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

Theory of the Loschmidt echo and dynamical quantum phase transitions in disordered Fermi systems