  We uncover a novel dynamical quantum phase transition, using random matrix
theory and its associated notion of planar limit. We study it for the isotropic
XY Heisenberg spin chain. For this, we probe its real-time dynamics through the
Loschmidt echo. This leads to the study of a random matrix ensemble with a
complex weight, whose analysis requires novel technical considerations, that we
develop. We obtain three main results: 1) There is a third order phase
transition at a rescaled critical time, that we determine. 2) The third order
phase transitions persists away from the thermodynamic limit. 3) For times
below the critical value, the difference between the thermodynamic limit and a
finite chain decreases exponentially with the system size. All these results
depend in a rich manner on the parity of the number of flipped spins of the
quantum state conforming the fidelity.

Quantum Physics;Quantum Gases;Statistical Mechanics;Strongly Correlated Electrons