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arXiv:1701.05755 · Annalen der Physik

Classical and quantum systems: transport due to rare events

François Huveneers

Published Mar 2, 2017• 13 citations
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Abstract

We review possible mechanisms for energy transfer based on ‘rare’ or ‘non‐perturbative’ effects, in physical systems that present a many‐body localized phenomenology. The main focus is on classical systems, with or without quenched disorder. For non‐quantum systems, the breakdown of localization is usually not regarded as an issue, and we thus aim at identifying the fastest channels for transport. Next, we contemplate the possibility of applying the same mechanisms in quantum systems, including disorder free systems (e.g. Bose‐Hubbard chain), disordered many‐body localized systems with mobility edges at energies below the edge, and strongly disordered lattice systems in . For quantum mechanical systems, the relevance of these considerations for transport is currently a matter of debate.

DOI
10.1002/andp.201600384

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We review possible mechanisms for energy transfer based on ‘rare’ or ‘non‐perturbative’ effects, in physical systems that present a many‐body localized phenomenology. The main focus is on classical systems, with or without quenched disorder. For non‐quantum systems, the breakdown of localization is usually not regarded as an issue, and we thus aim at identifying the fastest channels for transport. Next, we contemplate the possibility of applying the same mechanisms in quantum systems, including disorder free systems (e.g. Bose‐Hubbard chain), disordered many‐body localized systems with mobility edges at energies below the edge, and strongly disordered lattice systems in . For quantum mechanical systems, the relevance of these considerations for transport is currently a matter of debate.
We review possible mechanisms for energy transfer based on 'rare' or 'non-perturbative' effects, in physical systems that present a many-body localized phenomenology. The main focus is on classical systems, with or without quenched disorder. For non-quantum systems, the breakdown of localization is usually not regarded as an issue, and we thus aim at identifying the fastest channels for transport. Next, we contemplate the possibility of applying the same mechanisms in quantum systems, including disorder free systems (e.g. Bose-Hubbard chain), disordered many-body localized systems with mobility edges at energies below the edge, and strongly disordered lattice systems in d>1. For quantum mechanical systems, the relevance of these considerations for transport is currently a matter of debate.
Disorder and interactions both play crucial roles in quantum transport. Decades ago, Mott showed that electron-electron interactions can lead to insulating behavior in materials that conventional band theory predicts to be conducting. Soon thereafter, Anderson demonstrated that disorder can localize a quantum particle through the wave interference phenomenon of Anderson localization. Although interactions and disorder both separately induce insulating behavior, the interplay of these two ingredients is subtle and often leads to surprising behavior at the periphery of our current understanding. Modern experiments probe these phenomena in a variety of contexts (e.g. disordered superconductors, cold atoms, photonic waveguides, etc.); thus, theoretical and numerical advancements are urgently needed. In this thesis, we report progress on understanding two contexts in which the interplay of disorder and interactions is especially important. The first is the so-called “dirty” or random boson problem. In the past decade, a strong-disorder renormalization group (SDRG) treatment by Altman, Kafri, Polkovnikov, and Refael has raised the possibility of a new unstable fixed point governing the superfluid-insulator transition in the one-dimensional dirty boson problem. This new critical behavior may take over from the weak-disorder criticality of Giamarchi and Schulz when disorder is sufficiently strong. We analytically determine the scaling of the superfluid susceptibility at the strong-disorder fixed point and connect our analysis to recent Monte Carlo simulations by Hrahsheh and Vojta. We then shift our attention to two dimensions and use a numerical implementation of the SDRG to locate the fixed point governing the superfluid-insulator transition there. We identify several universal properties of this transition, which are fully independent of the microscopic features of the disorder. The second focus of this thesis is the interplay of localization and interactions in systems with high energy density (i.e., far from the usual low energy limit of condensed matter physics). Recent theoretical and numerical work indicates that localization can survive in this regime, provided that interactions are sufficiently weak. Stronger interactions can destroy localization, leading to a so-called many-body localization transition. This dynamical phase transition is relevant to questions of thermalization in isolated quantum systems: it separates a many-body localized phase, in which localization prevents transport and thermalization, from a conducting (“ergodic”) phase in which the usual assumptions of quantum statistical mechanics hold. Here, we present evidence that many-body localization also occurs in quasiperiodic systems that lack true disorder.
We establish rigorously that transport is slower than diffusive for a class of disordered one-dimensional Hamiltonian chains. This is done by deriving quantitative bounds on the variance in equilibrium of the energy or particle current, as a function of time. The slow transport stems from the presence of rare insulating regions (Griffiths regions). In many-body disordered quantum chains, they correspond to regions of anomalously high disorder, where the system is in a localized phase. In contrast, we deal with quantum and classical disordered chains where the interactions, usually referred to as anharmonic couplings in classical systems, are sparse. The system hosts thus rare regions with no interactions and, since the chain is Anderson localized in the absence of interactions, the non-interacting rare regions are insulating. Part of the mathematical interest of our model is that it is one of the few non-integrable models where the diffusion constant can be rigorously proven not to be infinite.
The study of disordered systems in statistical mechanics is a topic that garners strong interest from the mathematical and physics communities. In this dissertation, we present two problems dealing with some type of strongly disordered interacting particle system. In particular, we study properties related to transport of energy in these systems. The first model deals with the spreading of wave packets in a 1D lattice governed by the disordered discrete non-linear Schrödinger (DNLS) equation. We study the rate of decay of the maximum of a wave packet initially supported around the origin, and prove slower than power law spreading. The second model deals with many-body localization (MBL), a phenomenon in which certain strongly disordered interacting particle systems fail to reach thermal equilibrium. We prove the vanishing of the heat conductivity for a robust 1D spin chain model. This result serves to constitute a first step toward a rigorous proof of many-body localization.
We investigate the possibility of many-body localization in translation-invariant Hamiltonian systems, which was recently brought up by several authors. A key feature of many-body localized disordered systems is recovered, namely the fact that resonant spots are rare and far-between. However, we point out that resonant spots are mobile, unlike in models with strong quenched disorder, and that these mobile spots constitute a possible mechanism for delocalization, albeit possibly only on very long timescales. In some models, this argument for delocalization can be made very explicit in first order of perturbation theory in the hopping. For models where this does not work, we present instead a nonperturbative argument that relies solely on ergodicity inside the resonant spots.
Quantum interference lies at the heart of several surprising equilibrium and non‐equilibrium phenomena in many‐body Physics. Here we discuss two recently explored non‐equilibrium scenarios where external periodic drive applied to closed (i.e., not attached to any external bath) quantum many‐body systems have apparently opposite effects in respective cases. In one case it freezes/localizes a disorder free system dynamically, while in the other it delocalizes a disordered many‐body localized system, and quantum interference is responsible for both the effects. We review these in the perspective of more general questions of ergodicity, energy absorption, asymptotic behavior, and finally the essential role of quantum mechanics in understanding these issues in periodically driven closed many‐body systems. In this article we intend to deliver a non‐technical account of some recent developments in this field in a manner accessible to a broad readership. image