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 consider the dynamics of an XY spin chain subjected to an external transverse field which is periodically quenched between two values. By deriving an exact expression of the Floquet Hamiltonian for this out-of-equilibrium protocol with arbitrary driving frequencies, we show how, after an unfolding of the Floquet spectrum, the parameter space of the system is characterized by alternations between local and non-local regions, corresponding respectively to the absence and presence of Floquet resonances. The boundary lines between regions are obtained analytically from avoided crossings in the Floquet quasi-energies and are observable as phase transitions in the synchronized state. The transient behaviour of dynamical averages of local observables similarly undergoes a transition, showing either a rapid convergence towards the synchronized state in the local regime, or a rather slow one exhibiting persistent oscillations in the non-local regime, where explicit decay coefficients are presented.

No abstract