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The theory of dynamical quantum phase transitions represents an attempt toextend the concept of phase transitions to the far from equilibrium regime.While there are many formal analogies to conventional transitions, it is amajor question to which extent it is possible to formulate a nonequilibriumcounterpart to a Landau-Ginzburg theory. In this work we take a first step inthis direction by constructing an effective free energy for continuousdynamical quantum phase transitions appearing after quantum quenches in thetransverse-field Ising chain. Due to unitarity of quantum time evolution thiseffective free energy becomes a complex quantity transforming the conventionalminimization principle of the free energy into a saddle-point equation in thecomplex plane of the order parameter, which as in equilibrium is themagnetization. We study this effective free energy in the vicinity of thedynamical quantum phase transition by performing an expansion in terms of thecomplex magnetization and discuss the connections to the equilibrium case.Furthermore, we study the influence of perturbations and signatures of thesedynamical quantum phase transitions in spin correlation functions.

Constructing effective free energies for dynamical quantum phase transitions in the transverse-field Ising chain