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 generator \hat{\mathcal{L}} ℒ ̂ of the dissipative Kitaev chain are found to be linked to the presence of Majorana edge master modes – localized eigenmodes of \hat{\mathcal{L}} ℒ ̂ . 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.
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 generator \hat{\mathcal{L}} ℒ ̂ of the dissipative Kitaev chain are found to be linked to the presence of Majorana edge master modes – localized eigenmodes of \hat{\mathcal{L}} ℒ ̂ . 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.
Abstract One-dimensional (1D) Floquet topological superconductors possess two types of degenerate Majorana edge modes at zero andπquasienergies, leaving more room for the design of boundary time crystals and quantum computing schemes than their static counterparts. In this work, we discover Floquet superconducting phases with large topological invariants and arbitrarily many Majorana edge modes in periodically driven Kitaev chains (KCs). Topological winding numbers defined for the Floquet operator and Floquet entanglement Hamiltonian are found to generate consistent predictions about the phase diagram, bulk-edge correspondence and numbers of zero andπMajorana edge modes of the system under different driving protocols. The bipartite entanglement entropy further shows non-analytic behaviors around the topological transition point between different Floquet superconducting phases. These general features are demonstrated by investigating the KC with periodically kicked pairing or hopping amplitudes. Our discovery reveals the rich topological phases and many Majorana edge modes that could be brought about by periodic driving fields in 1D superconducting systems. It further introduces a unified description for a class of Floquet topological superconductors from their quasienergy bands and entanglement properties.
We study the bulk and edge properties of a driven Kitaev chain, where the driving is performed as instantaneous quenches of the on-site energies. We identify three periodic driving regimes: low period, which is equivalent to a static model, with renormalized parameters obtained from the Baker-Campbell-Hausdorff (BCH) expansion; intermediate period, where the first order BCH expansion breaks down; and high period when the quasienergy gap at $\ensuremath{\omega}/2$ closes. We investigate the dynamical localization properties for the case of quasiperiodic potential driving as a function of its amplitude and the pairing strength, obtaining regimes with extended, critical, and localized bulk states, if the driving is performed at high frequencies. In these we characterize wave-packet propagation, obtaining ballistic, subdiffusive, and absence of spreading, respectively. In the intermediate period regime, we find an additional region in the phase diagram with a mobility edge between critical and localized states. Furthermore, we investigate the stability of these phases under time aperiodicity on the drivings, observing that the system eventually thermalizes: It results in featureless random states which can be described by the symmetry of the Hamiltonian. In a system with open edges, we find that both Majorana and fermionic localized edge modes can be engineered with a spatially quasiperiodic potential, in similarity with the case of homogeneous on-site energies. Besides, we demonstrate the possibility of creating multiple Majorana 0 and $\ensuremath{\pi}$ modes in a driven setting, even if the underlying static Hamiltonian is in its trivial phase. Lastly, we study the robustness of the Majorana modes against the aperiodicity in the driving period, showing that the ones created via quasiperiodic potential are more robust to the decoherence. Moreover, we find an example where a Majorana mode displays high robustness, provided that it is chosen from a special point in the topological region.
A core characteristic of dissipative non-Hermitian topology is that the relaxation dynamics tracks the non-Bloch bulk-boundary correspondence, rendering an algebraic decay in the gapless regime and an exponential falloff in the gapped phase, so that local observables directly diagnose the topology. We show that this correspondence breaks down in a dissipative topological superconductor, where the local observables turn blind to the very topology they are expected to decipher. Via a bond-dissipative dimerized Kitaev chain in a third-quantized rapidity-matrix formulation, we find that at zero chemical potential the Majorana rapidity matrix decomposes into two independent non-Hermitian sectors, each with its own generalized Brillouin zone and non-Bloch winding number, thereby revealing a sector-resolved non-Bloch bulk-boundary correspondence. The local density is a cross-sector covariance and relaxes at the sum of the two sector rates, so it remains sector-blind even when one sector is gapless and topological. For balanced gain and loss, the finite-time zero events of the entanglement spectrum under purely periodic-boundary Lindblad evolution recover this hidden edge content sector by sector, serving as a dynamical invariant that returns the open-boundary edge rapidities without physically opening the chain.
Floquet Majorana edge modes capture the topological features of periodically driven superconductors. We present a Kitaev chain with multiple time periodic driving and demonstrate how the avoidance of bands crossing is altered, which gives rise to new regions supporting Majorana edge modes. A one dimensional generalized method was proposed to predict Majorana edge modes via the Zak phase of the Floquet bands. We also study the time independent effective Hamiltonian at high frequency limit and introduce diverse index to characterize topological phases with different relative phase between the multiple driving. Our work enriches the physics of driven system and paves the way for locating Majorana edge modes in larger parameter space.
Engineering dissipative dynamics in open quantum systems is under active focus, especially in topological settings where resilient edge modes are expected to exhibit decay rates distinct from the bulk. In this letter, we propose an efficient dynamical scheme to discern such long-lived excitations. Employing a Floquet-Lindblad framework, we explore how periodic driving reshapes the key features of a paradigmatic topological model, namely a Creutz ladder. Our results bear testimony to a drive-induced unipolar-bipolar transition in the Liouvillian skin effect, which gets dynamically manifested as a chiral-helical damping crossover. Such a transition effectively rescales the bulk localization length, giving rise to a polarization drift that we identify as a new invariant for efficient diagnosis of the nontrivial phases. As the transition becomes more gradual via tuning drive-rescaled parameters, we uncover signatures of a scale-free localization where skin and extended modes co-exist with distinct decay rates. The emergent hierarchy of the decay rates yields two disparate timescales: a chiral wavefront that rapidly empties the bulk followed by a long-lived regime dominated by robust edge modes. Overall, our results provide convincing evidence that periodic driving serves as a powerful handle to manipulate dissipative topological phases and dynamically isolate the boundary modes.
We study the generalized one-dimensional (1D) quantum dissipative models corresponding to a Majorana wire which can possess more than one Majorana bound state at each end. The system consists of a 1D fermionic open quantum system whose dynamics is governed by a quadratic Lindblad equation. Using the adjoint Lindblad equation for the fermionic two-point correlations, we find the gaps in the damping and purity spectra of a generic 1D model. Then, using the symmetry-based classification, we show that a winding number as the topological invariant can be defined which distinguishes different steady states of the system in the presence of damping and purity gaps. Then we focus on certain models with different Lindblad quantum jump terms and explore their phase diagrams by calculating the damping and the purity gaps as well as the winding number. In particular, we show that by inclusion of quantum jumps between next-nearest-neighbor sites, higher winding numbers and equivalently more Majorana bound states can be achieved. Also, by introducing imbalanced couplings we can switch between states with negative and positive winding numbers. Finally, we should mention that since our formulation is based on the fermionic correlations rather than the Majorana operators, it can be easily extended to the dissipative topological phases belonging to other symmetry classes.