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arXiv:2212.06502 · Physical review. B./Physical review. B

Resilience of topological superconductivity under particle current

Alfonso Maiellaro, Francesco Romeo, Fabrizio Illuminati, R. Citro

Published Feb 13, 2023• 7 citations
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Abstract

We investigate the robustness of topological superconductors under the perturbing influence of a finite charge current. To this aim, we introduce a modified Kitaev Hamiltonian parametrically dependent on the quasiparticle momentum induced by the current. Using different quantifiers of the topological phase, such as the Majorana polarization and the edge state quantum conditional mutual information, we prove the existence of a finite critical value of the quasiparticle momentum below which edge modes and topological superconductivity survive. We also discuss how a finite current breaks time reversal symmetry and changes the topological class in the Altland-Zirnbauer classification scheme compared to the case of isolated systems. Our findings provide a nontrivial example of the interplay between topology and the nonequilibrium physics of open quantum systems, a relation of crucial importance in the quest to a viable topological quantum electronics.

DOI
10.1103/physrevb.107.064505

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We investigate the robustness of topological superconductors under the perturbing influence of a finite charge current. To this aim, we introduce a modified Kitaev Hamiltonian parametrically dependent on the quasiparticle momentum induced by the current. Using different quantifiers of the topological phase, such as the Majorana polarization and the edge state quantum conditional mutual information, we prove the existence of a finite critical value of the quasiparticle momentum below which edge modes and topological superconductivity survive. We also discuss how a finite current breaks time reversal symmetry and changes the topological class in the Altland-Zirnbauer classification scheme compared to the case of isolated systems. Our findings provide a nontrivial example of the interplay between topology and the nonequilibrium physics of open quantum systems, a relation of crucial importance in the quest to a viable topological quantum electronics.
A time-reversal invariant Kitaev-type model is introduced in which spins (Dirac matrices) on the square lattice interact via anisotropic nearest-neighbor and next-nearest-neighbor exchange interactions. The model is exactly solved by mapping it onto a tight-binding model of free Majorana fermions coupled with static Z2{\mathbb{Z}}_{2}Z2​ gauge fields. The Majorana fermion model can be viewed as a model of time-reversal-invariant superconductor and is classified as a member of symmetry class DIII in the Altland-Zirnbauer classification. The ground-state phase diagram has two topologically distinct gapped phases which are distinguished by a Z2{\mathbb{Z}}_{2}Z2​ topological invariant. The topologically nontrivial phase supports both a Kramers' pair of gapless Majorana edge modes at the boundary and a Kramers' pair of zero-energy Majorana states bound to a 0-flux vortex in the $\ensuremath{\pi}$-flux background. Power-law decaying correlation functions of spins along the edge are obtained by taking the gapless Majorana edge modes into account. The model is also defined on the one-dimension ladder, in which case again the ground-state phase diagram has Z2{\mathbb{Z}}_{2}Z2​ trivial and nontrivial phases.
We study the properties of one-dimensional topological superconductors under the influence of generic boundary conditions mimicking the coupling with external environments. We identify a general four-parameter classification of the boundary effects and show that particle-hole and reflection symmetries can be broken or preserved by appropriately fixing the boundary parameters. When the particle-hole symmetry is broken, the topological protection of the edge modes is lost due to the hybridization with the external degrees of freedom. We assess the robustness of the edge modes in the various regimes by considering different quantifiers of topological properties. In particular, we investigate the resilience of the long-distance, edge-to-edge quantum mutual information and squashed entanglement, measuring the nonlocal correlations of the Majorana excitations. Besides their relevance for the open dynamics of topological systems, these results may provide a useful guide to the appropriate embedding of low-dimensional topological systems on nanodevices in realistic conditions.
Identifying entanglement-based order parameters characterizing topological systems, in particular topological superconductors and topological insulators, has remained a major challenge for the physics of quantum matter in the last two decades. Here we show that the end-to-end, long-distance, bipartite squashed entanglement between the edges of a many-body system, defined in terms of the edge-to-edge quantum conditional mutual information, is the natural nonlocal order parameter for topological superconductors in one dimension as well as in quasi-one-dimensional geometries. For the Kitaev chain in the entire topological phase, the edge squashed entanglement is quantized to ln(2)/2ln(2)/2ln(2)/2, half the maximal Bell-state entanglement, and vanishes in the trivial phase. Such a topological squashed entanglement exhibits the correct scaling at the quantum phase transition, is stable in the presence of interactions, and is robust against disorder and local perturbations. Edge quantum conditional mutual information and edge squashed entanglement defined with respect to different multipartitions discriminate topological superconductors from symmetry breaking magnets, as shown by comparing the fermionic Kitaev chain and the spin-1/2 Ising model in transverse field. For systems featuring multiple topological phases with different numbers of edge modes, like the quasi-1D Kitaev ladder, topological squashed entanglement counts the number of Majorana excitations and distinguishes the different topological phases of the system. In fact, we show that the edge quantum conditional mutual information and the edge squashed entanglement remain valid detectors of topological superconductivity even for systems, like the Kitaev tie with long-range hopping, featuring geometrical frustration and a suppressed bulk-edge correspondence.
We discover novel topological effects in the one-dimensional Kitaev chain modified by long-range Hamiltonian deformations in the hopping and pairing terms. This class of models display symmetry-protected topological order measured by the Berry/Zak phase of the lower-band eigenvector and the winding number of the Hamiltonians. For exponentially decaying hopping amplitudes, the topological sector can be significantly augmented as the penetration length increases, something experimentally achievable. For power-law decaying superconducting pairings, the massless Majorana modes at the edges get paired together into a massive nonlocal Dirac fermion localized at both edges of the chain: a new topological quasiparticle that we call topological massive Dirac fermion. This topological phase has fractional topological numbers as a consequence of the long-range couplings. Possible applications to current experimental setups and topological quantum computation are also discussed.
Superfluids and superconductors can be either fully gapped or gapless with nodal structures in momentum space. Both the gapped and nodal phases can be topologically protected and possess nontrivial topological invariants. Topological quantum phase transitions exist at zero temperature between different gapped phases or between gapped and nodal phases. These phase transitions can be driven by chemical potential and/or spin exchange fields. The two phases separated by the phase transition can have the same local order but differ in global topology. Thus, these topological quantum phase transitions cannot be described by the Landau paradigm of symmetry breaking. Although some aspects of these transitions, such as the change of topological invariants and gapless boundary states, have been discussed before, a complete theory of these transitions has yet to be developed. In this dissertation, we construct effective field theories to study the universality and thermodynamic signatures of these transitions. We find four different universality classes in superfluids and superconductors. Certain thermodynamic quantities, such as compressibility or spin susceptibility, change non-analytically across the transitions. For certain time-reversal symmetry breaking fields that lead to bulk phase transitions, there also exist topological phase transitions on the surface. All the topological phase transitions studied in this dissertation only exist at zero temperature. At finite temperature, different states are connected by smooth crossovers. There exists a quantum critical region at finite temperature near the quantum critical point (QCP). In this quantum critical region, thermodynamic quantities have universal scaling dependence on temperature dictated by the universality class of the QCP. We argue that these scaling properties can be used to probe and differentiate these QCPs. These bulk and surface topological quantum phase transitions are discussed in various concrete models, such as chiral and time-reversal invariant p-wave superfluids, topological superconductors of emergent Dirac fermions, and topological superconducting model of CuₓBi₂Se₃.
A characteristic feature of topological systems is the presence of robust gapless edge states. In this work the effect of time-dependent perturbations on the edge states is considered. Specifically we consider perturbations that can be understood as changes of the parameters of the Hamiltonian. These changes may be sudden or carried out at a fixed rate. In general, the edge modes decay in the thermodynamic limit, but for finite systems a revival time is found that scales with the system size. The dynamics of fermionic edge modes and Majorana modes are compared. The effect of periodic perturbations is also referred allowing the appearance of edge modes out of a topologically trivial phase.