Excited states of interacting one-dimensional systems are many-body localized only at sufficiently strong disorder, and melt continuously into an ergodic, high temperature, classical liquid as disorder is weakened. In contrast to conventional classical or quantum phase transitions, this dynamical delocalization transition represents an entirely new type of critical point in which statistical mechanics and thermalization break down sharply at a continuous phase transition. We proposed a novel renormalization group approach to derive an effective quantum percolation model by accounting for collective resonant tunneling processes. This leads to a picture of delocalization driven via a percolating resonant backbone. In this way, we computed scaling properties of entanglement, dynamics, and energy transport and find an unusual critical structure that has no precedent in more conventional classical or quantum phase transitions. We proposed a scaling theory of entanglement near the transition, revealing some discontinuous aspects and a bimodal structure. Recently, we constructed a unifying picture of the MBL transition in one dimension in terms of a two-parameter Kosterlitz-Thouless scaling, relying on the fact that the transition is driven by `avalanches’. We showed that this simple picture is compatible with many different microscopically motivated RG models for the transition.

The concepts of scaling and universality near a critical point are central to the modern understanding of equilibrium statistical mechanics and condensed matter physics. Using a quantitative and analytically tractable renormalization group method, we showed that universality and quantum criticality can also emerge at “infinite temperature”, far from thermal equilibrium. Our approach provides us with an infinite family of systems whose excited states exhibit properties characteristic of quantum critical ground states. These “quantum critical glasses” can therefore be thought of as critical variants of MBL, and represent an entirely new class of non-equilibrium dynamical phases and phase-transitions with no equilibrium counterparts.

The concept of symmetry plays a crucial role in our understanding of phases of matter. The interplay of symmetry and dimensionality leads to very general constraints on possible types of symmetry-breaking phases and phase transitions, such as the Peierls or Mermin-Wagner theorems. We derived similar constraints for quantum systems far-from-equilibrium, and showed that MBL is not possible with symmetry groups that protect multiplets (e.g., all non-Abelian symmetry groups). Our results rule out the existence of many MBL symmetry-protected topological phases including electronic topological insulators. We investigated the consequences of this no-go theorem for random-bond XXZ spin chains, and for the protection of topological parafermionic zero modes using MBL. More recently, we also analyzed the instability of tentative nonergodic phases in non-Abelian quantum chains, and derived expressions for the thermalization timescales and length scales.