Slow dissipation and spreading in disordered classical systems: A direct comparison between numerics and mathematical bounds.

We study the breakdown of Anderson localization in the one-dimensional nonlinear Klein-Gordon chain, a prototypical example of a disordered classical many-body system. A series of numerical works indicate that an initially localized wave packet spreads polynomially in time, while analytical studies rather suggest a much slower spreading. Here, we focus on the decorrelation time in equilibrium. On the one hand, we provide a mathematical theorem establishing that this time is larger than any inverse power law in the effective anharmonicity parameter λ, and on the other hand our numerics show that it follows a power law for a broad range of values of λ. This numerical behavior is fully consistent with the power law observed numerically in spreading experiments, and we conclude that the state-of-the-art numerics may well be unable to capture the long-time behavior of such classical disordered systems.

Many-Body Delocalization as a Quantum Avalanche.

We propose a multiscale diagonalization scheme to study disordered one-dimensional chains, in particular, the transition between many-body localization (MBL) and the ergodic phase, expected to be governed by resonant spots. Our scheme focuses on the dichotomy of MBL versus validity of the eigenstate thermalization hypothesis. We show that a few natural assumptions imply that the system is localized with probability one at criticality. On the ergodic side, delocalization is induced by a quantum avalanche seeded by large ergodic spots, whose size diverges at the transition. On the MBL side, the typical localization length tends to the inverse of the maximal entropy density at the transition, but there is a divergent length scale related to the response to an inclusion of large ergodic spots. A mean-field approximation analytically illustrates these results and predicts a power-law distribution for thermal inclusions at criticality.

Many-body localization: stability and instability.

Rare regions with weak disorder (Griffiths regions) have the potential to spoil localization. We describe a non-perturbative construction of local integrals of motion (LIOMs) for a weakly interacting spin chain in one dimension, under a physically reasonable assumption on the statistics of eigenvalues. We discuss ideas about the situation in higher dimensions, where one can no longer ensure that interactions involving the Griffiths regions are much smaller than the typical energy-level spacing for such regions. We argue that ergodicity is restored in dimension d>1, although equilibration should be extremely slow, similar to the dynamics of glasses.This article is part of the themed issue 'Breakdown of ergodicity in quantum systems: from solids to synthetic matter'.

How a Small Quantum Bath Can Thermalize Long Localized Chains.

We investigate the stability of the many-body localized phase for a system in contact with a single ergodic grain modeling a Griffiths region with low disorder. Our numerical analysis provides evidence that even a small ergodic grain consisting of only three qubits can delocalize a localized chain as soon as the localization length exceeds a critical value separating localized and extended regimes of the whole system. We present a simple theory, consistent with De Roeck and Huveneers's arguments in [Phys. Rev. B 95, 155129 (2017)PRBMDO2469-995010.1103/PhysRevB.95.155129] that assumes a system to be locally ergodic unless the local relaxation time determined by Fermi's golden rule is larger than the inverse level spacing. This theory predicts a critical value for the localization length that is perfectly consistent with our numerical calculations. We analyze in detail the behavior of local operators inside and outside the ergodic grain and find excellent agreement of numerics and theory.

Exponentially Slow Heating in Periodically Driven Many-Body Systems.

We derive general bounds on the linear response energy absorption rates of periodically driven many-body systems of spins or fermions on a lattice. We show that, for systems with local interactions, the energy absorption rate decays exponentially as a function of driving frequency in any number of spatial dimensions. These results imply that topological many-body states in periodically driven systems, although generally metastable, can have very long lifetimes. We discuss applications to other problems, including the decay of highly energetic excitations in cold atomic and solid-state systems.

Symmetries of the ratchet current.

Recent advances in nonequilibrium statistical mechanics shed new light on the ratchet effect. The ratchet motion can thus be understood in terms of symmetry (breaking) considerations. We introduce an additional symmetry operation besides time reversal, that switches between two modes of operation. That mode reversal combined with time reversal decomposes the nonequilibrium action so as to clarify under what circumstances the ratchet current is a second order effect around equilibrium, what is the direction of the ratchet current, and what are possibly the symmetries in its fluctuations.

Quantum version of free-energy--irreversible-work relations.

We give a quantum version of the Jarzynski relation between the distribution of work done over a certain time-interval on a system and the difference of equilibrium free energies. The main ingredient is the identification of work depending on the quantum history of the system and the proper definition of various quantum ensembles over which the averages should be made. We also discuss a number of different regimes that have been considered by other authors and which are unified in the present set-up. In all cases, quantum or classical, it is a general relation between heat and time-reversal that makes the Jarzynski relation so universally valid.