Boundaries of quantum chaos

Introduction

Physical systems are both universal and special, depending on the physical property under consideration and the corresponding scale, such as the energy, time, or length scale. From the perspective of quantum dynamics, it has been recently established that the ability of isolated quantum systems to thermalize after being driven away from equilibrium is related to the emergence of universal properties that comply with random matrix theory.

Indicators of Ergodicity and Quantum Chaos

Specific indicators for the onset of ergodicity and quantum chaos are related to:

1. The statistical properties of the energy spectrum.
2. Hamiltonian eigenfunction properties.
3. The expectation values of observables in these states.

At the same time, however, these indicators also carry fingerprints of nonuniversal properties of a given system. Remarkable examples of the latter include, e.g., information on the nature of energy and charge transport, and the scaling of characteristic relaxation times.

Main Conjectures

One of the main conjectures of this ERC project is that these indicators, despite complying with the universal predictions of random matrix theory, also carry information about the proximity of phase transitions. Here we focus on ergodicity breaking phase transitions, which represent a novel type of phase transitions at the boundaries of quantum chaos.

Extension of the Project Scope

We then extend the scope of the project to the critical properties at the ergodicity breaking transitions. We conjecture that they also exhibit certain universal properties, yet likely different from those described by the conventional random matrix theory.

Project Outcome

The outcome of the project is to establish a phenomenological theory of ergodicity breaking transitions that applies to a broad class of quantum systems. Additionally, we aim to clarify the impact of:

• Dimensionality
• Symmetries
• The nature of interactions
• Other mechanisms

on universal properties of ergodicity breaking transitions.