We investigate the effects of dissipation and driving on topological order in superconducting nanowires. Rather than studying the non-equilibrium steady state, we propose a method to classify and detect dynamical signatures of topological order in open quantum systems. Bulk winding numbers for the Lindblad gen-eratorL of the dissipative Kitaev chain are found to be linked to the presence of Majorana edge master modes -localized eigenmodes ofL. Despite decaying in time, these modes provide dynamical fingerprints of the topological phases of the closed system, which are now separated by intermediate regions where winding numbers are ill-defined and the bulk-boundary correspondence breaks down. Combining these techniques with the Floquet formalism reveals higher winding numbers and different types of edge modes under periodic driving. Finally, we link the presence of edge modes to a steady state current. Contents arXiv:1812.02126v2 [cond-mat.str-el] 25 Jan 2019 SciPost Physics Submission 4.2 Lindblad-Floquet theory 15 4.3 Numerical results 16 5 Conclusions and outlook 20 References 21 A Geometric interpretation of the winding number 25 B Computing the winding number 25 C Topology of X vs. A 26 D Edge modes of X 27 E Time evolution of observables 28 F Off-diagonal Floquet blocks and their importance 29

We study the topological Casimir effect, in which extra vacuum energy emerges as a result of the topological features of the theory, rather than due to the conventional fluctuations of the physical propagating degrees of freedom. We compute the corresponding topological term in quantum Maxwell theory defined on a compact manifold. Numerically, the topological effect is much smaller than the conventional Casimir effect. However, we argue that the topological Casimir effect is highly sensitive to an external magnetic field, which may help to discriminate it from the conventional Casimir effect. It is quite amazing that the external magnetic field plays the role of the state, similar to a vacuum in QCD, or ¼ in topological insulators.

We compute the effect of Markovian bulk dephasing noise on the staggered magnetization of the spin-1 2 XXZ Heisenberg chain, as the system evolves after a Néel quench. For sufficiently weak system-bath coupling, the unitary dynamics are found to be preserved up to a single exponential damping factor. This is a consequence of the interplay between PT symmetry and weak symmetries, which strengthens previous predictions for PT-symmetric Liouvillian dynamics. Requirements are a non-degenerate PT-symmetric generator of time evolutionL, a weak parity symmetry and an observable that is anti-symmetric under this parity transformation. The spectrum ofL then splits up into symmetry sectors, yielding the same decay rate for all modes that contribute to the observable's time evolution. This phenomenon may be realized in trapped ion experiments and has possible implications for the control of decoherence in out-of-equilibrium many-body systems.

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