Two-dimensional Topological Superconductors Research Articles

Surface Bogoliubov-Dirac cones and helical Majorana hinge modes in superconducting Dirac semimetals

In the presence of certain symmetries, three-dimensional Dirac semimetals can harbor not only surface Fermi arcs, but also surface Dirac cones. Motivated by the experimental observation of rotation-symmetry-protected Dirac semimetal states in iron-based superconductors, we investigate the potential intrinsic topological phases in a $C_{4z}$-rotational invariant superconducting Dirac semimetal with $s_{\pm}$-wave pairing. When the normal state harbors only surface Fermi arcs on the side surfaces, we find that an interesting gapped superconducting state with a quartet of Bogoliubov-Dirac cones on each side surface can be realized, even though the first-order topology of its bulk is trivial. When the normal state simultaneously harbors surface Fermi arcs and surface Dirac cones, we find that a second-order time-reversal invariant topological superconductor with helical Majorana hinge states can be realized. The criteria for these two distinct topological phases have a simple geometric interpretation in terms of three characteristic surfaces in momentum space. By reducing the bulk material to a thin film normal to the axis of rotation symmetry, we further find that a two-dimensional first-order time-reversal invariant topological superconductor can be realized if the inversion symmetry is broken by applying a gate voltage. Our work reveals that diverse topological superconducting phases and types of Majorana modes can be realized in superconducting Dirac semimetals.

Two-Dimensional Topological Superconductors with Spin-Orbit Coupling and Zeeman Term

Topological superconductors have attracted much attention for their potential applications in realizing topological quantum computing. In this paper, we show a system with 8 × 8 Bogoliubov-de Gennes Hamiltonian. The system has particle-hole symmetry. By adding a Zeeman term to the model, we discuss this system from two situations. In the first case of breaking the inversion symmetry while preserving the mirror symmetry, it is obtained from the topological phase diagram of the system that it is a topological superconductor. In another case where the mirror symmetry is broken while preserving the inversion symmetry, the system has two nodes connected by a flat band of zero-energy Andreev edge states and the topological number is non-zero . It can be concluded that the system is a topological nodal superconductor.

Possible topological phases in quantum anomalous Hall insulator/unconventional superconductor hybrid systems

Quantum anomalous Hall insulator (QAH)/$s$-wave superconductor (SC) hybrid systems are known to be an ideal platform for realizing two-dimensional topological superconductors with chiral Majorana edge modes. In this paper we study QAH/unconventional SC hybrid systems whose pairing symmetry is $p$-wave, $d$-wave, chiral $p$-wave, or chiral $d$-wave. The hybrid systems are a generalization of the QAH/$s$-wave SC hybrid system. In view of symmetries of the QAH and pairings, we introduce three topological numbers to classify topological phases of the hybrid systems. One is the Chern number that characterizes chiral Majorana edge modes and the others are topological numbers associated with crystalline symmetries. We numerically calculate the topological numbers and associated surface states for three characteristic regimes that feature an influence of unconventional SCs on QAHs. Our calculation shows a rich variety of topological phases and unveils the following topological phases that are no counterpart of the $s$-wave case: crystalline symmetry-protected helical Majorana edge modes, a line node phase (crystalline-symmetry-protected Bogoliubov Fermi surface), and multiple chiral Majorana edge modes. The phenomena result from a nontrivial topological interplay between the QAH and unconventional SCs. Finally, we discuss tunnel conductance in a junction between a normal metal and the hybrid systems, and show that the chiral and helical Majorana edge modes are distinguishable in terms of the presence/absence of zero-bias conductance peak.

Superconductors with anomalous Floquet higher-order topology

We develop a general theory for two-dimensional (2D) anomalous Floquet higher-order topological superconductors (AFHOTSC), which are dynamical Majorana-carrying phases of matter with no static counterpart. Despite the triviality of its bulk Floquet bands, an AFHOTSC generically features the simultaneous presence of corner-localized Majorana modes at both zero and $\pi/T$ quasi-energies, a phenomenon beyond the scope of any static topological band theory. We show that the key to AFHOTSC is its unavoidable singular behavior in the phase spectrum of the bulk time-evolution operator. By mapping such evolution-phase singularities to the stroboscopic boundary signatures, we classify all 2D AFHOTSCs that are protected by a rotation group symmetry in symmetry class D. We further extract a higher-order topological index for unambiguously predicting the presence of Floquet corner Majorana modes, which we confirm numerically. Our theory serves as a milestone towards a dynamical topological theory for Floquet superconducting systems.

Hierarchy of higher-order topological superconductors in three dimensions

After exploring much on two-dimensional higher-order topological superconductors (HOTSCs) hosting Majorana corner modes (MCMs) only, we propose a simple fermionic model based on a three-dimensional topological insulator proximized with $s$-wave superconductor to realize Majorana hinge modes (MHMs) followed by MCMs under the application of appropriate Wilson-Dirac perturbations. We interestingly find that the second-order topological superconductor, hosting MHMs, appears above a threshold value of the first type perturbation while the third-order topological superconducting phase, supporting MCMs, immediately arises incorporating infinitesimal perturbation of the second kind. Thus, a hierarchy of HOTSC phases can be realized in a single three-dimensional model. Additionally, the application of bulk magnetic field is found to be instrumental in manipulating the number of MHMs, leaving the number for MCMs unaltered. We analytically understand these above-mentioned numerical findings by resorting to the low energy model. We further characterize these topological phases with a distinct structure of the Wannier spectra. From the practical point of view, we manifest quantized transport signatures of these higher-order modes. Finally, we construct Floquet engineering to generate the hierarchy of HOTSC phases by kicking the same perturbations as considered in their static counterpart.

Robustness of vortex-bound Majorana zero modes against correlated disorder

We investigate the effect of correlated disorder on Majorana zero modes (MZMs) bound to magnetic vortices in two-dimensional topological superconductors. By starting from a lattice model of interacting fermions with a $p_x \pm i p_y$ superconducting ground state in the disorder-free limit, we use perturbation theory to describe the enhancement of the Majorana localization length at weak disorder and a self-consistent numerical solution to understand the breakdown of the MZMs at strong disorder. We find that correlated disorder has a much stronger effect on the MZMs than uncorrelated disorder and that it is most detrimental if the disorder correlation length $\ell$ is on the same order as the superconducting coherence length $\xi$. In contrast, MZMs can survive stronger disorder for $\ell \ll \xi$ as random variations cancel each other within the length scale of $\xi$, while an MZM may survive up to very strong disorder for $\ell \gg \xi$ if it is located in a favorable domain of the given disorder realization.

Optical Responses of Chiral Majorana Edge States in Two-Dimensional Topological Superconductors.

Majorana fermions exist on the boundaries of two-dimensional topological superconductors (TSCs) as charge-neutral quasiparticles. The neutrality makes the detection of such states challenging from both experimental and theoretical points of view. Current methods largely rely on transport measurements in which Majorana fermions manifest themselves by inducing electron-pair tunneling at the lead-contacting point. Here we show that chiral Majorana fermions in TSCs generate enhanced local optical response. The features of local optical conductivity distinguish them not only from trivial superconductors or insulators but also from normal fermion edge states such as those in quantum Hall systems. Our results provide a new applicable method to detect dispersive Majorana fermions and may lead to a novel direction of this research field.

Majorana zero mode in the vortex of artificial topological superconductor

Majorana zero mode, the bound state of Majorana fermion in a condensed matter system, plays a critical role in fault-tolerant topological quantum computation. It can be realized in the vortex of a two-dimensional topological superconductor as a zero-energy excitation. A topological superconductor can be constructed by coupling superconductivity to the topological surface states by the proximity effect. In this review article, we discuss the fabrication of such artificially engineered topological superconductors by molecular beam epitaxy. The topological phase and superconductivity are exhibited in Bi2Se3/NbSe2 and Bi2Te3/NbSe2 heterostructures simultaneously. Several characteristic features of Majorana zero mode have been revealed in the vortex by a low-temperature scanning tunneling microscope and corroborated by theoretical results. The discovery of Majorana zero mode may pave the way for further applications in topological quantum computing.

Tunable Majorana corner modes in noncentrosymmetric superconductors: Tunneling spectroscopy and edge imperfections

Majorana corner modes appearing in two-dimensional second-order topological superconductors have great potential applications for fault-tolerant topological quantum computations. We demonstrate that in the presence of an in-plane magentic field two-dimensional ($s+p$)-wave superconductors host Majorana corner modes, whose location can be manipulated by the direction of the magnetic field. In addition, we discuss the effects of edge imperfections on the Majorana corner modes. We describe how different edge shapes and edge disorder affect the number and controllability of the Majorana corner modes, which is of relevance for the implementation of topological quantum computations. We also discuss tunneling spectroscopy in the presence of the Majorana corner modes, where a lead-wire is attached to the corner of the noncentrosymmetric superconductor. The zero-bias differential conductance shows a distinct periodicity with respect to the direction of the magnetic field, which demonstrates the excellent controllability of the Majorana corner modes in this setup. Our results lay down the theoretical groundwork for observing and tuning Majoran corner modes in experiments on ($s+p$)-wave superconductors.

Anderson disorder-induced nontrivial topological phase transitions in two-dimensional topological superconductors

We numerically investigate the quantum phase transitions induced by Anderson disorder in a topological superconductor (TSC), which is composed of a quantum anomalous Hall insulator (QAHI) and a proximity coupled $s$-wave superconductor (SC). From the transport phenomena presented, we deduce that with the increase of Anderson disorder strength, the topological quantum phase changes from Chern number $\mathcal{N}=0$ to $\mathcal{N}=1$ and finally to $\mathcal{N}=2$. Then we use the effective-medium theory to verify our numerical results and conclude that the phase transitions should ascribe to the negative correction of topological mass.

Two-dimensional topological superconductivity candidate in a van der Waals layered material

Two-dimensional (2D) topological superconductors are highly desired because they not only offer opportunities for exploring novel exotic quantum physics, but also possesses potential applications in quantum computation. However, there are few reports on 2D superconductors, let alone topological superconductors. Here, we find a 2D monolayer W$_2$N$_3$, which can be exfoliated from its real van der Waals bulk material with much lower exfoliation energy than MoS$_2$, to be a topological metal with exotic topological states at different energy level. Due to the Van Hove singularities, the density of states near Fermi level are high, making the monolayer a compensate metal. Moreover, the monolayer W$_2$N$_3$ is unveiled to be a superconductor with the superconducting transition temperature Tc $\sim$ 22 K and a superconducting gap of about 5 meV based on the anisotropic Migdal-Eliashberg formalism, arising from the strong electron-phonon coupling around the $\Gamma$ point. Because of the strong electron and lattice coupling, the monolayer displays a non-Fermi liquid behavior in its normal states at temperatures lower than 80 K, where the specific heat exhibit T$^3$ behavior and the Wiedemann-Franz law dramatically violates. Our findings not only provide the platform to study the emergent phenomena in 2D topological superconductors, but also open a door to discover more 2D high-temperature topological superconductors in van der Waals materials.

Ab initio simulation of non-Abelian braiding statistics in topological superconductors

We numerically investigate non-Abelian braiding dynamics of vortices in two-dimensional topological superconductors, such as $s$-wave superconductors with Rashba spin-orbit coupling. Majorana zero modes (MZMs) hosted by the vortices constitute a topological qubit, which offers a fundamental building block of topological quantum computation. As the MZMs are protected by $\mathbb{Z}_2$ invariant, however, the Majorana qubit and quantum gate operations may be sensitive to intrinsic decoherence caused by quasiparticle interference. Numerically simulating the time-dependent Bogoliubov-de Gennes equation without assuming ${\it a \ priori}$ existence of MZMs, we examine quantum noises on the unitary operators of non-abelian braiding dynamics due to interactions with neighboring MZMs and other quasiparticle states. We demonstrate that after the interchange of two vortices, the lowest vortex-bound states accumulate the geometric phase $\pi/2$, and errors stemming from dynamical phases are negligibly small, irrespective of interactions of MZMs. Furthermore, we numerically simulate the braiding dynamics of four vortices in two-dimensional topological superconductors, and discuss an optimal braiding condition for realizing the high performance of non-Abelian statistics and quantum gates operations of Majorana-based qubits.

Marginally localized edges of time-reversal symmetric topological superconductors

We demonstrate that the one-dimensional helical Majorana edges of two-dimensional time-reversal symmetric topological superconductors (class DIII) can become gapless and insulating by a combination of random edge velocity and interaction. Such a gapless insulating edge breaks time-reversal symmetry inhomogeneously, and the local symmetry broken regions can be regarded as static mass potentials or dynamical Ising spins. In both limits, we find that such gapless insulating Majorana edges are generically exponentially localized and trap Majorana zero modes. Interestingly, for a statistically time-reversal symmetric edge (symmetry is broken locally, but the symmetry breaking order parameter is zero on average), the low-energy theory can be mapped to a Dyson model at zero energy, manifesting a diverging density of states and exhibiting marginal localization (i.e., a diverging localization length). Although the ballistic edge state transport is absent, the localized Majorana zero modes reflect the nontrivial topology in the bulk. Experimental signatures are also discussed.

Braiding Majorana corner modes in a second-order topological superconductor

We propose the concept of a device based on a square-shaped sample of a two-dimensional second-order topological helical superconductor which hosts two zero-dimensional Majorana quasiparticles at the corners. The two zero-energy modes rely on particle-hole symmetry (PHS) and their spacial position can be shifted by rotating an in-plane magnetic field and tuning proximity-induced spin-singlet pairing. We consider an adiabatic cycle performed on the degenerate ground-state manifold and show that it realizes the braiding of the two modes whereby they accumulate a non-trivial statistical phase $\pi$ within one cycle. Alongside with the PHS-ensured operator algebra, the fractional statistics confirms the Majorana nature of the zero-energy excitations. A schematic design for a possible experimental implementation of such a device is presented, which could be a step towards realizing non-Abelian braiding.

Evidence for topological superconductivity: Topological edge states in Bi2Te3/FeTe heterostructure**Project supported by the National Natural Science Foundation of China (Grant Nos. 61734008 and 11774143), the National Key Research and Development Program of China (Grant Nos. 2018YFA0307100, 2016YFA0301703, and 2016YFA0300300), the Natural Science Foundation of Guangdong Province, China (Grant Nos. 2015A030313840 and 2017A030313033), the State Key Laboratory of Low-Dimensional Quantum Physics

Majorana fermions have been predicted to exist at the edge states of a two-dimensional topological superconductor. We fabricated single quintuple layer (QL) Bi2Te3/FeTe heterostructure with the step-flow epitaxy method and studied the topological properties of this system by using angle-resolved photoemission spectroscopy and scanning tunneling microscopy/spectroscopy. We observed the coexistence of robust superconductivity and edge states on the single QL Bi2Te3 islands which can be potential evidence for topological superconductor.

Transport signatures of a junction between a quantum spin Hall system and a chiral topological superconductor

We investigate transport through a normal-superconductor (NS) junction made from a quantum spin Hall (QSH) system with helical edge states and a two-dimensional (2D) chiral topological superconductor (TSC) having a chiral Majorana edge mode. We employ a two-dimensional extended four-band model for HgTe-based quantum wells in a magnetic (Zeeman) field and subject to s-wave superconductivity. We show using the Bogoliubov-de Gennes scattering formalism that this structure provides a striking transport signal of a 2D TSC. As a function of the sample width (or Fermi energy) the conductance resonances go through a sequence of $2e^2/h$ (non-trivial phase) and $4e^2/h$ plateaux (trivial phase) which fall within the region of a non-zero Chern number (2D limit) as the sample width becomes large. These signatures are a manifestation of the topological nature of the QSH effect and the TSC.

Second-order bulk-boundary correspondence in rotationally symmetric topological superconductors from stacked Dirac Hamiltonians

Two-dimensional second-order topological superconductors host zero-dimensional Majorana bound states at their boundaries. In this work, focusing on rotation-invariant crystalline topological superconductors, we establish a bulk-boundary correspondence linking the presence of such Majorana bound states to bulk topological invariants introduced by Benalcazar et al. We thus establish when a topological crystalline superconductor protected by rotational symmetry displays second-order topological superconductivity. Our approach is based on stacked Dirac Hamiltonians, using which we relate transitions between topological phases to the transformation properties between adjacent gapped boundaries. We find that in addition to the bulk rotational invariants, the presence of Majorana boundary bound states in a given geometry depends on the interplay between weak topological invariants and the location of the rotation center relative to the lattice. We provide numerical examples for our predictions and discuss possible extensions of our approach.

Topological Braiding of Non-Abelian Midgap Defects in Classical Metamaterials.

Nontrivial braid-group representations appear as non-Abelian quantum statistics of emergent Majorana zero modes in one- and two-dimensional topological superconductors. Here, we generate such representations with topologically protected domain-wall modes in a classical analog of the Kitaev superconducting chain, with a particle-holelike symmetry and a Z_{2} topological invariant. The midgap modes are found to exhibit distinct fusion channels and rich non-Abelian braiding properties, which are investigated using a T-junction setup. We employ the adiabatic theorem to explicitly calculate the braiding matrices for one and two pairs of these midgap topological defects.

Magnetic flux periodicity in second order topological superconductors

The magnetic flux periodicity of $\frac{hc}{2e}$ is a well known manifestation of Cooper pairing in typical s-wave superconductors. In this paper we theoretically show that the flux periodicity of a two-dimensional second-order topological superconductor, which features zero-energy Majorana modes localized at the corners of the sample, is $\frac{hc}{e}$ instead. We further show that the periodicity changes back to $\frac{hc}{2e}$ at the transition to a topologically trivial superconductor, where the Majorana modes hybridize with the bulk states, demonstrating that the doubling of periodicity is a manifestation of the non-trivial topology of the state.

Majorana corner flat bands in two-dimensional second-order topological superconductors

In this paper we find that confining a second-order topological superconductor with a harmonic potential leads to a proliferation of Majorana corner modes. As a consequence, this results in the formation of Majorana corner flat bands which have a fundamentally different origin from that of the conventional mechanism. This is due to the fact that they arise solely from the one-dimensional gapped boundary states of the hybrid system that become gapless without the bulk gap closing under the increase of the trapping potential magnitude. The Majorana corner states are found to be robust against the strength of the harmonic trap and the transition from Majorana corner states to Majorana flat bands is merely a smooth crossover. As a harmonic trap can potentially be realized in heterostructures, this proposal paves a way to observe these Majorana corner flat bands in an experimental context.