Two-dimensional Topological Superconductors Research Articles

Edge states of two-dimensional time-reversal invariant topological superconductors with strong interactions and disorder: A view from the lattice

Edge states of two-dimensional time-reversal invariant topological superconductors with strong interactions and disorder: A view from the lattice

Controllable superconducting to semiconducting phase transition in topological superconductor 2M-WS2

The investigation of exotic properties in two-dimensional (2D) topological superconductors has garnered increasing attention in condensed matter physics, particularly for applications in topological qubits. Despite this interest, a reliable way of fabricating topological Josephson junctions (JJs) utilizing topological superconductors has yet to be demonstrated. Controllable structural phase transition presents a unique approach to achieving topological JJs in atomically thin 2D topological superconductors. In this work, we report the pioneering demonstration of a structural phase transition from the superconducting to the semiconducting phase in the 2D topological superconductor 2M-WS2. We reveal that the metastable 2M phase of WS2 remains stable in ambient conditions but transitions to the 2H phase when subjected to temperatures above 150 °C. We further locally induced the 2H phase within 2M-WS2 nanolayers using laser irradiation. Notably, the 2H phase region exhibits a hexagonal shape, and scanning tunneling microscopy uncovers an atomically sharp crystal structural transition between the 2H and 2M phase regions. Moreover, the 2M to 2H phase transition can be induced at the nanometer scale by a 200 kV electron beam. The electrical transport measurements further confirmed the superconductivity of the pristine 2M-WS2 and the semiconducting behavior of the laser-irradiated 2M-WS2. Our results establish a novel approach for controllable topological phase change in 2D topological superconductors, significantly impacting the development of atomically scaled planar topological JJs.

Hybrid-order topological odd-parity superconductors via Floquet engineering

Having the potential for performing quantum computation, topological superconductors have been generalized to the second-order case. The hybridization of different orders of topological superconductors is attractive because it facilitates the simultaneous utilization of their respective advantages. However, previous studies found that they cannot coexist in one system due to the constraint of symmetry. We propose a Floquet engineering scheme to generate two-dimensional (2D) hybrid-order topological superconductors in an odd-parity superconductor system. Exotic hybrid-order phases exhibiting the coexisting gapless chiral edge states and gapped Majorana corner states not only in two different quasienergy gaps but also in one single quasienergy gap are created by periodic driving. The generalization of this scheme to the 3D system allows us to discover a second-order Dirac superconductor featuring coexisting surface and hinge Majorana Fermi arcs. Our results break a fundamental limitation on topological phases and open a feasible avenue to realize topological superconductor phases without static analogs.

Tunable zero-energy Dirac and Luttinger nodes in a two-dimensional topological superconductor

Cooper pairing in ultrathin films of topological insulators, induced intrinsically or by proximity effect, can produce an energetically favorable spin-triplet superconducting state. The spin–orbit coupling acts as an SU(2) gauge field and stimulates the formation of a spin-current vortex lattice in this superconducting state. Here we study the Bogoliubov quasiparticles in such a state and find that the quasiparticle spectrum consists of a number of Dirac nodes pinned to zero energy by the particle-hole symmetry. Some nodes are ‘accidental’ and move through the first Brillouin zone along high-symmetry directions as the order parameter magnitude or the strength of the spin–orbit coupling are varied. At special parameter values, nodes forming neutral quadruplets merge and become gapped out, temporarily producing a quadratic band-touching spectrum. All these features are tunable by controlling the order parameter magnitude via a gate voltage in a heterostructure device. In addition to analyzing the spectrum at the mean-field level, we briefly discuss a few experimental signatures of this spectrum.

RKKY interaction at helical edges of topological superconductors

We study spin configurations of magnetic impurities placed close to the edge of a two-dimensional topological superconductor both analytically and numerically. First, we demonstrate that the spin of a single magnetic impurity close to the edge of a topological superconductor tends to align along the edge. The strong easy-axis spin anisotropy behind this effect originates from the interaction between the impurity and the gapless helical Majorana edge states. We then compute the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between two magnetic impurities placed close to the edge. We show that, in the limit of large interimpurity distances, the RKKY interaction between the two impurities is mainly mediated by the Majorana edge states and leads to a ferromagnetic alignment of both spins along the edge. This effect can be used to detect helical Majorana edge states.

Anderson localization at the boundary of a two-dimensional topological superconductor

A one-dimensional boundary of a two-dimensional topological superconductor can host a number of topologically protected chiral modes. Combining two topological superconductors with different topological indices, it is possible to achieve a situation when only a given number of channels ($m$) are topologically protected, while others are not and therefore are subject to Anderson localization in the presence of disorder. We study transport properties of such quasi-one-dimensional quantum wires with broken time-reversal and spin-rotational symmetries (class D) and calculate the average conductance, its variance and the third cumulant, as well as the average shot noise power. The results are obtained for arbitrary wire length, tracing a crossover from the diffusive Drude regime to the regime of strong localization where only $m$ protected channels conduct. Our approach is based on the nonperturbative treatment of the nonlinear supersymmetric sigma model of symmetry class D with two replicas developed in our recent publication [D. S. Antonenko et al., Phys. Rev. B 102, 195152 (2020)]. The presence of topologically protected modes results in the appearance of a topological Wess-Zumino-Witten term in the sigma-model action, which leads to an additional subsidiary series of eigenstates of the transfer-matrix Hamiltonian. The developed formalism can be applied to study the interplay of Anderson localization and topological protection in quantum wires of other symmetry classes.

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.

Detrimental effects of disorder in two-dimensional time-reversal invariant topological superconductors

The robustness against local perturbations, as long as the symmetry of the system is preserved, is a distinctive feature of topological quantum states. Magnetic impurities and defects break time-reversal invariance and, consequently, time-reversal invariant (TRI) topological superconductors are fragile against this type of disorder. Non-magnetic impurities, however, preserve time-reversal symmetry and one naively expects a TRI topological superconductor to persist in the presence of non-magnetic impurities. In this work, we study the effect of non-magnetic disorder on a TRI topological superconductor with extended $s$-wave pairing, which can be engineered at the interface of an Fe-based superconductor and a strongly spin-orbit coupled Rashba layer. We model two different types of non-magnetic random disorder and analyze both the bulk density of states and edge state spectrum. Contrary to naive expectations, we find that the disorder strongly affects the topological phase by closing the energy gap, while trivial superconducting phases remain stable and fully gapped. The disorder phase diagram reveals a strong expansion of a nodal phase with increasing disorder. We further show the decay of the helical Majorana edge states in the topological phase and how they eventually disappear with increasing disorder. These results alter our understanding of effects of impurities and disorder on TRI topological phases and may help explain the difficulty of experimental observation of TRI topological superconductors.

Quasidisorder-induced topology

We study the effects of quasidisorder and Anderson disorder on a two-dimensional topological superconductor with an applied external magnetic field. The cases of a $p$-wave superconductor and a noncentrosymmetric superconductor with mixed $p$- and $s$-wave pairings and the Rashba spin-orbit coupling are studied. We show that, for a perpendicular magnetic field, the introduction of quasidisorder leads to the appearance of topological phases in new regions, characterized by an integer value of the Chern number. For a parallel magnetic field, we identify regimes with the appearance of new Majorana flat bands and also new unidirectional Majorana edge states, as quasidisorder is introduced. We show that the Majorana flat bands have a quantized Berry phase of $\ensuremath{\pi}$ and identify it as a topological invariant. Two topological transitions are identified and the values of the critical exponents $z$ and $\ensuremath{\nu}$ are obtained. The fractal nature of the eigenstates is discussed both for Anderson and Aubry-Andr\'e disorders.

Wire construction of class DIII topological crystalline superconductors in two dimensions

Topological superconductors have drawn wide interest as they host Majorana zero modes potentially useful in quantum computation. The crystalline symmetries remarkably enrich the family of topological superconductors. Here, the authors present a real-space method, called the ``wire construction'', to construct two-dimensional topological superconductors. A complete classification of 2D topological crystalline superconductors in class DIII is obtained, with anomalous boundary states enumerated. The new method provides a real-space picture for understanding topological superconductors, and should guide the discovery of realistic topological superconducting materials.

Josephson junctions of two-dimensional time-reversal invariant superconductors: Signatures of the topological phase

We determine the current-phase relation (CPR) of two-terminal configurations of Josephson junctions containing two-dimensional (2D) time-reversal invariant topological superconductors (TRITOPS), including TRITOPS-TRITOPS, as well as junctions between topological and non-topological superconductors (TRITOPS-S). We focus on long junctions for which several channels intervene in the tunneling coupling through the junction. We present a description of the topological edge modes for different TRITOPS models including $p$-wave pairing and the combination of $s$-wave pairing with spin-orbit coupling. We derive effective low-energy Hamiltonians to describe the Josephson junction, which can be solved analytically to explain the contribution of the edge states to the Josephson current as a function of the phase bias. We find that edge-modes yield singular corrections to the CPR for both junction types. The primary effects occur for the response of the Majorana zero-modes at half-flux quantum phase $\phi\approx \pi$ in TRITOPS-TRITOPS junctions and for integer flux quantum phase $\phi \approx 0$ for TRITOPS-S junctions, respectively. The former effect is particularly strong two-component nematic superconductors. The latter effect leads to a spontaneously broken time-reversal symmetry in the TRITOPS-S junction and to a breakdown of the bulk-boundary correspondence.

Dephasing-enhanced Majorana zero modes in two-dimensional and three-dimensional higher-order topological superconductors

The 1D Kitaev model in the topological phase, with open boundary conditions, hosts strong Majorana zero modes. These are fermion parity-odd operators that almost commute with the Hamiltonian and manifest in long coherence times for edge degrees of freedom. We obtain higher-dimensional counterparts of such Majorana operators by explicitly computing their closed form expressions in models describing 2D and 3D higher-order superconductors. Due to the existence of such strong Majorana zero modes, the coherence time of the infinite temperature autocorrelation function of the corner Majorana operators in these models diverges with the linear system size. In the presence of a certain class of orbital-selective dissipative dynamics, the coherence times of half of the corner Majorana operators is enhanced, while the time correlations corresponding to the remaining corner Majoranas decay much faster as compared with the unitary case. We numerically demonstrate robustness of the coherence times to the presence of disorder.

Dispersive one-dimensional Majorana modes with emergent supersymmetry in one-dimensional proximitized superconductors via spatially modulated potentials and magnetic fields

In condensed matter systems, zero-dimensional or one-dimensional Majorana modes can be realized respectively as the end and edge states of one-dimensional and two-dimensional topological superconductors. In this $\textit{top-down}$ approach, $(d-1)$-dimensional Majorana modes are obtained as the boundary states of a topologically nontrivial $d$-dimensional bulk. In a $\textit{bottom-up}$ approach instead, $d$-dimensional Majorana modes in a $d$-dimensional system can be realized as the continuous limit of a periodic lattice of coupled $(d-1)$-dimensional Majorana modes. We illustrate this idea by considering one-dimensional proximitized superconductors with spatially-modulated potential or magnetic fields. The ensuing inhomogeneous topological state exhibits one-dimensional counterpropagating Majorana modes with finite dispersion, and with a Majorana gap that can be controlled by external fields. In the massless case, the Majorana modes have opposite Majorana polarizations and pseudospins, are conformally invariant, and realize centrally extended quantum mechanical supersymmetry. The supersymmetry exhibits spontaneous partial breaking. Consequently, the massless Majorana fermion can be identified as a Goldstino, i.e., the Nambu-Goldstone fermion associated with the spontaneously broken supersymmetry.

Shot noise and differential conductance as signatures of putative topological superconductivity in FeSe0.45Te0.55

We present a theory for the differential shot noise, $dS/dV$, as measured via shot-noise scanning tunneling spectroscopy, and the differential conductance, $dI/dV$, for tunneling into Majorana zero modes (MZMs) in the putative topological superconductor ${\mathrm{FeSe}}_{0.45}{\mathrm{Te}}_{0.55}$. We demonstrate that for tunneling into chiral Majorana edge modes near domain walls, as well as MZMs localized in vortex cores and at the end of defect lines, $dS/dV$ vanishes whenever $dI/dV$ reaches a quantized value being equal to the quantum of conductance. These results are independent of the particular orbital tunneling path, thus establishing a vanishing $dS/dV$ concomitant with a quantized $dI/dV$, as universal signatures for Majorana modes in two-dimensional topological superconductors, irrespective of the material's specific complex electronic band structure.

Tunable Majorana corner modes by orbital-dependent exchange interaction in a two-dimensional topological superconductor

We theoretically study the effect of orbital-dependent exchange field in the formation of second order topological superconductors. We demonstrate that changing the orbital difference can induce topological transition and the Majorana corner modes therein can be manipulated. We further propose to detect the corner modes via a normal probe terminal. The conductance quantization is found to be robust to changes of the relevant system parameters.

Role of topology and symmetry for the edge currents of a two-dimensional superconductor

The bulk-boundary correspondence guarantees topologically protected edge states in a two-dimensional topological superconductor. Unlike in topological insulators, these edge states are, however, not connected to a quantized (spin) current as the electron number is not conserved in a Bogolyubov--de Gennes Hamiltonian. Still, edge currents are in general present. Here we use the two-dimensional Rashba system as an example to systematically analyze the effect symmetry reductions have on the order-parameter mixing and the edge properties in a superconductor of Altland-Zirnbauer class DIII (time-reversal-symmetry preserving) and D (time-reversal-symmetry breaking). In particular, we employ both Ginzburg-Landau and microscopic modeling to analyze the bulk superconducting properties and edge currents appearing in a strip geometry. We find edge (spin) currents independent of bulk topology and associated topological edge states which evolve continuously even when going through a phase transition into a topological state. Our findings emphasize the importance of symmetry over topology for the understanding of the nonquantized edge currents.

Controllable synthesis and tunable properties of topological material WTe<sub>2</sub>

<p indent=0mm>The two-dimensional topological insulator is expected to exhibit the quantum spin Hall effect because of the topologically protected helical edge state. The topological superconductor becomes one of the major material candidates for topological quantum computation due to the existence of Majorana zero-energy mode obeying non-Abelian statistics. Both two-dimensional topological insulator (2DTI) and topological superconductor (TS) can be realized in the monolayer (for 2DTI) or bulk (for TS) WTe₂. Using molecular beam epitaxy, we were able to grow a high-quality 1<italic>T</italic>′-WTe₂ monolayer. We revealed its semimetallic electron band structures using scanning tunneling microscopy/spectroscopy, an insulating phase with a fully opened energy gap is driven in the epitaxial 1<italic>T</italic>′-WTe₂ monolayer that is still topological nontrivial. Under ambient pressure, we performed alkali (K atoms) intercalation to dope electrons and achieved superconductivity transition in bulk Td-WTe₂ (type II Weyl semimetal). The intercalation of K atoms results in a negligible change in the crystal structure of Td-WTe₂, implying that superconductivity in intercalated Td-WTe₂ is still topologically nontrivial, according to X-ray diffraction characterizations.

Tunneling characteristics of weakly coupled Majorana wire arrays

The Majorana modes in solid-state systems have the potential to be applied in fault tolerant quantum computing. It is well known that a semiconducting nanowire with strong Rashba coupling and in the proximity of a superconductor hosts Majorana edge modes. An array of such nanowires with inter-wire coupling gives an approximate description of a two-dimensional topological superconductor, where depending on the strength of the magnetic field and the chemical potential, a rich phase diagram hosting trivial and different types of non-trivial phases can be achieved. In this work, we theoretically consider such a two-dimensional assembly of spin–orbit coupled superconducting nanowires and calculate the collective tunneling conductance between normal electrodes and the wires in the topological regime. When the number of wires in the assembly is N, as a consequence of the way the Majorana bonding and anti-bonding states form, we find that N conductance peaks symmetric about the bias V=0 appear for even N. When N is odd, a zero-bias conductance peak (ZBCP) also appears. We have identified the region in the parameter space where this odd–even rule applies. The effects of finite temperature, finite dissipation, and the barrier potential on the conductance profiles have also been discussed in detail. A device comprising of an array of weakly coupled Majorana wires can be realized by standard nano-fabrication techniques where individual nanowires can be turned ON or OFF by using a mechanical switch (or local top gating) to make N either even or odd—thereby switching the ZBCP OFF or ON, respectively, modulo the control parameter being in the desired range. Hence, in principle, our results can be used to realize and detect topological superconductivity efficiently and in a controlled manner.

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.