Majorana modes at the ends of superconductor vortices in doped topological insulators

Stripe charge order and its interaction with Majorana bound states in 2M-WS2 topological superconductors

To achieve logic operations via Majorana braiding, positional control of the Majorana bound states (MBSs) must be established. Here we report the observation of a striped surface charge order coexisting with superconductivity and its interaction with the MBS in the topological superconductor 2M-WS2, using low-temperature scanning tunneling microscopy. By applying an out-of-plane magnetic field, we observe that MBSs are absent in vortices in the region with stripe order. This is in contrast to adjacent underlaying layers without charge order, where vortex-bound MBSs are observed. Via theoretical simulations, we show that the surface stripe order does not destroy the bulk topology, but it can effectively modify the spatial distribution of MBSs, i.e. it pushes them downward, away from the 2M-WS2 surface. Our findings demonstrate that the interplay of charge order and topological superconductivity can potentially be used to tune the positions of MBSs, and to explore new states of matter.

Topological band inversion and chiral Majorana mode in hcp thallium

The chiral Majorana fermion is an exotic particle that is its own antiparticle. It can arise in a one-dimensional edge of topological materials, and especially that in a topological superconductor can be exploited in non-Abelian quantum computation. While the chiral Majorana mode (CMM) remains elusive, a promising situation is realized when superconductivity coexists with a topologically non-trivial surface state. Here, we perform fully non-empirical calculation for the CMM considering superconductivity and surface relaxation, and show that hexagonal close-packed thallium (Tl) has an ideal electronic state that harbors the CMM. Thekz=0plane of Tl is a mirror plane, realizing a full-gap band inversion corresponding to a topological crystalline insulating phase. Its surface and hinge are stable and easy to make various structures. Another notable feature is that the surface Dirac point is very close to the Fermi level, so that a small Zeeman field can induce a topological transition. Our calculation indicates that Tl will provide a new platform of the Majorana fermion.

Dislocation Majorana bound states in iron-based superconductors

We show that lattice dislocations of topological iron-based superconductors such as FeTe1-xSex will intrinsically trap non-Abelian Majorana quasiparticles, in the absence of any external magnetic field. Our theory is motivated by the recent experimental observations of normal-state weak topology and surface magnetism that coexist with superconductivity in FeTe1-xSex, the combination of which naturally achieves an emergent second-order topological superconductivity in a two-dimensional subsystem spanned by screw or edge dislocations. This exemplifies a new embedded higher-order topological phase in class D, where Majorana zero modes appear around the "corners" of a low-dimensional embedded subsystem, instead of those of the full crystal. A nested domain wall theory is developed to understand the origin of these defect Majorana zero modes. When the surface magnetism is absent, we further find that s± pairing symmetry itself is capable of inducing a different type of class-DIII embedded higher-order topology with defect-bound Majorana Kramers pairs. We also provide detailed discussions on the real-world material candidates for our proposals, including FeTe1-xSex, LiFeAs, β-PdBi2, and heterostructures of bismuth, etc. Our work establishes lattice defects as a new venue to achieve high-temperature topological quantum information processing.

Chiral Topological Superconductivity in Superconductor-Obstructed Atomic Insulator-Ferromagnetic Insulator Heterostructures

Implementing topological superconductivity (TSC) and Majorana states (MSs) is one of the most significant and challenging tasks in both fundamental physics and topological quantum computations. In this work, taking the obstructed atomic insulator (OAI) Nb_{3}Br_{8}, s-wave superconductor (SC) NbSe_{2}, and ferromagnetic insulator (FMI) CrI_{3} as an example, we propose a new setup to realize the 2D chiral TSC and MSs in the SC/OAI/FMI heterostructure, which could avoid the subband problem effectively and has the advantage of huge Rashba spin-orbit coupling. As a result, the TSC phase can be stabilized in a wide region of chemical potential and Zeeman splitting, and four distinct TSC phases with superconducting Chern number N=-1,-2,-3, 3 can be achieved. Moreover, a 2D Bogoliubov-de Gennes Hamiltonian based on the triangular lattice of obstructed Wannier charge centers, combined with the s-wave superconductivity paring and Zeeman splitting, is constructed to understand the whole topological phase diagram analytically. These results expand the application of OAIs and pave a new way to realize the TSC and MSs with unique advantages.

Edge state behavior in a Su-Schrieffer-Heeger like model with periodically modulated hopping

Su-Schrieffer-Heeger (SSH) model is one of the simplest models to show topological end/edge states and the existence of Majorana fermions. Here we consider a SSH like model both in one and two dimensions where a nearest neighbor hopping features spatially periodic modulations. In the 1D chain, we witness appearance of new in-gap end states apart from a pair of Majorana zero modes (MZMs) when the hopping periodicity go beyond two lattice spacings. The pair of MZMs, that appear in the topological regime, characterize the end modes each existing in either end of the chain. These, however, crossover to both-end end modes for small hopping modulation strength in a finite chain. Contrarily in a 2D SSH model with symmetric hopping that we consider, both non-zero and zero energy topological states appear in a finite square lattice even with a simple staggered hopping, though the zero energy modes disappear in a ribbon configuration. Apart from edge modes, the 2D system also features corner modes as well as modes with satellite peaks distributed non-randomly within the lattice. In both the dimensions, an increase in the periodicity of hopping modulation causes the zero energy Majorana modes to become available for either sign of the modulation. But interestingly with different periodicity for hopping modulations in the two directions, the zero energy modes in a 2D model become rarer and does not appear for all strength and sign of the modulation.

Hunting for Majoranas

Over the past decade, there have been considerable efforts to observe non-abelian quasiparticles in novel quantum materials and devices. These efforts are motivated by the goals of demonstrating quantum statistics of quasiparticles beyond those of fermions and bosons and of establishing the underlying science for the creation of topologically protected quantum bits. In this Review, we focus on efforts to create topological superconducting phases that host Majorana zero modes. We consider the lessons learned from existing experimental efforts, which are motivating both improvements to present platforms and the exploration of new approaches. Although the experimental detection of non-abelian quasiparticles remains challenging, the knowledge gained thus far and the opportunities ahead offer high potential for discovery and advances in this exciting area of quantum physics.

Superconductor Vortex Spectrum Including Fermi Arc States in Time-Reversal Symmetric Weyl Semimetals

Using semiclassics to surmount the hurdle of bulk-surface inseparability, we derive the superconductor vortex spectrum in nonmagnetic Weyl semimetals and show that it stems from the Berry phase of orbits made of Fermi arcs on opposite surfaces and bulk chiral modes. Tilting the vortex transmutes it between bosonic, fermionic, and supersymmetric, produces periodic peaks in the density of states that signify novel nonlocal Majorana modes, and yields a thickness-independent spectrum at magic "magic angles." We propose (Nb,Ta)P as candidate materials and tunneling spectroscopy as the ideal experiment.

Topological Nodal Surface and Quadratic Dirac Semimetal States and van Hove Singularities in ScH3 and LuH3 Superconductors

The coexistence of non-trivial topology and superconductivity in a material may induce a novel physical phenomenon known as topological superconductivity. Topological superconductors have been the subject of intense research, yet there are severe limitations in their application due to a lack of suitable materials. Topological nodal surface semimetals with nearly flat nodal surfaces near the Fermi level can be promising materials to achieve topological superconductivity. Here, we use first-principles calculations to examine the topological electronic characteristics of two new superconductors, ScH₃ and LuH₃, at both ambient and high pressures. Our studies show that both ScH₃ and LuH₃ have van Hove singularities, which confirms their superconductivity. Interestingly, both materials host topological nodal surface states under the protection of time reversal and spatial inversion symmetries in the absence of spin-orbit coupling (SOC). These nodal surfaces are distinguished by a pair of unique drum-head-like surface states not previously observed in nodal surface semimetals. Moreover, the nodal surfaces transform into essential spin-orbit quadratic Dirac points when SOC is included. Our findings demonstrate that ScH₃ and LuH₃ are good candidates to investigate the exotic properties of both nodal surface semimetals (NSSMs) and quadratic Dirac semimetal states and also provide a platform to explore the coexistence of topology and superconductivity in NSSMs with promising applications in high-speed electronics and topological quantum computing.

Observation of Gapped Topological Surface States and Isolated Surface Resonances in PdTe2 Ultrathin Films

The superconductor PdTe₂ is known to host bulk Dirac bands and topological surface states. The coexistence of superconductivity and topological surface states makes PdTe₂ a promising platform for exploring topological superconductivity and Majorana bound states. In this work, we report the spectroscopic characterization of ultrathin PdTe₂ films with thickness down to three monolayers (ML). In the 3 ML PdTe₂ film, we observed spin-polarized surface resonance states, which are isolated from the bulk bands due to the quantum size effects. In addition, the hybridization of surface states on opposite faces leads to a thickness-dependent gap in the topological surface Dirac bands. Our photoemission results show clearly that the size of the hybridization gap increases as the film thickness is reduced. The observation of isolated surface resonances and gapped topological surface states sheds light on the applications of PdTe₂ quantum films in spintronics and topological quantum computation.

Topological superconducting vortex from trivial electronic bands

Superconducting vortices are promising traps to confine non-Abelian Majorana quasi-particles. It has been widely believed that bulk-state topology, of either normal-state or superconducting ground-state wavefunctions, is crucial for enabling Majorana zero modes in solid-state systems. This common belief has shaped two major search directions for Majorana modes, in either intrinsic topological superconductors or trivially superconducting topological materials. Here we show that Majorana-carrying superconducting vortex is not exclusive to bulk-state topology, but can arise from topologically trivial quantum materials as well. We predict that the trivial bands in superconducting HgTe-class materials are responsible for inducing anomalous vortex topological physics that goes beyond any existing theoretical paradigms. A feasible scheme of strain-controlled Majorana engineering and experimental signatures for vortex Majorana modes are also discussed. Our work provides new guidelines for vortex-based Majorana search in general superconductors.

Gapless vortex bound states in superconducting topological semimetals

We find that the vortex bound states in superconducting topological semimetals are gapless owing to topological massless excitations in their normal states. We demonstrate this universal result in a variety of semimetals, including Dirac and Weyl semimetals, three-fold degenerate spin-1 fermions, spin-3/2 Rarita-Schwinger-Weyl fermion semimetals and other exotic fermion semimetals. The formation of these gapless bound states is closely related to their Andreev specular reflection and propagating Andreev modes in π-phase superconductor-normal metal-superconductor junctions. We further demonstrate that these gapless states are topologically protected and can be derived from a topological pumping process.

Competing Vortex Topologies in Iron-Based Superconductors

In this Letter, we establish a new theoretical paradigm for vortex Majorana physics in the recently discovered topological iron-based superconductors (TFeSCs). While TFeSCs are widely accepted as an exemplar of topological insulators (TIs) with intrinsic s-wave superconductivity, our theory implies that such a common belief could be oversimplified. Our main finding is that the normal-state bulk Dirac nodes, usually ignored in TI-based vortex Majorana theories for TFeSCs, will play a key role of determining the vortex state topology. In particular, the interplay between TI and Dirac nodal bands will lead to multiple competing topological phases for a superconducting vortex line in TFeSCs, including an unprecedented hybrid topological vortex state that carries both Majorana bound states and a gapless dispersion. Remarkably, this exotic hybrid vortex phase generally exists in the vortex phase diagram for our minimal model for TFeSCs and is directly relevant to TFeSC candidates such as LiFeAs. When the fourfold rotation symmetry is broken by vortex-line tilting or curving, the hybrid vortex gets topologically trivialized and becomes Majorana free, which could explain the puzzle of ubiquitous trivial vortices observed in LiFeAs. The origin of the Majorana signal in other TFeSC candidates such as FeTe_{x}Se_{1-x} and CaKFe_{4}As_{4} is also interpreted within our theory framework. Our theory sheds new light on theoretically understanding and experimentally engineering Majorana physics in high-temperature iron-based systems.

Nodeless superconductivity in topologically nontrivial materials HfRuP and ZrRuAs

Topologically nontrivial electronic states are recently found in a family of noncentrosymmetric transition metal pnictidesTRuX(T=Zr, Hf;X=P, As), presenting a unique platform for superconductivity to interplay with topological electronic states and asymmetric spin-orbit coupling. Here, we investigate the superconducting order parameter of HfRuP and ZrRuAs by measuring the magnetic penetration depth changeΔλ(T)using a method based on the tunnel-diode oscillator. Both compounds show clear exponential temperature dependence inΔλ(T)at low temperatures, suggesting fully-gapped superconductivity. Moreover, the superfluid densities in both HfRuP and ZrRuAs can be reasonably described by ans-wave superconducting model.

Controllable Majorana vortex states in iron-based superconducting nanowires

To reveal the non-Abelian braiding statistics of Majorana zero modes (MZMs), it is crucial to design a Majorana platform, in which MZMs can be easily manipulated in a broad topological nontrivial parameter space. This is also an essential step to confirm their existence. In this study, we propose an iron-based superconducting nanowire system with Majorana vortex states to satisfy desirable conditions. This system has a radius-induced topological phase transition, giving a lower bound for the nanowire radius. In the topological phase, the iron-based superconducting nanowires have only one pair of MZMs over a wide range of radii, chemical potential and external magnetic fields. The wave function of MZMs has a sizable distribution at the side edge of the nanowires. This property enables the control of the interaction of MZMs in neighboring vortex nanowires and paves the way for Majorana fusion and braiding.

Chiral Majorana edge modes and vortex Majorana zero modes in superconducting antiferromagnetic topological insulator

The antiferromagnetic topological insulator (AFTI) is topologically protected by the combined time-reversal and translational symmetryTc. In this paper we investigate the effects of thes-wave superconducting pairings on the multilayers of AFTI, which breaksTcsymmetry and can realize quantum anomalous Hall insulator with unit Chern number. For the weakly coupled pairings, the system corresponds to the topological superconductor (TSC) with the Chern numberC= ±2. We answer the following questions whether the local Chern numbers and chiral Majorana edge modes of such a TSC distribute around the surface layers. By the numerical calculations based on a theoretic model of AFTI, we find that when the local Chern numbers are always dominated by the surface layers, the wavefunctions of chiral Majorana edge modes must not localize on the surface layers and show a smooth crossover from spatially occupying all layers to only distributing near the surface layers, similar to the hinge states in a three dimensional second-order topological phases. The latter phase, denoted by the hinged TSC, can be distinguished from the former phase by the measurements of the local density of state. In addition we also study the superconducting vortex phase transition in this system and find that the exchange field in the AFTI not only enlarges the phase space of topological vortex phase but also enhances its topological stability. These conclusions will stimulate the investigations on superconducting effects of AFTI and drive the studies on chiral Majorana edge modes and vortex Majorana zero modes into a new era.

Fermi Arc Criterion for Surface Majorana Modes in Superconducting Time-Reversal Symmetric Weyl Semimetals

Many clever routes to Majorana fermions have been discovered by exploiting the interplay between superconductivity and band topology in metals and insulators. However, realizations in semimetals remain less explored. We ask, "Under what conditions do superconductor vortices in time-reversal symmetric Weyl semimetals-three-dimensional semimetals with only time-reversal symmetry-trap Majorana fermions on the surface?" If each constant-k_{z} plane, where z is the vortex axis, contains equal numbers of Weyl nodes of each chirality, we predict a generically gapped vortex and derive a topological invariant ν=±1 in terms of the Fermi arc structure that signals the presence or absence of surface Majorana fermions. In contrast, if certain constant-k_{z} planes contain a net chirality of Weyl nodes, the vortex is gapless. We analytically calculate ν within a perturbative scheme and provide numerical support with a lattice model. The criteria survive the presence of other bulk and surface bands and yield phase transitions between trivial, gapless, and topological vortices upon tilting the vortex. We propose Li(Fe_{0.91}Co_{0.09})As and Fe_{1+y}Se_{0.45}Te_{0.55} with broken inversion symmetry as candidates for realizing our proposals.

New types of topological superconductors under local magnetic symmetries

We classify gapped topological superconducting (TSC) phases of one-dimensional quantum wires with local magnetic symmetries, in which the time-reversal symmetry [Formula: see text] is broken, but its combinations with certain crystalline symmetries, such as [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], are preserved. Our results demonstrate that an equivalent BDI class TSC can be realized in the [Formula: see text] or [Formula: see text] superconducting wire, which is characterized by a chiral Z c invariant. More interestingly, we also find two types of totally new TSC phases in the [Formula: see text] and [Formula: see text] superinducting wires, which are beyond the known AZ class, and are characterized by a helical Z h invariant and Z h ⊕Z c invariants, respectively. In the Z h TSC phase, Z pairs of Majorana zero modes (MZMs) are protected at each end. In the [Formula: see text] case, the MZMs can be either chiral or helical, and even helical-chiral coexisting. The minimal models preserving [Formula: see text] or [Formula: see text] symmetry are presented to illustrate their novel TSC properties and MZMs.

Observation of magnetic adatom-induced Majorana vortex and its hybridization with field-induced Majorana vortex in an iron-based superconductor

Braiding Majorana zero modes is essential for fault-tolerant topological quantum computing. Iron-based superconductors with nontrivial band topology have recently emerged as a surprisingly promising platform for creating distinct Majorana zero modes in magnetic vortices in a single material and at relatively high temperatures. The magnetic field-induced Abrikosov vortex lattice makes it difficult to braid a set of Majorana zero modes or to study the coupling of a Majorana doublet due to overlapping wave functions. Here we report the observation of the proposed quantum anomalous vortex with integer quantized vortex core states and the Majorana zero mode induced by magnetic Fe adatoms deposited on the surface. We observe its hybridization with a nearby field-induced Majorana vortex in iron-based superconductor FeTe0.55Se0.45. We also observe vortex-free Yu-Shiba-Rusinov bound states at the Fe adatoms with a weaker coupling to the substrate, and discover a reversible transition between Yu-Shiba-Rusinov states and Majorana zero mode by manipulating the exchange coupling strength. The dual origin of the Majorana zero modes, from magnetic adatoms and external magnetic field, provides a new single-material platform for studying their interactions and braiding in superconductors bearing topological band structures.

Vortex and Surface Phase Transitions in Superconducting Higher-order Topological Insulators

Topological insulators, having intrinsic or proximity-coupled s-wave superconductivity, host Majorana zero modes (MZMs) at the ends of vortex lines. The MZMs survive up to a critical doping of the TI at which there is a vortex phase transition that eliminates the MZMs. In this work, we show that the phenomenology in higher-order topological insulators (HOTIs) can be qualitatively distinct. In particular, we find two distinct features. (i) We find that vortices placed on the gapped (side) surfaces of the HOTI, exhibit a pair of phase transitions as a function of doping. The first transition is a surface phase transition after which MZMs appear. The second transition is the well-known vortex phase transition. We find that the surface transition appears because of the competition between the superconducting gap and the local T-breaking gap on the surface. (ii) We present numerical evidence that shows strong variation of the critical doping for the vortex phase transition as the center of the vortex is moved toward or away from the hinges of the sample. We believe our work provides new phenomenology that can help identify HOTIs, as well as illustrating a promising platform for the realization of MZMs.

Vortex End Majorana Zero Modes in Superconducting Dirac and Weyl Semimetals

Time-reversal invariant Dirac and Weyl semimetals in three dimensions (3D) can host open Fermi arcs and spin-momentum locking Fermi loops on the surfaces. We find that when they become superconducting with s-wave pairing and the doping is lower than a critical level, straight π-flux vortex lines terminating at surfaces with Fermi arcs or spin-momentum locking Fermi loops can realize 1D topological superconductivity and harbor Majorana zero modes at their ends. Remarkably, we find that the vortex-generation-associated Zeeman field can open (when the surfaces have only Fermi arcs) or enhance the topological gap protecting Majorana zero modes, which is contrary to the situation in superconducting topological insulators. By studying the tilting effect of bulk Dirac and Weyl cones, we further find that type-I Dirac and Weyl semimetals in general have a much broader topological regime than type-II ones. Our findings build up a connection between time-reversal invariant Dirac and Weyl semimetals and Majorana zero modes in vortices.

Nodal line topological superfluid and multiply protected Majorana Fermi arc in a three-dimensional time-reversal-invariant superfluid model

We theoretically study a time-reversal-invariant three-dimensional superfluid model by stacking in z direction identical bilayer models with intralayer spin-orbit coupling and contrary Zeeman energy splitting for different layers, which has been suggested recently to realize two-dimensional time-reversal-invariant topological superfluid. We find that this model shows two kinds of topologically nontrivial phases: gapless phases with nodal lines in pairs protected by chiral symmetry and a gapped phase, both of which support a time-reversal-invariant Majorana Fermi arc (MFA) on the yz and xz side surface. These MFA abide by time-reversal and particle-hole symmetries and are topologically protected by the winding numbers in mirror subspaces and the Z ₂ numbers of two-dimensional DIII class topological superfluid, which are different from MFA in the time-reversal broken Weyl superfluid protected by nonzero Chern numbers. This important observation means that MFA in our model represents a new type of topological state not explored previously. The Zeeman field configuration in our model is relevant to the antiferromagnetic topological insulator MnBi₂Te₄, thus our work stimulates the further studies on superconducting effects in the realistic antiferromagnetic topological insulator.

Flux-induced topological superconductivity in full-shell nanowires

Hybrid semiconductor-superconductor nanowires have emerged as a promising platform for realizing topological superconductivity (TSC). Here, we present a route to TSC using magnetic flux applied to a full superconducting shell surrounding a semiconducting nanowire core. Tunneling into the core reveals a hard induced gap near zero applied flux, corresponding to zero phase winding, and a gapped region with a discrete zero-energy state around one applied flux quantum, corresponding to 2π phase winding. Theoretical analysis indicates that the winding of the superconducting phase can induce a transition to a topological phase supporting Majorana zero modes. Measured Coulomb blockade peak spacing around one flux quantum shows a length dependence that is consistent with the existence of Majorana modes at the ends of the nanowire.

Zero-Energy Modes, Fractional Fermion Numbers and The Index Theorem in a Vortex-Dirac Fermion System

Physics of topological materials has attracted much attention from both physicists and mathematicians recently. The index and the fermion number of Dirac fermions play an important role in topological insulators and topological superconductors. A zero-energy mode exists when Dirac fermions couple to objects with soliton-like structure such as kinks, vortices, monopoles, strings, and branes. We discuss a system of Dirac fermions interacting with a vortex and a kink. This kind of systems will be realized on the surface of topological insulators where Dirac fermions exist. The fermion number is fractionalized and this is related to the presence of fermion zero-energy excitation modes. A zero-energy mode can be regarded as a Majorana fermion mode when the chemical potential vanishes. Our discussion includes the case where there is a half-flux quantum vortex associated with a kink in a magnetic field in a bilayer superconductor. A normalizable wave function of fermion zero-energy mode does not exist in the core of the half-flux quantum vortex. The index of Dirac operator and the fermion number have additional contributions when a soliton scalar field has a singularity.

Topological vortex phase transitions in iron-based superconductors

We study topological vortex phases in iron-based superconductors. Besides the previously known vortex end Majorana zero modes (MZMs) phase stemming from the existence of a three dimensional (3D) strong topological insulator state, we show that there is another topologically nontrivial phase as iron-based superconductors can be doped superconducting 3D weak topological insulators (WTIs). The vortex bound states in a superconducting 3D WTI exhibit two different types of quantum states, a robust nodal superconducting phase with pairs of bulk MZMs and a full-gap topologically nontrivial superconducting phase which has single vortex end MZM in a certain range of doping level. Moreover, we predict and summarize various topological phases in iron-based superconductors, and find that carrier doping and interlayer coupling can drive systems to have phase transitions between these different topological phases.

Experimental evidence of anomalously large superconducting gap on topological surface state of β-Bi2Pd film

Connate topological superconductor (TSC) combines topological surface states with nodeless superconductivity in a single material, achieving effective p-wave pairing without interface complication. By combining angle-resolved photoemission spectroscopy and in-situ molecular beam epitaxy, we studied the momentum-resolved superconductivity in β-Bi₂Pd film. We found that the superconducting gap of topological surface state (Δ_TSS ∼ 3.8 meV) is anomalously enhanced from its bulk value (Δ_b ∼ 0.8 meV). The ratio of 2Δ_TSS/k_BT_c ∼ 16.3, is substantially larger than the BCS value. By measuring β-Bi₂Pd bulk single crystal as a comparison, we clearly observed the upward-shift of chemical potential in the film. In addition, a concomitant increasing of surface weight on the topological surface state was revealed by our first principle calculation, suggesting that the Dirac-fermion-mediated parity mixing may cause this anomalous superconducting enhancement. Our results establish β-Bi₂Pd film as a unique case of connate TSCs with a highly enhanced topological superconducting gap, which may stabilize Majorana zero modes at a higher temperature.

Zero-energy vortex bound state in the superconducting topological surface state of Fe(Se,Te)

Majorana quasiparticles in condensed matter are important for topological quantum computing^1-3, but remain elusive. Vortex cores of topological superconductors may accommodate Majorana quasiparticles that appear as the Majorana bound state (MBS) at zero energy^4,5. The iron-based superconductor Fe(Se,Te) possesses a superconducting topological surface state^6-9 that was investigated by scanning tunnelling microscopy (STM) studies, which suggest such a zero-energy vortex bound state (ZVBS)^10,11. Here we present ultrahigh energy-resolution spectroscopic imaging (SI)-STM to clarify the nature of the vortex bound states in Fe(Se,Te). We found the ZVBS at 0 ± 20 μeV, which constrained its MBS origin, and showed that some vortices host the ZVBS but others do not. We show that the fraction of vortices hosting the ZVBS decreases with increasing magnetic field and that local quenched disorders are not related to the ZVBS. Our observations elucidate the necessary conditions to realize the ZVBS, which paves the way towards controllable Majorana quasiparticles.

Quasi-1D Topological Nodal Vortex Line Phase in Doped Superconducting 3D Dirac Semimetals

We study vortex bound states in three-dimensional (3D) superconducting Dirac semimetals with time reversal symmetry. We find that there exist robust gapless vortex bound states propagating along the vortex line in the s-wave superconducting state. We refer to this newly found phase as the quasi-1D nodal vortex line phase. According to the Altland-Zirnbauer classification, the phase is characterized by a topological index (ν;N) at k_{z}=0 and k_{z}=π, where ν is the Z_{2} topological invariant for a 0D class-D system and N is the Z topological invariant for a 0D class-A system. Furthermore, we show that the vortex end Majorana zero mode can coexist with the quasi-1D nodal phase in certain types of Dirac semimetals. The possible experimental observations and material realization of such nodal vortex line states are discussed.

Z_{4} Topological Superconductivity in UCoGe

Topological nonsymmorphic crystalline superconductivity (TNCS) is an intriguing phase of matter, offering a platform to study the interplay between topology, superconductivity, and nonsymmorphic crystalline symmetries. Interestingly, some of TNCSs are classified into Z_{4} topological phases, which have unique surface states referred to as a Möbius strip or an hourglass, and they have not been achieved in symmorphic superconductors. However, material realization of Z_{4} TNCS has never been known, to the best of our knowledge. Here, we propose that the paramagnetic superconducting phase of UCoGe under pressure is a promising candidate of Z_{4}-nontrivial TNCS enriched by glide symmetry. We evaluate Z_{4} invariants of UCoGe by deriving the formulas relating Z_{4} invariants to the topology of Fermi surfaces. Applying the formulas and previous ab initio calculations, we clarify that three odd-parity representations out of four are Z_{4}-nontrivial TNCS, whereas the other is also Z_{2}-nontrivial TNCS. We also discuss possible Z_{4} TNCS in CrAs and related materials.

Crystalline-Symmetry-Protected Helical Majorana Modes in the Iron Pnictides

We propose that propagating one-dimensional Majorana fermions will develop in the vortex cores of certain iron-based superconductors, most notably Li(Fe_{1-x}Co_{x})As. A key ingredient of this proposal is the 3D Dirac cones recently observed in photoemission experiments [P. Zhang et al., Nat. Phys. 15, 41 (2019)NPAHAX1745-247310.1038/s41567-018-0280-z]. Using an effective Hamiltonian around the Γ-Z line we demonstrate the development of gapless one-dimensional helical Majorana modes, protected by C_{4} symmetry. A topological index is derived which links the helical Majorana modes to the presence of monopoles in the Berry curvature of the normal state. We present various experimental consequences of this theory and discuss its possible connections with cosmic strings.

Conductance Spectroscopy of Exfoliated Thin Flakes of Nb xBi2Se3

We study unconventional superconductivity in exfoliated single crystals of a promising three-dimensional (3D) topological superconductor candidate, Nb-doped Bi₂Se₃ through differential conductance spectroscopy and magneto-transport. The strong anisotropy of the critical field along the out-of-plane direction suggests that the thin exfoliated flakes are in the quasi-2D limit. Normal metal-superconductor (NS) contacts with either high or low transparencies made by depositing gold leads onto Nb-doped Bi₂Se₃ flakes both show significant enhancement in zero bias conductance and coherence dips at the superconducting energy gap. Such behavior is inconsistent with conventional Blonder-Tinkham-Klapwijk theory. Instead, we discuss how our results are consistent with p-wave pairing symmetry, supporting the possibility of topological superconductivity in Nb-doped Bi₂Se₃. Finally, we observe signatures of multiple superconducting energy gaps, which could originate from multiple Fermi surfaces reported earlier in bulk crystals.

Fractional Angular Momentum at Topological Insulator Interfaces

Recently, two fundamental topological properties of a magnetic vortex at the interface of a superconductor (SC) and a strong topological insulator (TI) have been established: The vortex carries both a Majorana zero mode relevant for topological quantum computation and, for a time-reversal invariant TI, a charge of e/4. This fractional charge is caused by the axion term in the electromagnetic Lagrangian of the TI. Here we determine the angular momentum J of the vortices, which in turn determines their mutual statistics. Solving the axion-London electrodynamic equations including screening in both a SC and a TI, we find that the elementary quantum of angular momentum of the vortex is -n^{2}ℏ/8, where n is the flux quantum of the vortex line. Exchanging two elementary fluxes thus changes the phase of the wave function by -π/4.

Evidence for Majorana bound states in an iron-based superconductor

The search for Majorana bound states (MBSs) has been fueled by the prospect of using their non-Abelian statistics for robust quantum computation. Two-dimensional superconducting topological materials have been predicted to host MBSs as zero-energy modes in vortex cores. By using scanning tunneling spectroscopy on the superconducting Dirac surface state of the iron-based superconductor FeTe_0.55Se_0.45, we observed a sharp zero-bias peak inside a vortex core that does not split when moving away from the vortex center. The evolution of the peak under varying magnetic field, temperature, and tunneling barrier is consistent with the tunneling to a nearly pure MBS, separated from nontopological bound states. This observation offers a potential platform for realizing and manipulating MBSs at a relatively high temperature.

Evidence for weak collective pinning and δl pinning in topological superconductor Cu x Bi2Se3

We investigated the vortex pinning behavior in the single crystal topological superconductor Cu_0.10Bi₂Se₃ with a pronounced anisotropic peak effect. A weak collective pinning regime is clarified from the power-law behavior in [Formula: see text] and the small critical current density ratio of [Formula: see text] ~ 10^-5 ([Formula: see text] is the critical current density, [Formula: see text] is the depairing current density). The spatial variation of the charge-carrier mean free path induced pinning is evidenced and probably results from the well-defined atomic defects. Within the framework of collective pinning theory, we computed the values of the correlated length and volume at 1.8 K, which start declining prior to the onset field of the peak effect [Formula: see text], demonstrating the vortex lattices already suffered a preferential collapse ahead of the peak effect turns up. Thus, the peak effect can be understood by elastic moduli softening near the upper critical field [Formula: see text]. We suggest Cu _x Bi₂Se₃ is a prototype topological material for investigating the vortex pinning dynamics associated with the peak effect phenomenon.

Magnetotransport study of topological superconductor Cu0.10Bi2Se3 single crystal

We report a magnetotransport study of vortex-pinning in Cu_0.10Bi₂Se₃ single crystal. The sample is demonstrated to be in clean limit and absent of Pauli spin-limiting effect. Interestingly, the resistivity versus magnetic field shows an anomalously pronounced increase when approaching the superconducting-normal state boundary for both [Formula: see text] and [Formula: see text] configurations. We have investigated the flux-flowing behavior under various magnetic fields and temperatures, enabling us to establish its anisotropic vortex phase diagram. Our results suggest the Cu_0.10Bi₂Se₃ can be served as one unique material for exploring exotic surface vortex states in topological superconductors.

Boundary states in the chiral symmetric systems with a spatial symmetry

We study topological systems with both a chiral and a spatial symmetry which result in an additional spatial chiral symmetry. We distinguish the topologically nontrivial states according to the chiral symmetries protecting them and study several models in 1D and 3D systems. The perturbations breaking the spatial symmetry can break only one of the two chiral symmetries while the perturbations preserving the spatial symmetry always break or preserve both of them. In 3D systems, besides the 3D symmetries, the topologically nontrivial boundary modes may also be protected by the hidden lower dimensional symmetries. We then figure out the corresponding topological invariants and connect them with the 3D invariants.

Full-gap superconductivity in spin-polarised surface states of topological semimetal β-PdBi2

A bulk superconductor possessing a topological surface state at the Fermi level is a promising system to realise long-sought topological superconductivity. Although several candidate materials have been proposed, experimental demonstrations concurrently exploring spin textures and superconductivity at the surface have remained elusive. Here we perform spectroscopic-imaging scanning tunnelling microscopy on the centrosymmetric superconductor β-PdBi2 that hosts a topological surface state. By combining first-principles electronic-structure calculations and quasiparticle interference experiments, we determine the spin textures at the surface, and show not only the topological surface state but also all other surface bands exhibit spin polarisations parallel to the surface. We find that the superconducting gap fully opens in all the spin-polarised surface states. This behaviour is consistent with a possible spin-triplet order parameter expected for such in-plane spin textures, but the observed superconducting gap amplitude is comparable to that of the bulk, suggesting that the spin-singlet component is predominant in β-PdBi2.Although several materials have been proposed as topological superconductors, spin textures and superconductivity at the surface remain elusive. Here, Iwaya et al. determine the spin textures at the surface of a superconductor β-PdBi2 and find the superconducting gap opening in all spin-polarised surface states.

Spontaneous breaking of time-reversal symmetry in topological superconductors

We study the behavior of spinless fermions in superconducting state, in which the phases of the superconducting order parameter depend on the direction of the link. We find that the energy of the superconductor depends on the phase differences of the superconducting order parameter. The solutions for the phases corresponding to the energy minimuma, lead to a topological superconducting state with the nontrivial Chern numbers. We focus our quantitative analysis on the properties of topological states of superconductors with different crystalline symmetry and show that the phase transition in the topological superconducting state is result of spontaneous breaking of time-reversal symmetry in the superconducting state. The peculiarities in the chiral gapless edge modes behavior are studied, the Chern numbers are calculated.

Engineering of many-body Majorana states in a topological insulator/s-wave superconductor heterostructure

We study a vortex chain in a thin film of a topological insulator with proximity-induced superconductivity-a promising platform to realize Majorana zero modes (MZMs)-by modeling it as a two-leg Majorana ladder. While each pair of MZMs hybridizes through vortex tunneling, we hereby show that MZMs can be stabilized on the ends of the ladder with the presence of tilted external magnetic field and four-Majorana interaction. Furthermore, a fruitful phase diagram is obtained by controlling the direction of magnetic field and the thickness of the sample. We reveal many-body Majorana states and interaction-induced topological phase transitions and also identify trivial-superconducting and commensurate/incommensurate charge-density-wave states in the phase diagram.

Observation of the spin-polarized surface state in a noncentrosymmetric superconductor BiPd

Recently, noncentrosymmetric superconductor BiPd has attracted considerable research interest due to the possibility of hosting topological superconductivity. Here we report a systematic high-resolution angle-resolved photoemission spectroscopy (ARPES) and spin-resolved ARPES study of the normal state electronic and spin properties of BiPd. Our experimental results show the presence of a surface state at higher-binding energy with the location of Dirac point at around 700 meV below the Fermi level. The detailed photon energy, temperature-dependent and spin-resolved ARPES measurements complemented by our first-principles calculations demonstrate the existence of the spin-polarized surface states at high-binding energy. The absence of such spin-polarized surface states near the Fermi level negates the possibility of a topological superconducting behaviour on the surface. Our direct experimental observation of spin-polarized surface states in BiPd provides critical information that will guide the future search for topological superconductivity in noncentrosymmetric materials.

Superconducting topological surface states in the noncentrosymmetric bulk superconductor PbTaSe2

The search for topological superconductors (TSCs) is one of the most urgent contemporary problems in condensed matter systems. TSCs are characterized by a full superconducting gap in the bulk and topologically protected gapless surface (or edge) states. Within each vortex core of TSCs, there exists the zero-energy Majorana bound states, which are predicted to exhibit non-Abelian statistics and to form the basis of the fault-tolerant quantum computation. To date, no stoichiometric bulk material exhibits the required topological surface states (TSSs) at the Fermi level (EF) combined with fully gapped bulk superconductivity. We report atomic-scale visualization of the TSSs of the noncentrosymmetric fully gapped superconductor PbTaSe2. Using quasi-particle scattering interference imaging, we find two TSSs with a Dirac point at E ≅ 1.0 eV, of which the inner TSS and the partial outer TSS cross EF, on the Pb-terminated surface of this fully gapped superconductor. This discovery reveals PbTaSe2 as a promising candidate for TSC.

Topological Superconductivity on the Surface of Fe-Based Superconductors

As one of the simplest systems for realizing Majorana fermions, the topological superconductor plays an important role in both condensed matter physics and quantum computations. Based on ab initio calculations and the analysis of an effective 8-band model with superconducting pairing, we demonstrate that the three-dimensional extended s-wave Fe-based superconductors such as Fe_{1+y}Se_{0.5}Te_{0.5} have a metallic topologically nontrivial band structure, and exhibit a normal-topological-normal superconductivity phase transition on the (001) surface by tuning the bulk carrier doping level. In the topological superconductivity (TSC) phase, a Majorana zero mode is trapped at the end of a magnetic vortex line. We further show that the surface TSC phase only exists up to a certain bulk pairing gap, and there is a normal-topological phase transition driven by the temperature, which has not been discussed before. These results pave an effective way to realize the TSC and Majorana fermions in a large class of superconductors.

Electronic structure and relaxation dynamics in a superconducting topological material

Topological superconductors host new states of quantum matter which show a pairing gap in the bulk and gapless surface states providing a platform to realize Majorana fermions. Recently, alkaline-earth metal Sr intercalated Bi2Se3 has been reported to show superconductivity with a Tc ~ 3 K and a large shielding fraction. Here we report systematic normal state electronic structure studies of Sr0.06Bi2Se3 (Tc ~ 2.5 K) by performing photoemission spectroscopy. Using angle-resolved photoemission spectroscopy (ARPES), we observe a quantum well confined two-dimensional (2D) state coexisting with a topological surface state in Sr0.06Bi2Se3. Furthermore, our time-resolved ARPES reveals the relaxation dynamics showing different decay mechanism between the excited topological surface states and the two-dimensional states. Our experimental observation is understood by considering the intra-band scattering for topological surface states and an additional electron phonon scattering for the 2D states, which is responsible for the superconductivity. Our first-principles calculations agree with the more effective scattering and a shorter lifetime of the 2D states. Our results will be helpful in understanding low temperature superconducting states of these topological materials.

Effect of Impurities on the Josephson Current through Helical Metals: Exploiting a Neutrino Paradigm

In this Letter we study the effect of time-reversal symmetric impurities on the Josephson supercurrent through two-dimensional helical metals such as on a topological insulator surface state. We show that, contrary to the usual superconducting-normal metal-superconducting junctions, the suppression of the supercurrent in the superconducting-helical metal-superconducting junction is mainly due to fluctuations of impurities in the junctions. Our results, which are a condensed matter realization of a part of the Mikheyev-Smirnov-Wolfenstein effect for neutrinos, show that the relationship between normal state conductance and the critical current of Josephson junctions is significantly modified for Josephson junctions on the surface of topological insulators. We also study the temperature dependence of the supercurrent and present a two fluid model which can explain some of the recent experimental results in Josephson junctions on the edge of topological insulators.

Evolution of Density of States and a Spin-Resolved Checkerboard-Type Pattern Associated with the Majorana Bound State

In terms of the Bogoliubov-de Gennes approach, we investigate the Majorana bound state (MBS) in a vortex of proximity-induced superconductivity on the surface of a topological insulator. Mapping out the local density of states (LDOS) of quasiparticle excitations as a function of energy and distance from the vortex center, it is found that the spectral distribution evolves from a V shape to a Y shape with the emergence of a MBS upon variation of the chemical potential, consistent with the STM/STS measurement in a very recent experiment [Xu et al., Phys. Rev. Lett. 114, 017001 (2015)] on a Bi(2)Te(3) thin layer on the top of NbSe(2). Moreover, we demonstrate that there is a checkerboard-type pattern in the relative LDOS between the spin-up and -down channels, where the quantum mechanical wave function of the MBS manifests itself clearly as a single quantum state. Therefore, a spin-resolved STM/STS technique is expected to be able to provide phase-sensitive evidence for a MBS in the vortex core of a topological superconductor.

Detecting Band Inversions by Measuring the Environment: Fingerprints of Electronic Band Topology in Bulk Phonon Linewidths

The interplay between topological phases of matter and dissipative baths constitutes an emergent research topic with links to condensed matter, photonic crystals, cold atomic gases, and quantum information. While recent studies suggest that dissipative baths can induce topological phases in intrinsically trivial quantum materials, the backaction of topological invariants on dissipative baths is overlooked. By exploring this backaction for a centrosymmetric Dirac insulator coupled to phonons, we show that the linewidths of bulk optical phonons can reveal electronic band inversions. This result is the first known example where topological phases of an open quantum system may be detected by measuring the bulk properties of the surrounding environment.

Topologically protected surface states in a centrosymmetric superconductor β-PdBi2

The topological aspects of electrons in solids can emerge in real materials, as represented by topological insulators. In theory, they show a variety of new magneto-electric phenomena, and especially the ones hosting superconductivity are strongly desired as candidates for topological superconductors. While efforts have been made to develop possible topological superconductors by introducing carriers into topological insulators, those exhibiting indisputable superconductivity free from inhomogeneity are very few. Here we report on the observation of topologically protected surface states in a centrosymmetric layered superconductor, β-PdBi₂, by utilizing spin- and angle-resolved photoemission spectroscopy. Besides the bulk bands, several surface bands are clearly observed with symmetrically allowed in-plane spin polarizations, some of which crossing the Fermi level. These surface states are precisely evaluated to be topological, based on the Z₂ invariant analysis in analogy to three-dimensional strong topological insulators. β-PdBi₂ may offer a solid stage to investigate the topological aspect in the superconducting condensate.

Materials possessing topologically non-trivial electronic surface states are predicted to host exotic Majorana fermion excitations in the superconducting state. Here, the authors demonstrate the existence of topologically-protected surface states in the centrosymmetric layered superconductor β-PdBi2.

Dirac surface states and nature of superconductivity in Noncentrosymmetric BiPd

In non-magnetic bulk materials, inversion symmetry protects the spin degeneracy. If the bulk crystal structure lacks a centre of inversion, however, spin-orbit interactions lift the spin degeneracy, leading to a Rashba metal whose Fermi surfaces exhibit an intricate spin texture. In superconducting Rashba metals a pairing wavefunction constructed from these complex spin structures will generally contain both singlet and triplet character. Here we examine the possible triplet components of the order parameter in noncentrosymmetric BiPd, combining for the first time in a noncentrosymmetric superconductor macroscopic characterization, atomic-scale ultra-low-temperature scanning tunnelling spectroscopy, and relativistic first-principles calculations. While the superconducting state of BiPd appears topologically trivial, consistent with Bardeen-Cooper-Schrieffer theory with an order parameter governed by a single isotropic s-wave gap, we show that the material exhibits Dirac-cone surface states with a helical spin polarization.

Induced topological phases at the boundary of 3D topological superconductors

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 realization of localized Majorana zero modes.

New class of topological superconductors protected by magnetic group symmetries

We study a new type of three-dimensional topological superconductor that exhibits Majorana zero modes (MZM) protected by a magnetic group symmetry, a combined antiunitary symmetry composed of a mirror reflection and time reversal. This new symmetry enhances the noninteracting topological classification of a superconducting vortex from Z2 to Z, indicating that multiple MZMs can coexist at the end of one magnetic vortex of unit flux. Especially, we show that a vortex binding two MZMs can be realized on the (001) surface of a topological crystalline insulator SnTe with proximity induced BCS Cooper pairing, or in bulk superconductor InxSn1-xTe.