We study quantum Hamiltonians with potentials defined by strictly ergodic dynamical systems. Our interest here are models where physical properties are understood in some regimes of disorder and the extent to which they vary in alternate regimes of disorder. For Schrödinger operators we show properties known to hold in the case of analytic potentials on the torus hold even for rough potentials only required to be Holder continuous. Specifically in this case we show, assuming a positive Lyapunov exponent, dynamical localization properties hold; as well as continuity of the measure of the spectrum for all rotations. For the quantum Ising model we show for phase structure that occur in the random regime, there are similar conditions for existence under the assumption of strictly ergodic dynamics. That is, moment conditions for random disorder are paralleled by conditions on the sampling functions in deterministic disorder. We obtain conditions for existence of phase transitions given any strictly egodically defined disorder. In addition, a new multiscale analysis method is developed to show the existence of stretched exponential decay in the random cluster model generalization of the quantum Ising model where only slower decay was obainable by previous methods.
|Commitee:||Cranston, Michael, Gorodetski, Anton|
|School:||University of California, Irvine|
|Department:||Mathematics - Ph.D.|
|School Location:||United States -- California|
|Source:||DAI-B 73/11(E), Dissertation Abstracts International|
|Subjects:||Mathematics, Quantum physics|
|Keywords:||Ergodic disorder, Ising model, Percolation, Schroedinger operators|
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