Quantum Ergodicity: Stability and Transitions

We shall develop methods and models for analyzing quantum ergodicity in many-body systems, proving its stability against small perturbations, and studying ergodicity-breaking transitions due to integrability, disorder, or localized impurities. Ergodicity is a cornerstone of statistical mechanics and a key manifestation of many-body quantum chaos, while manipulating ergodicity and engineering ergodicity-breaking transitions will have immense applications (cf.scarred states in Rydberg arrays, heating transitions in Floquet systems, time crystalline phases of matter). PI proposed groundbreaking methods for establishing quantum ergodicity by rigorous analysis of spectral statistics, correlation functions, dynamical complexity indicators and entanglement on the basis of space-time duality. Most of current understanding of many-body physics or quantum fields is based on perturbative expansions around free, integrable or localized models. Here we propose a twist of paradigm: We shall study weak perturbations of statistically exactly solvable ergodic models, such as dual-unitary chaotic quantum circuits proposed by PI. An intuitive expectation of structural stability of ergodic dynamics (in analogy to rigorous results in classical ergodic theory) implies that such expansions typically have, unlike expansions around free/integrable models, finite radii of convergence. Various order parameters of the ergodic phase shall be developed and compared in their utility to signal and characterize ergodicity-breaking transitions. A related goal is a construction of exactly solvable models in which the eigenstate thermalization hypothesis can be proven. Being of fundamental importance in mathematical and statistical physics, the results are expected to have widespread applications across fields: from studying localization transitions in disordered systems, benchmarking quantum simulators and certifying quantum supremacy, to rigorous proofs of chaos in holographic models of black holes.