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Interacting many-body systems that are driven far away from equilibrium canexhibit phase transitions between dynamically emerging quantum phases, whichmanifest as singularities in the Loschmidt echo. Whether and under whichconditions such dynamical transitions occur in higher-dimensional systems withspontaneously broken continuous symmetries is largely elusive thus far. Here,we study the dynamics of the Loschmidt echo in the three dimensional O(N) modelfollowing a quantum quench from a symmetry breaking initial state. The O(N)model exhibits a dynamical transition in the asymptotic steady state,separating two phases with a finite and vanishing order parameter, that isassociated with the broken symmetry. We analytically calculate the ratefunction of the Loschmidt echo and find that it exhibits periodic kinksingularities when this dynamical steady-state transition is crossed. Thesingularities arise exactly at the zero-crossings of the oscillating orderparameter. As a consequence, the appearance of the kink singularities in thetransient dynamics is directly linked to a dynamical transition in the orderparameter. Furthermore, we argue, that our results for dynamical quantum phasetransitions in the O(N) model are general and apply to generic systems withcontinuous symmetry breaking.

Dynamical Quantum Phase Transitions in Systems with Continuous Symmetry Breaking