  We consider a new class of unconventional critical phenomena that is
characterized by singularities only in dynamical quantities and has no
thermodynamic signatures. A possible example is the recently proposed many-body
localization transition, in which transport coefficients vanish at a critical
temperature. Describing this unconventional quantum criticality has been
technically challenging as understanding the finite-temperature dynamics
requires the knowledge of a large number of many-body eigenstates. Here we
develop a real-space renormalization group method for excited state (RSRG-X),
that allow us to overcome this challenge, and establish the existence and
universal properties of such temperature-tuned dynamical phase transitions. We
characterize a specific example: the 1D disordered transverse field Ising model
with interactions. Using RSRG-X, we find a finite-temperature transition,
between two localized phases, characterized by non-analyticities of the dynamic
spin correlation function and the low frequency heat conductivity.
