Fading ergodicity in the ensembles of random matrices

Eigenstate thermalization hypothesis (ETH) has left a mark in many-body physics since it allows the linking of thermalization of physical observables with the applicability of random matrix theory (RMT). Recent work has shown counterexamples to thermalization, ranging from integrability, single-particle chaos, many-body localization, many-body scars, to Hilbert-space fragmentation. In all of these, the conventional ETH is violated. Until recently, it remained unanswered how does ETH break down when approaching the boundaries of ergodicity, and whether the range of validity of ETH coincides with the validity of RMT-like spectral statistics. In our work, we set up a novel scenario of the ETH breakdown, dubbed the fading ergodicity regime, which, while still ergodic, forms a bridge between ETH and nonergodic behavior. We provided numerical and analytical arguments for its validity in the quantum sun model of ergodicity-breaking phase transition (EBT). We conjectured this regime to be an inherent part of the EBT. Following the same phenomenology, we present more arguments in favor of our conjecture. We focus on the ensembles of random matrices that lean towards the EBT and illustrate the relevance of fading ergodicity regime. We extend our findings with further evidence that breakdown of ETH does not coincide with the breakdown of RMT-like spectral statistics.