  In this work we develop the theory of the Loschmidt echo and dynamical phase
transitions in non-interacting strongly disordered Fermi systems after a
quench. In finite systems the Loschmidt echo displays zeros in the complex time
plane that depend on the random potential realization. Remarkably, the zeros
coalesce to form a 2D manifold in the thermodynamic limit, atypical for 1D
systems, crossing the real axis at a sharply-defined critical time. We show
that this dynamical phase transition can be understood as a transition in the
distribution function of the smallest eigenvalue of the Loschmidt matrix, and
develop a finite-size scaling theory. Contrary to expectations, the notion of
dynamical phase transitions in disordered systems becomes decoupled from the
equilibrium Anderson localization transition. Our results highlight the
striking qualitative differences of quench dynamics in disordered and
non-disordered many-fermion systems.
