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Birkhäuser Basel eBooks

Statistical Properties of Disordered Quantum Systems

Francesco Fidaleo, Carlangelo Liverani

Published Jan 1, 2004• 4 citations
Publisher

Abstract

We discuss two different approaches to the study of the long-time behavior of some disordered quantum anharmonic chains.
DOI
10.1007/3-7643-7314-8_7

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We investigate the long-time behavior of a wave packet initially localized at a single site n0{n}_{0}n0​ in translationally invariant harmonic and anharmonic chains with random interactions. In the harmonic case, the energy profile ⟨en(t)⟩‾\overline{⟨{e}_{n}(t)⟩}⟨en​(t)⟩​ averaged on time and disorder decays for large $|n\ensuremath{-}{n}_{0}|$ as a power law $\overline{⟨{e}_{n}(t)⟩}\ensuremath{\approx}C{|n\ensuremath{-}{n}_{0}|}^{\ensuremath{-}\ensuremath{\eta}}$, where $\ensuremath{\eta}=5/2$ and 3/2 for initial displacement and momentum excitations, respectively. The prefactor CCC depends on the probability distribution of the harmonic coupling constants and diverges in the limit of weak disorder. As a consequence, the moments $⟨{m}_{\ensuremath{\nu}}(t)⟩$ of the energy distribution averaged with respect to disorder diverge in time as ${t}^{\ensuremath{\beta}(\ensuremath{\nu})}$ for $\ensuremath{\nu}\ensuremath{\ge}2$, where $\ensuremath{\beta}=\ensuremath{\nu}+1\ensuremath{-}\ensuremath{\eta}$ for $\ensuremath{\nu}>\ensuremath{\eta}\ensuremath{-}1$. Molecular-dynamics simulations yield good agreement with these theoretical predictions. Therefore, in this system, the second moment of the wave packet diverges as a function of time despite the wave packet is not spreading. Thus, this only criterion, often considered earlier as proving the spreading of a wave packet, cannot be considered as sufficient in any model. The anharmonic case is investigated numerically. It is found for intermediate disorder that the tail of the energy profile becomes very close to those of the harmonic case. For weak and strong disorders, our results suggest that the crossover to the harmonic behavior occurs at much larger $|n\ensuremath{-}{n}_{0}|$ and larger time.
A central goal in the study of modern condensed matter physics is the characterization of the dynamical properties of quantum systems. Many decades of effort towards this goal, studying a diverse range of (near-equilibrium) quantum matter, from Fermi liquids, to quantum two-level systems, to interacting spin models, and more, has revealed a remarkable pervasiveness of the simple dynamical description of these complex systems in terms of quasi-particles that carry spin, charge, and heat, and that are generally able to equilibrate systems. \n\nThis thesis is an examination of some exceptions to this rule. Specifically, we study a number of instances of quantum matter where equilibration phenomena happens at rather long time scales, or does not occur at all. Particular emphasis is laid on the role of dimensionality, disorder, and dissipation in engendering such novel dynamical behavior.\n\nFirst, we consider non-equilibrium dynamics in one-dimensional quasi-condensates. Low dimensionality inhibits scattering in these systems, and low-energy excitations are long-lived phase fluctuations that exhibit an enriched conformal symmetry. Utilizing this symmetry, we generalize sudden quenches typically used to study non-equilibrium dynamics to quenches along general relativistic and conformal trajectories. Gases never truly equilibrate after such a quench; instead, they evolve into a `prethermal' state with thermal-looking correlations and a chiral asymmetry. \n\nWe then study the problem of the dynamical transition driven by disorder, from an ergodic to a non-ergodic phase, in one-dimensional quantum spin chains. In particular, in XXZ chains with on-site disorder, we find a unique intermediate phase straddling the boundary of the dynamical phase transition, wherein rare-region effects lead to long-time tails in equilibration and vanishing DC conduction before the onset of non-ergodicity. We propose generalizations of such `Griffiths' behavior to arbitrary dimensions. We also study the dynamics of random-bond Heisenberg chains by developing a strong-disorder renormalization group protocol for these systems. We discuss how magnetic noise from such disordered systems contains signatures of their anomalous dynamical properties. \n\nNext, we re-examine the phenomenological theory of two-level systems in amorphous materials in the light of new experimental evidence that these states have large electric/magnetic dipole moments. We propose and justify an interpretation of the model as one of tunneling electrons slowed down by a large phonon drag and discuss the dynamical consequences of such polaronic effects. \n\nFinally, we discuss how magnetic noise measurements can be used to non-invasively access the anomalous properties of systems such as those discussed above. In particular, we examine how scattering properties of isolated magnetic impurities and non-local transport in a variety of two-dimensional materials can be probed experimentally using NV centers as noise magnetometers.