Second-order Topological Superconductors Research Articles

Majorana bound states and phase transitions in first- and second-order topological superconductors

Majorana bound states and phase transitions in first- and second-order topological superconductors

Mixed higher-order topology, and nodal and nodeless flat band topological phases in a superconducting multiorbital model

We investigate the topological phases that appear in an orbital version of the Benalcazar-Bernevig-Hughes (BBH) model in the presence of conventional spin-singlet s-wave superconductivity and with the possibility of tuning an in-plane magnetic field. We chart out the phase diagram by considering different boundary conditions, with the topology of the individual phases further examined by considering both the Wannier and entanglement spectra, as well as the Majorana polarization. For weak to moderate values of magnetic field and superconducting pairing amplitude, we find a second-order topological superconductor phase with eight zero-energy corner modes. Further increasing field or pairing, half of the corner states can be turned into zero-energy edge-localized modes, thus forming what we name hybrid-order phase. Then, we find two different putative first-order topological phases, a nodal and a nodeless phase, both with zero-energy flat bands localized along mirror-symmetric open edges. For the nodal phase, the flat bands are, as expected, localized between the nodes in reciprocal space, while in the nodeless phase, the zero-energy boundary flat band instead spans the whole Brillouin zone and appears disjoint from the fully gapped bulk spectrum. As a consequence, this model present several unexpected phases with unusual surface states that can be tuned via an external magnetic field. Published by the American Physical Society 2024

Erratum: Braiding Majorana corner modes in a second-order topological superconductor [Phys. Rev. Research 2 , 032068(R) (2020)

Erratum: Braiding Majorana corner modes in a second-order topological superconductor [Phys. Rev. Research <b>2</b> , 032068(R) (2020)

Non-Hermitian second-order topological superconductors

Non-Hermitian second-order topological superconductors

Magnetically controlled topological braiding with Majorana corner states in second-order topological superconductors

Magnetically controlled topological braiding with Majorana corner states in second-order topological superconductors

Sublattice-enriched tunability of bound states in second-order topological insulators and superconductors

Bound states at sharp corners have been widely viewed as the hallmark of two-dimensional second-order topological insulators and superconductors. In this work, we show that the existence of sublattice degrees of freedom can enrich the tunability of bound states on the boundary and hence lift the constraint on their locations. We take the Kane-Mele model with honeycomb-lattice structure to illustrate the underlying physics. With the introduction of an in-plane exchange field to the model, we find that the boundary Dirac mass induced by the exchange field has a sensitive dependence on the boundary sublattice termination. We find that the sensitive sublattice dependence can lead bound states to emerge at a specific type of boundary defects named as sublattice domain walls if the exchange field is of ferromagnetic nature, even in the absence of any sharp corner on the boundary. Remarkably, this sensitive dependence of the boundary Dirac mass on the boundary sublattice termination allows the positions of bound states to be manipulated to any place on the boundary for an appropriately-designed sample. With a further introduction of conventional s-wave superconductivity to the model, we find that, no matter whether the exchange field is ferromagnetic, antiferromagnetic, or ferrimagnetic, highly controllable Majorana zero modes can be achieved at the sublattice domain walls. Our work reshapes the understanding of boundary physics in second-order topological phases, and meanwhile opens potential avenues to realize highly controllable bound states for potential applications.

Time evolution of Majorana corner modes in a Floquet second-order topological superconductor

We propose a practically feasible time-periodic sinusoidal drive protocol in onsite mass term to generate the two-dimensional~(2D) Floquet second-order topological superconductor, hosting both the regular $0$- and anomalous $\pi$-Majorana corner modes~(MCMs) while starting from a static 2D topological insulator/$d$-wave superconductor heterostructure setup. We theoretically study the local density spectra and the time dynamics of MCMs in the presence of such drive. The dynamical MCMs are topologically characterized by employing the average quadrupolar motion. Furthermore, we employ the Floquet perturbation theory~(FPT) in the strong driving amplitude limit to provide analytical insight into the problem. We compare our exact (numerical), and the FPT results in terms of the eigenvalue spectra and the time dynamics of the MCMs. We emphasize that the agreement between the exact numerical and the FPT results are more prominent in the higher frequency regime for close to the $0$-quasi-energy mode.

Generating Many Majorana Corner Modes and Multiple Phase Transitions in Floquet Second-Order Topological Superconductors

A d-dimensional, nth-order topological insulator or superconductor has localized eigenmodes at its (d−n)-dimensional boundaries (n≤d). In this work, we apply periodic driving fields to two-dimensional superconductors, and obtain a wide variety of Floquet second-order topological superconducting (SOTSC) phases with many Majorana corner modes at both zero and π quasienergies. Two distinct Floquet SOTSC phases are found to be separated by three possible kinds of transformations, i.e., a topological phase transition due to the closing/reopening of a bulk spectral gap, a topological phase transition due to the closing/reopening of an edge spectral gap, or an entirely different phase in which the bulk spectrum is gapless. Thanks to the strong interplay between driving and intrinsic energy scales of the system, all the found phases and transitions are highly controllable via tuning a single hopping parameter of the system. Our discovery not only enriches the possible forms of Floquet SOTSC phases, but also offers an efficient scheme to generate many coexisting Majorana zero and π corner modes, which may find applications in Floquet quantum computation.

Non-Hermitian higher-order topological superconductors in two dimensions: Statics and dynamics

Being motivated by intriguing phenomena such as the breakdown of conventional bulk boundary correspondence and emergence of skin modes in the context of non-Hermitian (NH) topological insulators, we here propose a NH second-order topological superconductor (SOTSC) model that hosts Majorana zero modes (MZMs). Employing the non-Bloch form of NH Hamiltonian, we topologically characterize the above modes by biorthogonal nested polarization and resolve the apparent breakdown of the bulk boundary correspondence. Unlike the Hermitian SOTSC, we notice that the MZMs inhabit only one corner out of four in the two-dimensional NH SOTSC. We extend the static MZMs into the realm of Floquet drive. We find anomalous $\pi$-mode following low-frequency mass-kick in addition to the regular $0$-mode that is usually engineered in a high-frequency regime. We further characterize the regular $0$-mode with biorthogonal Floquet nested polarization. Our proposal is not limited to the $d$-wave superconductivity only and can be realized in the experiment with strongly correlated optical lattice platforms.

Higher-order topological phases on fractal lattices

Electronic materials harbor a plethora of exotic quantum phases, ranging from unconventional superconductors to non-Fermi liquids, and, more recently, topological phases of matter. While these quantum phases in integer dimensions are well characterized by now, their presence in fractional dimensions remains vastly unexplored. Here, we theoretically show that a special class of crystalline phases, namely, higher-order topological phases that via an extended bulk-boundary correspondence feature robust gapless modes on lower dimensional boundaries, such as corners and hinges, can be found on a representative family of fractional materials: \emph{quantum fractals}. To anchor this general proposal, we demonstrate realizations of second-order topological insulators and superconductors, supporting charged and neutral Majorana corner modes, on planar Sierpi\'{n}ski carpet and triangle fractals, respectively. These predictions can be experimentally tested on designer electronic fractal materials, as well as on various highly tunable metamaterial platforms, such as photonic and acoustic lattices.

Theory of topological superconductivity in doped IV-VI semiconductors

We theoretically study potential unconventional superconductivity in doped $AB$-type IV-VI semiconductors, based on a minimal effective model with interaction up to the next-nearest neighbors. According to the experimental implications, we focus on the spin-triplet channels and obtain the superconducting phase diagram with respect to the anisotropy of the Fermi surfaces and the interaction strength. Abundant nodal and nodeless states with different symmetry breaking appear in the phase diagram, and all the states are time-reversal invariant and topologically nontrivial. Specifically, the various nodal superconducting ground states, dubbed as the topological Dirac superconductors, are featured by Dirac nodes in the bulk and Majorana arcs on the surface; among the full-gap states, there exist a mirror-symmetry-protected second-order topological superconductor state favoring helical Majorana hinge cones, and different first-order topological superconductor states supporting four surface Majorana cones. The experimental verification of the different kinds of superconducting ground states is also discussed.

Topological Phases of an Interacting Majorana Benalcazar–Bernevig–Hughes Model

We study the effects of Coulomb repulsive interactions on a Majorana Benalcazar–Bernevig–Huges (MBBH) model. The MBBH model belongs to the class of second-order topological superconductors (HOTSC2), featuring robust Majorana corner modes. We consider an interacting strip of four chains of length L and perform a density matrix renormalization group (DMRG) numerical simulation based on a tensor-network approach. Study of the non-local fermionic correlations and the degenerate entanglement spectrum indicates that the topological phases are robust in the presence of interactions, even in the strongly interacting regime.

Higher-order topological superconductors from Weyl semimetals

We propose that doped Weyl semimetals with {time-reversal and certain crystalline symmetries} are natural candidates to realize higher-order topological superconductors, which exhibit a fully gapped bulk while the surface hosts robust gapless chiral hinge states. We show that in such a doped Weyl semimetal, a featureless finite-range attractive interaction favors a p+ip pairing symmetry. By analyzing its topological properties, we identify such a chiral pairing state as a higher-order topological superconductor, which depending on the existence of a four-fold rotoinversion symmetry, is either intrinsic, {meaning that the corresponding hinge states can only be removed by closing the bulk gap, rather than modifying the surface states}, or extrinsic. We achieve this understanding via various methods recently developed for higher-order topology, including Wannier representability, Wannier spectrum, and defect classification approaches. For the four-fold rotoinversion symmetric case, we provide a complete classification of the higher-order topological superconductors. We show that such second-order topological superconductors exhibit chiral hinge modes that are robust in the absence of interaction effects but can be eliminated at the cost of introducing surface topological order.

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.

Higher-order topological superconductors based on weak topological insulators

High-order topological phases host robust boundary states at the boundary of the boundary, which can be interpreted from their boundary topology. In this work, considering the interplay between superconductors and magnetic fields to gap the surface states of three-dimensional weak topological insulators, we show that second-order topological superconductors (TSCs) featuring helical or chiral Majorana hinge modes and third-order TSC featuring Majorana corner modes can be realized. Remarkably, the higher-order TSCs in our models can be attributed to their certain boundaries, surfaces, or hinges, which naturally behave as first-order TSC in DIII or D symmetry class. Correspondingly, these higher-order TSCs can be characterized by the boundary first-order topological invariants, such as surface Chern numbers or surface $Z_2$ topological invariants for surface TSCs. Our models can effectively capture the topology of iron-based superconductors with desired inverted band structures and superconducting pairings.

Majorana bound states in semiconducting nanostructures

In this Tutorial, we give a pedagogical introduction to Majorana bound states (MBSs) arising in semiconducting nanostructures. We start by briefly reviewing the well-known Kitaev chain toy model in order to introduce some of the basic properties of MBSs before proceeding to describe more experimentally relevant platforms. Here, our focus lies on simple “minimal” models where the Majorana wave functions can be obtained explicitly by standard methods. In the first part, we review the paradigmatic model of a Rashba nanowire with strong spin–orbit interaction (SOI) placed in a magnetic field and proximitized by a conventional s-wave superconductor. We identify the topological phase transition separating the trivial phase from the topological phase and demonstrate how the explicit Majorana wave functions can be obtained in the limit of strong SOI. In the second part, we discuss MBSs engineered from proximitized edge states of two-dimensional (2D) topological insulators. We introduce the Jackiw–Rebbi mechanism leading to the emergence of bound states at mass domain walls and show how this mechanism can be exploited to construct MBSs. Due to their recent interest, we also include a discussion of Majorana corner states in 2D second-order topological superconductors. This Tutorial is mainly aimed at graduate students—both theorists and experimentalists—seeking to familiarize themselves with some of the basic concepts in the field.

Chiral Majorana hinge modes in superconducting Dirac materials

Chiral Majorana hinge modes are characteristic of a second-order topological superconductor in three dimensions. Here we systematically study pairing symmetry in the point group D_{2h}, and find that the leading pairing channels can be of s-, d-, and s+id-wave pairing in Dirac materials. Except for the odd-parity s-wave pairing superconductivity, the s+id-wave pairing superconductor is topologically nontrivial and possesses Majorana hinge and surface modes. The chiral Majorana hinge modes can be characterized by a winding number of the quadrupole moment, or quantized quadruple moment at the symmetrically invariant point. Our findings suggest the strong spin-orbital coupling, crystalline symmetries and electron-electron interaction in the Dirac materials may provide a microscopic mechanism to realize chiral Majorana hinge modes without utilizing the proximity effect or external fields.

Topological Edge States of a Majorana BBH Model

We investigate a Majorana Benalcazar–Bernevig–Hughes (BBH) model showing the emergence of topological corner states. The model, consisting of a two-dimensional Su–Schrieffer–Heeger (SSH) system of Majorana fermions with π flux, exhibits a non-trivial topological phase in the absence of Berry curvature, while the Berry connection leads to a non-trivial topology. Indeed, the system belongs to the class of second-order topological superconductors (HOTSC2), exhibiting corner Majorana states protected by C4 symmetry and reflection symmetries. By calculating the 2D Zak phase, we derive the topological phase diagram of the system and demonstrate the bulk-edge correspondence. Finally, we analyze the finite size scaling behavior of the topological properties. Our results can serve to design new 2D materials with non-zero Zak phase and robust edge states.

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.

Odd-Parity Spin-Triplet Superconductivity in Centrosymmetric Antiferromagnetic Metals.

We propose a route to achieve odd-parity spin-triplet (OPST) superconductivity in metallic collinear antiferromagnets with inversion symmetry. Owing to the existence of hidden antiunitary symmetry, which we call the effective time-reversal symmetry (eTRS), the Fermi surfaces of ordinary antiferromagnetic metals are generally spin degenerate, and spin-singlet pairing is favored. However, by introducing a local inversion symmetry breaking perturbation that also breaks the eTRS, we can lift the degeneracy to obtain spin-polarized Fermi surfaces. In the weak-coupling limit, the spin-polarized Fermi surfaces constrain the electrons to form spin-triplet Cooper pairs with odd parity. Interestingly, all the odd-parity superconducting ground states we obtained host nontrivial band topologies manifested as chiral topological superconductors, second-order topological superconductors, and nodal superconductors. We propose that double perovskite oxides with collinear antiferromagnetic or ferrimagnetic ordering, such as SrLaVMoO_{6}, are promising candidate systems where our theoretical ideas can be applied to.