Gapless topological superconductors: Model Hamiltonian and realization (original) (raw)

Induced Topological Phases at the Boundary of 3D Topological Superconductors

Physical Review Letters, 2015

We present tight-binding models of 3D topological superconductors in class DIII that support a variety of winding numbers. We show that gapless Majorana surface states emerge at their boundary in agreement with the bulk-boundary correspondence. At the presence of a Zeeman field the surface states become gapped and the boundary behaves as a 2D superconductor in class D. Importantly, the 2D and 3D winding numbers are in agreement signifying that the topological phase of the boundary is induced by the phase of the 3D bulk. Hence, the boundary of a 3D topological superconductor in class DIII can be used for the robust realisation of localised Majorana zero modes.

Majorana flat bands in s-wave gapless topological superconductors

We demonstrate how the non-trivial interplay between spin-orbit coupling and nodeless s-wave superconductivity can drive a fully gapped two-band topological insulator into a time-reversal invariant gapless topological superconductor supporting symmetry-protected Majorana flat bands. We characterize topological phase diagrams by a Z2 × Z2 partial Berry-phase invariant, and show that, despite the trivial crystal geometry, no unique bulk-boundary correspondence exists. We trace this behavior to the anisotropic quasiparticle bulk gap closing, linear vs. quadratic, and argue that this provides a unifying principle for gapless topological superconductivity. Experimental implications for tunneling conductance measurements are addressed, relevant for lead chalcogenide materials.

Unconventional Superconductivity on a Topological Insulator

Physical Review Letters, 2010

We study proximity-induced superconductivity on the surface of a topological insulator (TI), focusing on unconventional pairing. We find that the excitation spectrum becomes gapless for any spin-triplet pairing, such that both subgap bound states and Andreev reflection is strongly suppressed. For spin-singlet pairing, the zeroenergy surface state in the d xy -wave case becomes a Majorana fermion, in contrast to the situation realized in the topologically trivial high-T c cuprates. We also study the influence of a Zeeman field on the surface states. Both the magnitude and direction of this field is shown to strongly influence the transport properties, in contrast to the case without TI. We predict an experimental signature of the Majorana states via conductance spectroscopy.

Edge-mode superconductivity in a two-dimensional topological insulator

Nature Nanotechnology, 2015

Topological superconductivity is an exotic state of matter that supports Majorana zero-modes, which are surface modes in 3D, edge modes in 2D or localized end states in 1D [1, 2]. In the case of complete localization these Majorana modes obey non-Abelian exchange statistics making them interesting building blocks for topological quantum computing [3, 4]. Here we report superconductivity induced into the edge modes of semiconducting InAs/GaSb quantum wells, a two-dimensional topological insulator [5-10]. Using superconducting quantum interference, we demonstrate gatetuning between edge-dominated and bulk-dominated regimes of superconducting transport. The edge-dominated regime arises only under conditions of high-bulk resistivity, which we associate with the 2D topological phase. These experiments establish InAs/GaSb as a robust platform for further confinement of Majoranas into localized states enabling future investigations of non-Abelian statistics.

Exactly solvable model for two-dimensional topological superconductors

Physical Review B, 2018

In this paper, we present an exactly solvable model for two dimensional topological superconductor with helical Majorana edge modes protected by time-reversal symmetry. Our construction is based on the idea of decorated domain walls and makes use of the Kasteleyn orientation on a two dimensional lattice, which were used for the construction of the symmetry protected fermion phase with Z2 symmetry by Tarantino et al. and Ware et al. By decorating the time-reversal domain walls with spinful Majorana chains, we are able to construct a commuting projector Hamiltonian with zero correlation length ground state wave function that realizes a strongly interacting version of the two dimensional topological superconductor. From our construction, it can be seen that the T 2 = −1 transformation rule for the fermions is crucial for the existence of such a nontrivial phase; with T 2 = 1, our construction does not work.

Evidence of topological boundary modes with topological nodal-point superconductivity

Nature Physics, 2021

The extension of the topological classification of band insulators to topological semimetals gave way to the topology classes of Dirac, Weyl and nodal line semimetals with their unique Fermi arc and drum head boundary modes [1-3]. Similarly, there are several suggestions to employ the classification of topological superconductors for topological nodal superconductors with Majorana boundary modes [4-6]. Here, we show that the surface 1H termination of the transition metal dichalcogenide compound 4Hb-TaS 2 , in which 1T-TaS 2 and 1H-TaS 2 layers are interleaved, has the phenomenology of a topological nodal point superconductor. We find in scanning tunneling spectroscopy a residual density of states within the superconducting gap. An exponentially decaying bound mode is imaged within the superconducting gap along the boundaries of the exposed 1H layer characteristic of a gapless Majorana edge mode. The anisotropic nature of the localization length of the edge mode aims towards topological nodal superconductivity. A zero-bias conductance peak is further imaged within fairly isotropic vortex cores. All our observations are accommodated by a theoretical model of a two-dimensional nodal Weyl-like superconducting state, which ensues from inter-orbital Cooper pairing. The observation of an intrinsic topological nodal superconductivity in a layered material will pave the way for further studies of Majorana edge modes and its applications in quantum information processing [7-10]. I. MAIN Topological superconductors are extensively explored with the aim to induce and manipulate Majorana zero modes that are essential to realize topological quantum information processing. The nonlocal nature of these zero modes and their intrinsic non-Abelian braiding statistics allow for quantum information to be stored and manipulated in a nonlocal manner, which protects it from the influence of the local environment [7-9]. A growing body of realizations of topological superconductivity in one-dimensional (1D) systems, including hybrid nanowires [11-14], atomic chains [15, 16], proximitized helical edge modes [17] and planar Josephson junctions [18, 19] have been reported so far. In these systems, Majorana zero

Vortex lattices in the superconducting phases of doped topological insulators and heterostructures

Physical Review B, 2013

Majorana fermions are predicted to play a crucial role in condensed matter realizations of topological quantum computation. These heretofore undiscovered quasiparticles have been predicted to exist at the cores of vortex excitations in topological superconductors and in heterostructures of superconductors and materials with strong spin-orbit coupling. In this work we examine topological insulators with bulk s-wave superconductivity in the presence of a vortex-lattice generated by a perpendicular magnetic field. Using self-consistent Bogoliubov-de Gennes, calculations we confirm that beyond the semi-classical, weak-pairing limit that the Majorana vortex states appear as the chemical potential is tuned from either side of the band edge so long as the density of states is sufficient for superconductivity to form. Further, we demonstrate that the previously predicted vortex phase transition survives beyond the semi-classical limit. At chemical potential values smaller than the critical chemical potential, the vortex lattice modes hybridize within the top and bottom surfaces giving rise to a dispersive low-energy mid-gap band. As the chemical potential is increased, the Majorana states become more localized within a single surface but spread into the bulk toward the opposite surface. Eventually, when the chemical potential is sufficiently high in the bulk bands, the Majorana modes can tunnel between surfaces and eventually a critical point is reached at which modes on opposite surfaces can freely tunnel and annihilate leading to the topological phase transition previously studied in the work of Hosur et al. 1 .

Topological quantum phase transitions in topological superconductors

EPL (Europhysics Letters), 2010

In this paper we show that BF topological superconductors (insulators) exibit phase transitions between different topologically ordered phases characterized by different ground state degeneracy on manifold with non-trivial topology. These phase transitions are induced by the condensation (or lack of) of topological defects. We concentrate on the (2+1)-dimensional case where the BF model reduce to a mixed Chern-Simons term and we show that the superconducting phase has a ground state degeneracy k and not k 2. When the symmetry is U (1) × U (1), namely when both gauge fields are compact, this model is not equivalent to the sum of two Chern-Simons term with opposite chirality, even if naively diagonalizable. This is due to the fact that U (1) symmetry requires an ultraviolet regularization that make the diagonalization impossible. This can be clearly seen using a lattice regularization, where the gauge fields become angular variables. Moreover we will show that the phase in which both gauge fields are compact is not allowed dynamically.

Multiband<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">mml:mis-wave topological superconductors: Role of dimensionality and magnetic field response

Physical Review B, 2013

We further investigate a class of time-reversal-invariant two-band s-wave topological superconductors introduced in Phys. Rev. Lett. 108, 036803 (2012). Provided that a sign reversal between the two superconducting pairing gaps is realized, the topological phase diagram can be determined exactly (within mean field) in one and two dimensions, as well as in three dimensions upon restricting to the excitation spectrum of time-reversal invariant momentum modes. We show how, in the presence of time-reversal symmetry, Z2 invariants that distinguish between trivial and non-trivial quantum phases can be constructed by considering only one of the Kramers' sectors in which the Hamiltonian decouples into. We find that the main features identified in our original two-dimensional setting remain qualitatively unchanged, with non-trivial topological superconducting phases supporting an odd number of Kramers' pairs of helical Majorana modes on each boundary, as long as the required π phase difference between gaps is maintained. We also analyze the consequences of time-reversal symmetry-breaking either due to the presence of an applied or impurity magnetic field or to a deviation from the intended phase matching between the superconducting gaps. We demonstrate how the relevant notion of topological invariance must be modified when time-reversal symmetry is broken, and how both the persistence of gapless Majorana modes and their robustness properties depend in general upon the way in which the original Hamiltonian is perturbed. Interestingly, a topological quantum phase transition between helical and chiral superconducting phases can be induced by suitably tuning a Zeeman field in conjunction with a phase mismatch between the gaps. Recent experiments in doped semiconducting crystals, of potential relevance to the proposed model, and possible candidate material realizations in superconductors with s± pairing symmetry are discussed.

Nested Defects on the Boundary of Topological Superconductors

Helical Majorana edge states at the 2D boundaries of 3D topological superconductors can be gapped by a surface Zeeman field. Here we study the effect nested defects imprinted on the Zeeman field can have on the edge states. We demonstrate that depending on the configuration of the field we can induce dimensional reduction of gapless Majorana modes from 2D to 1D or quasi-0D at magnetic domain walls. We determine the nature of the Majorana localisation on these defects as a function of the magnitude and configuration of the Zeeman field. Finally, we observe a generalisation of the index theorem governing the number of gapless modes at the interface between topologically non-trivial systems with partial Chern numbers.