The Hilbert-glass transitions: new universality of temperature-tuned many-body dynamical quantum criticality

We study a new class of unconventional critical phenomena that is characterized by singularities only in dynamical quantities and has no thermodynamic signatures. Describing this purely dynamical quantum criticality is technically challenging as understanding the finite-temperature dynamics necessarily requires averaging over a large number of matrix elements between many-body eigenstates. Here we develop a real-space renormalization group method for excited state (RSRG-X) that allows us to overcome this challenge in a large class of models. We characterize a specific example: the 1D disordered transverse field Ising model with generic interactions. While thermodynamic phase transitions are generally forbidden in this model, using RSRG-X we find a finite-temperature dynamical transition between two localized phases. The transition is characterized by non-analyticities in the low frequency heat conductivity and in the long-time (dynamic) spin correlation function. The latter is a consequence of an up-down spin symmetry that results in the appearance of an Edwards-Anderson-like order parameter in one of the localized phases.