Breakdown of the adiabatic limit in low dimensional gapless systems

It is generally believed that a generic system can be reversibly transformed from one state into another by sufficiently slow change of parameters. A standard argument favoring this assertion is based on a possibility to expand the energy or the entropy of the system into the Taylor series in the ramp speed. Here we show that this argumentation is only valid in high enough dimensions and can break down in low-dimensional gapless systems. We identify three generic regimes of a system response to a slow ramp: (A) mean-field, (B) non-analytic, and (C) non-adiabatic. In the last regime the limits of the ramp speed going to zero and the system size going to infinity do not commute and the adiabatic process does not exist in the thermodynamic limit. We support our results by numerical simulations. Our findings can be relevant to condensed-matter, atomic physics, quantum computing, quantum optics, cosmology and others.Comment: 11 pages, 5 figures, to appear in Nature Physics (originally submitted version

Breakdown of the adiabatic limit in low dimensional gapless systems

It is generally believed that a generic system can be reversibly transformed from one state into another by sufficiently slow change of parameters. A standard argument favoring this assertion is based on a possibility to expand the energy or the entropy of the system into the Taylor series in the ramp speed. Here we show that this argumentation is only valid in high enough dimensions and can break down in low-dimensional gapless systems. We identify three generic regimes of a system response to a slow ramp: (A) mean-field, (B) non-analytic, and (C) non-adiabatic. In the last regime the limits of the ramp speed going to zero and the system size going to infinity do not commute and the adiabatic process does not exist in the thermodynamic limit. We support our results by numerical simulations. Our findings can be relevant to condensed-matter, atomic physics, quantum computing, quantum optics, cosmology and others.Comment: 11 pages, 5 figures, to appear in Nature Physics (originally submitted version

Transport in quasiperiodic interacting systems: from superdiffusion to subdiffusion

Using a combination of numerically exact and renormalization-group techniques we study the nonequilibrium transport of electrons in an one-dimensional interacting system subject to a quasiperiodic potential. For this purpose we calculate the growth of the mean-square displacement as well as the melting of domain walls. While the system is nonintegrable for all studied parameters, there is no on finite region default of parameters for which we observe diffusive transport. In particular, our model shows a rich dynamical behavior crossing over from superdiffusion to subdiffusion. We discuss the implications of our results for the general problem of many-body localization, with a particular emphasis on the rare region Griffiths picture of subdiffusion.Comment: 6 pages, 5 figures. A more detailed analysis of the dynamical exponents extraction and discussion of the relevant times. Adds a log-derivative for the FRG sectio

Entropies for severely contracted configuration space

We demonstrate that dual entropy expressions of the Tsallis type apply naturally to statistical-mechanical systems that experience an exceptional contraction of their configuration space. The entropic index $\alpha>1$ describes the contraction process, while the dual index $\alpha ^{\prime }=2-\alpha<1$ defines the contraction dimension at which extensivity is restored. We study this circumstance along the three routes to chaos in low-dimensional nonlinear maps where the attractors at the transitions, between regular and chaotic behavior, drive phase-space contraction for ensembles of trajectories. We illustrate this circumstance for properties of systems that find descriptions in terms of nonlinear maps. These are size-rank functions, urbanization and similar processes, and settings where frequency locking takes place

On the stability of many-body localization in d>1d>1

Recent work by De Roeck et al. [Phys. Rev. B 95, 155129 (2017)] has argued that many-body localization (MBL) is unstable in two and higher dimensions due to a thermalization avalanche triggered by rare regions of weak disorder. To examine these arguments, we construct several models of a finite ergodic bubble coupled to an Anderson insulator of non-interacting fermions. We first describe the ergodic region using a GOE random matrix and perform an exact diagonalization study of small systems. The results are in excellent agreement with a refined theory of the thermalization avalanche that includes transient finite-size effects, lending strong support to the avalanche scenario. We then explore the limit of large system sizes by modeling the ergodic region via a Hubbard model with all-to-all random hopping: the combined system, consisting of the bubble and the insulator, can be reduced to an effective Anderson impurity problem. We find that the spectral function of a local operator in the ergodic region changes dramatically when coupling to a large number of localized fermionic states---this occurs even when the localized sites are weakly coupled to the bubble. In principle, for a given size of the ergodic region, this may arrest the avalanche. However, this back-action effect is suppressed and the avalanche can be recovered if the ergodic bubble is large enough. Thus, the main effect of the back-action is to renormalize the critical bubble size.Comment: v3: Published version. Expanded the discussion in Section IV to include a new calculation and figure (Fig. 7

The mixmaster universe: A chaotic Farey tale

When gravitational fields are at their strongest, the evolution of spacetime is thought to be highly erratic. Over the past decade debate has raged over whether this evolution can be classified as chaotic. The debate has centered on the homogeneous but anisotropic mixmaster universe. A definite resolution has been lacking as the techniques used to study the mixmaster dynamics yield observer dependent answers. Here we resolve the conflict by using observer independent, fractal methods. We prove the mixmaster universe is chaotic by exposing the fractal strange repellor that characterizes the dynamics. The repellor is laid bare in both the 6-dimensional minisuperspace of the full Einstein equations, and in a 2-dimensional discretisation of the dynamics. The chaos is encoded in a special set of numbers that form the irrational Farey tree. We quantify the chaos by calculating the strange repellor's Lyapunov dimension, topological entropy and multifractal dimensions. As all of these quantities are coordinate, or gauge independent, there is no longer any ambiguity--the mixmaster universe is indeed chaotic.Comment: 45 pages, RevTeX, 19 Figures included, submitted to PR