The notion of phases and phase transitions is essential for our understanding of equilibrium states of matter and it has proven to be insightful to expand these concepts for the characterization of quantum matter out of equilibrium. In a collaboration with Pei Wang (Zhejiang Normal University) and Stefan Kehrein we analyzed the behavior of the Hall conductance of topological insulators in asymptotic non-equilibrium states. Although the unitary evolution conserves the initial topological invariant of the wave function, we found that the Hall conductance exhibits a universal non-analytic transition point whenever the control parameter crosses a topological phase boundary; this indicates a topologically driven non-equilibrium phase transition [1]. In a subsequent work we extended this analysis to Floquet topological insulators, demonstrating that in these experimentally relevant systems, the same holds with slight modifications [2].

Critical behavior can, however, as well occur in the transient dynamics after the system was pushed out of equilibrium. For so-called dynamical quantum phase transitions (DQPTs) time plays the role of the control parameter and the phenomenon bears surprising analogies to equilibrium phase transitions. In my Master’s project and continuing as a PhD student, I investigated DQPTs in the Kitaev model on the honeycomb lattice [3]. This work was one of the first studies of DQPTs in a two-dimensional system and it comprises a detailed analysis of the characteristics of the critical points based on the distribution of Fisher zeros in the complex time plane. In a subsequent work in collaboration with Markus Heyl we devised a Monte Carlo scheme to investigate DQPTs two- and three-dimensional quantum Ising models [4].

[1] P. Wang, M. Schmitt, and S. Kehrein, *Universal nonanalytic behavior of the Hall con- ductance in a Chern insulator at the topologically driven nonequilibrium phase transition*, Phys. Rev. B 93 085134 (2016)

[2] M. Schmitt and P. Wang, *Universal non-analytic behavior of the non-equilibrium Hall con- ductance in Floquet topological insulators*, Phys. Rev. B 96, 054306 (2017)

[3] M. Schmitt and S. Kehrein, *Dynamical Quantum Phase Transitions in the Kitaev Honeycomb Model*, Phys. Rev. B 92 075114 (2015)

[4] M. Schmitt and M. Heyl, *Quantum dynamics in transverse-field Ising models from classical networks*, SciPost Phys. 4, 013 (2018)