Speaker
Description
The random matrix theory (RMT) has been a hallmark in the study of quantum thermalization as it sets a link between properties of random matrix eigenstates and observable matrix elements. Nevertheless, this description is incomplete for lattice models. A more relevant picture is given by the eigenstate thermalization hypothesis (ETH), which takes into account the local structure of the Hamiltonian and introduces an energy scale, dubbed Thouless energy, separating universal RMT dynamics from local Hamiltonian dynamics. Recently, this picture has been extended to systems in the vicinity of an ergodicity breaking phase transition, where the Thouless energy becomes vanishingly small. This phenomenon, denoted fading ergodicity [\href{https://journals.aps.org/prb/abstract/10.1103/PhysRevB.110.134206}{Phys. Rev. B 110, 134206}], shows how fluctuations of matrix elements soften whence an eigenstate transition is approached. Here, following the approach in the seminal work by Deutsch we show that within RMT one can derive the softening of matrix elements fluctuations from the properties of eigenstate coefficients. Remarkably, there exist a direct relation to the fractal nature of its eigenstates in the unperturbed basis. We then extend our analysis to the Ultrametric matrix and Rosenzweig-Porter matrix ensembles. We argue that these RMT models, share some remarkably similar dynamical properties of physical observables before the onset of localization. Our results reveal that the fading ergodicity gives rise to universal features of anomalously slow dynamics and unveil a potentially accessible measure, the noise spectrum, as a probe of emergent nonthermal behaviour.
