A uniform area law for the entanglement of eigenstates in the disordered XY chain
• An Area Law for the Bipartite Entanglement of Disordered Oscillator Systems
• Many-Body Localization: Concepts and Simple Models
• Entanglement of a class of non-Gaussian states in disordered harmonic oscillator systems
• Bounds on the bipartite entanglement entropy for oscillator systems with or without disorder
• On the Regime of Localized Excitations for Disordered Oscillator Systems
• First order sensitivity analysis of symplectic eigenvalues
• Localization in the random XXZ quantum spin chain
• Many-body localization in the droplet spectrum of the random XXZ quantum spin chain
• Dynamical Evolution of Entanglement in Disordered Oscillator Systems
• Derivation of Euler equations from quantum and classical microscopic dynamics