The Hilbert-glass transition: 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.