Review “Ergodicity, Entanglement and Many-Body Localization”

Quantum systems of many particles, which satisfy the ergodicity hypothesis, are conventionally described by statistical mechanics. However, not all quantum systems are ergodic, with many-body localization providing a generic mechanism of ergodicity breaking by disorder. Many-body localized (MBL) systems remain perfect insulators at non-zero temperature, which do not thermalize and therefore cannot be described using statistical mechanics. In the review “Ergodicity, Entanglement and Many-Body Localization” arXiv:1804.11065 written together with D. Abanin, E. Altman, and I. Bloch, we summarize recent theoretical and experimental advances in studies of MBL systems, focusing on the new perspective provided by entanglement and non-equilibrium experimental probes such as quantum quenches.

Weak ergodicity breaking from quantum many-body scars

Quantum scars present an example of weak ergodicity breaking in a context of quantum chaos. Recently, we also generalized this concept to the many-body case.

Quantum scars in a billiard

Imagine a ball bouncing around in an oval stadium. It will bounce around chaotically, back and forth through the available space. As its motion is random, it will sooner or later visit every place in the stadium as is illustrated in the example below:

animation of chaotic trajectory in a billiard

Amidst all the chaos, however, there might be a potential for order: if the ball happens to hit the wall at a special spot and at the “correct” angle of incidence, it might end up in a periodic orbit, visiting the same places in the stadium over and over and not visiting the others. Such a periodic orbit is extremely unstable as the slightest perturbation will divert the ball off its track and back into chaotic pondering around the stadium:

The same idea is applicable to quantum systems, except that instead of a ball bouncing around, we are looking at a wave, and instead of a trajectory, we are observing a probability function. Classical periodic orbits can cause a quantum wave to be concentrated in its vicinity, causing a “scar”-like feature in a probability that would otherwise be uniform. Such imprints of classical orbits on the probability function have been named “quantum scars”. Below we compare the “scarred“ eigenstate in the stadium with the more typical state:

 

Quantum many-body scars

We observed quantum-scarred eigenstates in the theoretical model that describes a chain of Rydberg atoms. All atoms in the chain can be in two possible states: excited and ground state. Moreover there exist a constraint that prohibits two excited atoms to be adjacent to each other. We found a coherent oscillations in such a system which underlie the quantum-scarred eigenstates. Below we show the animated cartoon of this trajectory for L=8 atoms.

Here the graph shows the space of all possible configurations, and the bottom shows the average density of excitation on each site. Animation shows oscillations between two patterns.

Probing dynamics by matrix elements

In order to determine generic non-equilibrium dynamics of the generic local observable O one needs two ingredients. The spectrum of the many-body Hamiltonian and the matrix elements of the operator O in the basis of eigenstates completely specify evolution of the local observable O. Moreover, the matrix elements at small energy differences correspond to the long-time dynamics that may be very hard to access in the thermalizing system.

Using these insights, we studied the properties of the matrix elements across the many-body localization transition in the recent preprint arXiv:1610.02389. In particular, we used matrix elements to define the (many-body) Thouless energy ETh which corresponds to the inverse time scale of relaxation. The behavior of this Thouless energy reveals the critical fan preceding localization transition. In addition, we associate the critical region with the onset of broad distributions and typicality breakdown.