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  • 1
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    Local perturbations perturb—exponentially–locally (2015)
    American Institute of Physics (AIP)
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    Publication Date: 2015-06-19
    Description: We elaborate on the principle that for gapped quantum spin systems with local interaction, “local perturbations [in the Hamiltonian] perturb locally [the groundstate].” This principle was established by Bachmann et al. [Commun. Math. Phys. 309 , 835–871 (2012)], relying on the “spectral flow technique” or “quasi-adiabatic continuation” [M. B. Hastings, Phys. Rev. B 69 , 104431 (2004)] to obtain locality estimates with sub-exponential decay in the distance to the spatial support of the perturbation. We use ideas of Hamza et al. [J. Math. Phys. 50 , 095213 (2009)] to obtain similarly a transformation between gapped eigenvectors and their perturbations that is local with exponential decay. This allows to improve locality bounds on the effect of perturbations on the low lying states in certain gapped models with a unique “bulk ground state” or “topological quantum order.” We also give some estimate on the exponential decay of correlations in models with impurities where some relevant correlations decay faster than one would naively infer from the global gap of the system, as one also expects in disordered systems with a localized groundstate.
    Print ISSN: 0022-2488
    Electronic ISSN: 1089-7658
    Topics: Mathematics , Physics
    Published by American Institute of Physics (AIP)
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  • 2
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    Locality and nonlocality of classical restrictions of quantum spin systems with applications to quantum large deviations and entanglement (2015)
    American Institute of Physics (AIP)
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    Publication Date: 2015-02-04
    Description: We study the projection on classical spins starting from quantum equilibria. We show Gibbsianness or quasi-locality of the resulting classical spin system for a class of gapped quantum systems at low temperatures including quantum ground states. A consequence of Gibbsianness is the validity of a large deviation principle in the quantum system which is known and here recovered in regimes of high temperature or for thermal states in one dimension. On the other hand, we give an example of a quantum ground state with strong nonlocality in the classical restriction, giving rise to what we call measurement induced entanglement and still satisfying a large deviation principle.
    Print ISSN: 0022-2488
    Electronic ISSN: 1089-7658
    Topics: Mathematics , Physics
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  • 3
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    Quantum diffusion with drift and the Einstein relation. I (2014)
    American Institute of Physics (AIP)
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    Publication Date: 2014-06-26
    Description: We study the dynamics of a quantum particle hopping on a simple cubic lattice and driven by a constant external force. It is coupled to an array of identical, independent thermal reservoirs consisting of free, massless Bose fields, one at each site of the lattice. When the particle visits a site x of the lattice it can emit or absorb field quanta of the reservoir at x. Under the assumption that the coupling between the particle and the reservoirs and the driving force are sufficiently small, we establish the following results: The ergodic average over time of the state of the particle approaches a non-equilibrium steady state describing a non-zero mean drift of the particle. Its motion around the mean drift is diffusive, and the diffusion constant and the drift velocity are related to one another by the Einstein relation.
    Print ISSN: 0022-2488
    Electronic ISSN: 1089-7658
    Topics: Mathematics , Physics
    Published by American Institute of Physics (AIP)
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  • 4
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    Quantum diffusion with drift and the Einstein relation. II (2014)
    American Institute of Physics (AIP)
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    Publication Date: 2014-06-27
    Description: This paper is a companion to Paper I [W. De Roeck, J. Fröhlich, and K. Schnelli, “Quantum diffusion with drift and the Einstein relation. I,” J. Math. Phys.55, 075206 (2014)]. The purpose of this paper is to describe and prove a certain number of technical results used in Paper I, but not proven there. Both papers concern long-time properties (diffusion, drift) of the motion of a driven quantum particle coupled to an array of thermal reservoirs. The main technical results derived in the present paper are: (1) an asymptotic perturbation theory applicable for small driving force, and (2) the construction of time-dependent correlation functions of particle observables.
    Print ISSN: 0022-2488
    Electronic ISSN: 1089-7658
    Topics: Mathematics , Physics
    Published by American Institute of Physics (AIP)
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