Many-body localization provides an example of strong ergodicity breaking. System has a memory of its initial state for any initial conditions. We also aim to understand what are the other possible weaker forms of ergodicity breaking. Below we show some recent results in this direction.

# Mixed phase space in quantum many-body systems

In the earlier work we identified the quantum scars as a mechanism of weak ergodicity breaking in interacting many-body systems. The name “quantum many-body scars” was inspired by the presence of unstable TDVP trajectory reported in Ref. [Phys. Rev. Lett. 122, 040603 (2019)], thus allowing to make parallels with quantum scars in single-particle billiards.

However, in a few-body quantum systems quantum scarring is not the only possible mechanism of dependence of relaxation on initial state. In fact, the mixed phase space in a few body systems presents another common mechanism. In the most recent work arXiv:1905.08564 we generalized the concept of mixed phase space to strongly interacting quantum systems.

# Weak ergodicity breaking from quantum many-body scars

Quantum scars present an example of weak ergodicity breaking in a context of quantum chaos. In our recent work in Nature Physics, 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:

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.