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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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We characterize the Majorana zero modes in topological hybrid superconductor-semiconductor wires with spin-orbit coupling and magnetic field, in terms of generalized Bloch coordinates $varphi, theta, delta$, and analyze their transformation under SU(2) rotations. We show that, when the spin-orbit coupling and the magnetic field are perpendicular, $varphi$ and $delta$ are universal in an appropriate coordinate system. We use these geometric properties to explain the behavior of the Josephson current in junctions of two wires with different orientations of the magnetic field and/or the spin-orbit coupling. We show how to extract from there, the angle $theta$, hence providing a full description of the Majorana modes.
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.
The importance of models with an exact solution for the study of materials with non-trivial topological properties has been extensively demonstrated. Among these, the Kitaev model of a one-dimensional $p$-wave superconductor plays a guiding role in the search for Majorana modes in condensed matter systems. Also, the $sp$ chain, with an anti-symmetric mixing among the $s$ and $p$ bands provides a paradigmatic example of a topological insulator with well understood properties. There is an intimate relation between these two models and in particular their topological quantum phase transitions share the same universality class. Here we consider a two-band $sp$ model of spinless fermions with an attractive (inter-band) interaction. Both the interaction and hybridization between the $s$ and $p$ fermions are anti-symmetric. The zero temperature phase diagram of the model presents a variety of phases including a Weyl superconductor, topological insulator and trivial phases. The quantum phase transitions between these phases can be either continuous or discontinuous. We show that the transition from the topological superconducting phase to the trivial one has critical exponents different from those of an equivalent transition in Kitaevs model.
We study a one-dimensional topological superconductor, the Kitaev chain, under the influence of a non-Hermitian but $mathcal{PT}$-symmetric potential. This potential introduces gain and loss in the system in equal parts. We show that the stability of the topological phase is influenced by the gain/loss strength and explicitly derive the bulk topological invariant in a bipartite lattice as well as compute the corresponding phase diagram using analytical and numerical methods. Furthermore we find that the edge state is exponentially localized near the ends of the wire despite the presence of gain and loss of probability amplitude in that region.
We study a one-dimensional wire with strong Rashba and Dresselhaus spin-orbit coupling (SOC), which supports Majorana fermions when subject to a Zeeman magnetic field and in proximity of a superconductor. Using both analytical and numerical techniques we calculate the electronic spin texture of the Majorana end states. We find that the spin polarization of these states depends on the relative magnitude of the Rashba and Dresselhaus SOC components. Moreover, we define and calculate a local Majorana polarization and Majorana density and argue that they can be used as order parameters to characterize the topological transition between the trivial system and the system exhibiting Majorana bound modes. We find that the local Majorana polarization is correlated to the transverse spin polarization, and we propose to test the presence of Majorana fermions in a 1D system by a spin-polarized density of states measurement.
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