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Register a new userComplex-band-structure eigenvalue method adapted to Floquet systems: topological superconducting wires as a case study
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Andres Alejandro Reynoso
Publication date
2013
fields
Physics
and research's language is
English
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For systems that can be modeled as a single-particle lattice extended along a privileged direction as, e.g., quantum wires, the so-called eigenvalue method provides full information about the propagating and evanescent modes as a function of energy. This complex-band structure method can be applied either to lattices consisting of an infinite succession of interconnected layers described by the same local Hamiltonian or to superlattices: Systems in which the spatial periodicity involves more than one layer. Here, for time-dependent systems subject to a periodic driving, we present an adapted version of the superlattice scheme capable of obtaining the Floquet states and the Floquet quasienergy spectrum. Within this scheme the time periodicity is treated as existing along spatial dimension added to the original system. The solutions at a single energy for the enlarged artificial system provide the solutions of the original Floquet problem. The method is suited for arbitrary periodic excitations including strong and anharmonic drivings. We illustrate the capabilities of the methods for both time-independent and time-dependent systems by discussing: (a) topological superconductors in multimode quantum wires with spin-orbit interaction and (b) microwave driven quantum dot in contact with a topological superconductor.
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We calculate the phase diagram of a model for topological superconducting wires with local s-wave pairing, spin-orbit coupling $vec{lambda}$ and magnetic field $vec{B}$ with arbitrary orientations. This model is a generalized lattice version of the one proposed by Lutchyn $textit{et al.}$ [Phys. Rev. Lett. $textbf{105}$ 077001 (2010)] and Oreg $textit{et al.}$ [Phys. Rev. Lett. $textbf{105}$ 177002 (2010)], who considered $vec{lambda}$ perpendicular to $vec{B}$. The model has a topological gapped phase with Majorana zero modes localized at the ends of the wires. We determine analytically the boundary of this phase. When the directions of the spin-orbit coupling and magnetic field are not perpendicular, in addition to the topological phase and the gapped non topological phase, a gapless superconducting phase appears.
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The topological characterization of nonequilibrium topological matter is highly nontrivial because familiar approaches designed for equilibrium topological phases may not apply. In the presence of crystal symmetry, Floquet topological insulator states cannot be easily distinguished from normal insulators by a set of symmetry eigenvalues at high symmetry points in the Brillouin zone. This work advocates a physically motivated, easy-to-implement approach to enhance the symmetry analysis to distinguish between a variety of Floquet topological phases. Using a two-dimensional inversion-symmetric periodically-driven system as an example, we show that the symmetry eigenvalues for anomalous Floquet topological states, of both first-order and second-order, are the same as for normal atomic insulators. However, the topological states can be distinguished from one another and from normal insulators by inspecting the occurrence of stable symmetry inversion points in their microscopic dynamics. The analysis points to a simple picture for understanding how topological boundary states can coexist with localized bulk states in anomalous Floquet topological phases.
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