Helical Hinge Majorana Modes in Iron-Based Superconductors

Higher-order topological phases in crystalline and non-crystalline systems: a review

In recent years, higher-order topological phases have attracted great interest in various fields of physics. These phases have protected boundary states at lower-dimensional boundaries than the conventional first-order topological phases due to the higher-order bulk-boundary correspondence. In this review, we summarize current research progress on higher-order topological phases in both crystalline and non-crystalline systems. We firstly introduce prototypical models of higher-order topological phases in crystals and their topological characterizations. We then discuss effects of quenched disorder on higher-order topology and demonstrate disorder-induced higher-order topological insulators. We also review the theoretical studies on higher-order topological insulators in amorphous systems without any crystalline symmetry and higher-order topological phases in non-periodic lattices including quasicrystals, hyperbolic lattices, and fractals, which have no crystalline counterparts. We conclude the review by a summary of experimental realizations of higher-order topological phases and discussions on potential directions for future study.

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.

Topological Superconductors from a Materials Perspective

Topological superconductors (TSCs) have garnered significant research and industry attention in the past two decades. By hosting Majorana bound states which can be used as qubits that are robust against local perturbations, TSCs offer a promising platform toward (nonuniversal) topological quantum computation. However, there has been a scarcity of TSC candidates, and the experimental signatures that identify a TSC are often elusive. In this Perspective, after a short review of the TSC basics and theories, we provide an overview of the TSC materials candidates, including natural compounds and synthetic material systems. We further introduce various experimental techniques to probe TSCs, focusing on how a system is identified as a TSC candidate and why a conclusive answer is often challenging to draw. We conclude by calling for new experimental signatures and stronger computational support to accelerate the search for new TSC candidates.

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.

Majorana corner states in an attractive quantum spin Hall insulator with opposite in-plane Zeeman energy at two sublattice sites

Higher-order topological superconductors and superfluids (SFs) host lower-dimensional Majorana corner and hinge states since novel topology exhibitions on boundaries. While such topological nontrivial phases have been explored extensively, more possible schemes are necessary for engineering Majorana states. In this paper we propose Majorana corner states could be realized in a two-dimensional attractive quantum spin-Hall insulator with opposite in-plane Zeeman energy at two sublattice sites. The appropriate Zeeman field leads to the opposite Dirac mass for adjacent edges of a square sample, and naturally induce Majorana corner states. This topological phase can be characterized by Majorana edge polarizations, and it is robust against perturbations on random potentials and random phase fluctuations as long as the edge gap remains open. Our work provides a new possibility to realize a second-order topological SF in two dimensions and engineer Majorana corner states.

Emergent Multifunctional Magnetic Proximity in van der Waals Layered Heterostructures

Proximity effect, which is the coupling between distinct order parameters across interfaces of heterostructures, has attracted immense interest owing to the customizable multifunctionalities of diverse 3D materials. This facilitates various physical phenomena, such as spin order, charge transfer, spin torque, spin density wave, spin current, skyrmions, and Majorana fermions. These exotic physics play important roles for future spintronic applications. Nevertheless, several fundamental challenges remain for effective applications: unavoidable disorder and lattice mismatch limits in the growth process, short characteristic length of proximity, magnetic fluctuation in ultrathin films, and relatively weak spin-orbit coupling (SOC). Meanwhile, the extensive library of atomically thin, 2D van der Waals (vdW) layered materials, with unique characteristics such as strong SOC, magnetic anisotropy, and ultraclean surfaces, offers many opportunities to tailor versatile and more effective functionalities through proximity effects. Here, this paper focuses on magnetic proximity, i.e., proximitized magnetism and reviews the engineering of magnetism-related functionalities in 2D vdW layered heterostructures for next-generation electronic and spintronic devices. The essential factors of magnetism and interfacial engineering induced by magnetic layers are studied. The current limitations and future challenges associated with magnetic proximity-related physics phenomena in 2D heterostructures are further discussed.

Second-order topological insulator in periodically driven optical lattices

The higher-order topological insulator (HOTI) is a new type of topological system which has special bulk-edge correspondence compared with conventional topological insulators. In this work, we propose a scheme to realize Floquet HOTI with ultracold atoms in optical lattices. With the combination of periodically spin-dependent driving of the superlattices and a long-range coupling term, a Floquet second-order topological insulator with four zero-energy corner states emerges, whose Wannier bands are gapless and exhibit interesting bulk topology. Furthermore, the nearest-neighbor anisotropic coupling term also induced other intriguing topological phenomena, e.g. non-topologically protected corner states and topological semimetal for two different types of lattice structures respectively. Our scheme may give insight into the construction of different types of higher-order topological insulators in synthetic systems. It also provides an experimentally feasible platform to research the relations between different types of topological states and may have a wide range of applications in future.

High-T c superconductor Fe(Se,Te) monolayer: an intrinsic, scalable and electrically tunable Majorana platform

Iron-based superconductors have been identified as a novel platform for realizing Majorana zero modes (MZMs) without heterostructures, due to their intrinsic topological properties and high-T c superconductivity. In the two-dimensional limit, the FeTe1-x Se x monolayer, a topological band inversion has recently been experimentally observed. Here, we propose to create MZMs by applying an in-plane magnetic field to the FeTe1-x Se x monolayer and tuning the local chemical potential via electric gating. Owing to the anisotropic magnetic couplings on edges, an in-plane magnetic field drives the system into an intrinsic high-order topological superconductor phase with Majorana corner modes. Furthermore, MZMs can occur at the domain wall of chemical potentials at either one edge or certain type of tri-junction in the two-dimensional bulk. Our study not only reveals the FeTe1-x Se x monolayer as a promising Majorana platform with scalability and electrical tunability and within reach of contemporary experimental capability, but also provides a general principle to search for realistic realization of high-order topological superconductivity.

Phase-Manipulation-Induced Majorana Mode and Braiding Realization in Iron-Based Superconductor Fe(Te,Se)

A recent experiment reported the evidence of dispersing one-dimensional Majorana mode trapped by the crystalline domain walls in FeSe_{0.45}Te_{0.55}. Here, we perform the first-principles calculations to show that iron atoms in the domain wall spontaneously form the ferromagnetic order in line with orientation of the wall. The ferromagnetism can impose a π phase difference between the domain-wall-separated surface superconducting regimes under the appropriate width and magnetization of the wall. Accordingly, the topological surface superconducting state of FeSe_{0.45}Te_{0.55} can give rise to one-dimensional Majorana modes trapped by the wall. More interestingly, we further propose a surface junction in the form of FeSe_{0.45}Te_{0.55}-ferromagnet-FeSe_{0.45}Te_{0.55}, which can be adopted to create and fuse the Majorana zero modes through controlling the width or magnetization of the interior ferromagnetic barrier. The braiding and readout of Majorana zero modes can be realized by the designed device. Such surface junction has the potential application in the superconducting topological quantum computation.

Iron pnictides and chalcogenides: a new paradigm for superconductivity

Superconductivity is a remarkably widespread phenomenon that is observed in most metals cooled to very low temperatures. The ubiquity of such conventional superconductors, and the wide range of associated critical temperatures, is readily understood in terms of the well-known Bardeen-Cooper-Schrieffer theory. Occasionally, however, unconventional superconductors are found, such as the iron-based materials, which extend and defy this understanding in unexpected ways. In the case of the iron-based superconductors, this includes the different ways in which the presence of multiple atomic orbitals can manifest in unconventional superconductivity, giving rise to a rich landscape of gap structures that share the same dominant pairing mechanism. In addition, these materials have also led to insights into the unusual metallic state governed by the Hund's interaction, the control and mechanisms of electronic nematicity, the impact of magnetic fluctuations and quantum criticality, and the importance of topology in correlated states. Over the fourteen years since their discovery, iron-based superconductors have proven to be a testing ground for the development of novel experimental tools and theoretical approaches, both of which have extensively influenced the wider field of quantum materials.

Higher-Order Weyl-Exceptional-Ring Semimetals

For first-order topological semimetals, non-Hermitian perturbations can drive the Weyl nodes into Weyl exceptional rings having multiple topological structures and no Hermitian counterparts. Recently, it was discovered that higher-order Weyl semimetals, as a novel class of higher-order topological phases, can uniquely exhibit coexisting surface and hinge Fermi arcs. However, non-Hermitian higher-order topological semimetals have not yet been explored. Here, we identify a new type of topological semimetal, i.e., a higher-order topological semimetal with Weyl exceptional rings. In such a semimetal, these rings are characterized by both a spectral winding number and a Chern number. Moreover, the higher-order Weyl-exceptional-ring semimetal supports both surface and hinge Fermi-arc states, which are bounded by the projection of the Weyl exceptional rings onto the surface and hinge, respectively. Noticeably, the dissipative terms can cause the coupling of two exceptional rings with opposite topological charges, so as to induce topological phase transitions. Our studies open new avenues for exploring novel higher-order topological semimetals in non-Hermitian systems.

Strong Correlation Between Superconductivity and Ferromagnetism in an Fe-Chalcogenide Superconductor

The interplay among topology, superconductivity, and magnetism promises to bring a plethora of exotic and unintuitive behaviors in emergent quantum materials. The family of Fe-chalcogenide superconductors FeTe_xSe_1-x are directly relevant in this context due to their intrinsic topological band structure, high-temperature superconductivity, and unconventional pairing symmetry. Despite enormous promise and expectation, the local magnetic properties of FeTe_xSe_1-x remain largely unexplored, which prevents a comprehensive understanding of their underlying material properties. Exploiting nitrogen vacancy (NV) centers in diamond, here we report nanoscale quantum sensing and imaging of magnetic flux generated by exfoliated FeTe_xSe_1-x flakes, demonstrating strong correlation between superconductivity and ferromagnetism in FeTe_xSe_1-x. The coexistence of superconductivity and ferromagnetism in an established topological superconductor opens up new opportunities for exploring exotic spin and charge transport phenomena in quantum materials. The demonstrated coupling between NV centers and FeTe_xSe_1-x may also find applications in developing hybrid architectures for next-generation, solid-state-based quantum information technologies.

Simulation of higher-order topological phases and related topological phase transitions in a superconducting qubit

Higher-order topological phases give rise to new bulk and boundary physics, as well as new classes of topological phase transitions. While the realization of higher-order topological phases has been confirmed in many platforms by detecting the existence of gapless boundary modes, a direct determination of the higher-order topology and related topological phase transitions through the bulk in experiments has still been lacking. To bridge the gap, in this work we carry out the simulation of a two-dimensional second-order topological phase in a superconducting qubit. Owing to the great flexibility and controllability of the quantum simulator, we observe the realization of higher-order topology directly through the measurement of the pseudo-spin texture in momentum space of the bulk for the first time, in sharp contrast to previous experiments based on the detection of gapless boundary modes in real space. Also through the measurement of the evolution of pseudo-spin texture with parameters, we further observe novel topological phase transitions from the second-order topological phase to the trivial phase, as well as to the first-order topological phase with nonzero Chern number. Our work sheds new light on the study of higher-order topological phases and topological phase transitions.

Floquet Second-Order Topological Phases in Momentum Space

Higher-order topological phases (HOTPs) are characterized by symmetry-protected bound states at the corners or hinges of the system. In this work, we reveal a momentum-space counterpart of HOTPs in time-periodic driven systems, which are demonstrated in a two-dimensional extension of the quantum double-kicked rotor. The found Floquet HOTPs are protected by chiral symmetry and characterized by a pair of topological invariants, which could take arbitrarily large integer values with the increase of kicking strengths. These topological numbers are shown to be measurable from the chiral dynamics of wave packets. Under open boundary conditions, multiple quartets Floquet corner modes with zero and π quasienergies emerge in the system and coexist with delocalized bulk states at the same quasienergies, forming second-order Floquet topological bound states in the continuum. The number of these corner modes is further counted by the bulk topological invariants according to the relation of bulk-corner correspondence. Our findings thus extend the study of HOTPs to momentum-space lattices and further uncover the richness of HOTPs and corner-localized bound states in continuum in Floquet systems.

Intrinsic Time-Reversal-Invariant Topological Superconductivity in Thin Films of Iron-Based Superconductors

We establish quasi-two-dimensional thin films of iron-based superconductors (FeSCs) as a new high-temperature platform for hosting intrinsic time-reversal-invariant helical topological superconductivity (TSC). Based on the combination of Dirac surface state and bulk extended s-wave pairing, our theory should be directly applicable to a large class of experimentally established FeSCs, opening a new TSC paradigm. In particular, an applied electric field serves as a "topological switch" for helical Majorana edge modes in FeSC thin films, allowing for an experimentally feasible design of gate-controlled helical Majorana circuits. Applying an in-plane magnetic field drives the helical TSC phase into a higher-order TSC carrying corner-localized Majorana zero modes. Our proposal should enable the experimental realization of helical Majorana fermions.

Observation of robust edge superconductivity in Fe(Se,Te) under strong magnetic perturbation

The iron-chalcogenide high temperature superconductor Fe(Se,Te) (FST) has been reported to exhibit complex magnetic ordering and nontrivial band topology which may lead to novel superconducting phenomena. However, the recent studies have so far been largely concentrated on its band and spin structures while its mesoscopic electronic and magnetic response, crucial for future device applications, has not been explored experimentally. Here, we used scanning superconducting quantum interference device microscopy for its sensitivity to both local diamagnetic susceptibility and current distribution in order to image the superfluid density and supercurrent in FST. We found that in FST with 10% interstitial Fe, whose magnetic structure was heavily disrupted, bulk superconductivity was significantly suppressed whereas edge still preserved strong superconducting diamagnetism. The edge dominantly carried supercurrent despite of a very long magnetic penetration depth. The temperature dependences of the superfluid density and supercurrent distribution were distinctively different between the edge and the bulk. Our Heisenberg modeling showed that magnetic dopants stabilize anti-ferromagnetic spin correlation along the edge, which may contribute towards its robust superconductivity. Our observations hold implication for FST as potential platforms for topological quantum computation and superconducting spintronics.

Inversion-Protected Higher-Order Topological Superconductivity in Monolayer WTe_{2}

Monolayer WTe_{2}, a centrosymmetric transition metal dichacogenide, has recently been established as a quantum spin Hall insulator and found superconducting upon gating. Here we study the pairing symmetry and topological nature of superconducting WTe_{2} with a microscopic model at mean-field level. Surprisingly, we find that the spin-triplet phases in our phase diagram all host Majorana modes localized on two opposite corners. Even when the conventional pairing is favored, we find that an intermediate in-plane magnetic field exceeding the Pauli limit stabilizes an unconventional equal-spin pairing aligning with the field, which also hosts Majorana corner modes. Motivated by our findings, we obtain a recipe for two-dimensional superconductors featuring "higher-order topology" from the boundary perspective. Generally, a superconducting inversion-symmetric quantum spin Hall material whose normal-state Fermi surface is away from high-symmetry points, such as gated monolayer WTe_{2}, hosts Majorana corner modes if the superconductivity is parity-odd. We further point out that this higher-order phase is an inversion-protected topological crystalline superconductor and study the bulk-boundary correspondence. Finally, we discuss possible experiments for probing the Majorana corner modes. Our findings suggest superconducting monolayer WTe_{2} is a playground for higher-order topological superconductivity and possibly the first material realization for inversion-protected Majorana corner modes without utilizing proximity effect.

On the Remarkable Superconductivity of FeSe and Its Close Cousins

Emergent electronic phenomena in iron-based superconductors have been at the forefront of condensed matter physics for more than a decade. Much has been learned about the origins and intertwined roles of ordered phases, including nematicity, magnetism, and superconductivity, in this fascinating class of materials. In recent years, focus has been centered on the peculiar and highly unusual properties of FeSe and its close cousins. This family of materials has attracted considerable attention due to the discovery of unexpected superconducting gap structures, a wide range of superconducting critical temperatures, and evidence for nontrivial band topology, including associated spin-helical surface states and vortex-induced Majorana bound states. Here, we review superconductivity in iron chalcogenide superconductors, including bulk FeSe, doped bulk FeSe, FeTe1−xSex, intercalated FeSe materials, and monolayer FeSe and FeTe1−xSex on SrTiO3. We focus on the superconducting properties, including a survey of the relevant experimental studies, and a discussion of the different proposed theoretical pairing scenarios. In the last part of the paper, we review the growing recent evidence for nontrivial topological effects in FeSe-related materials, focusing again on interesting implications for superconductivity.

Leakage of Majorana bound states in an inhomogeneous topological nanowire

We present an exact solution of the continuum Bogolyubov-de-Gennes Hamiltonian for Majorana bound states (MBSs) generated in a superconductor-semiconductor hybrid topological nanowire. The full energy spectra that include the band states and in-gap states could be obtained. We show that for relatively short wire length, the zero energy mode could be induced even in the topological trivial regime, which also indicates oscillatory dependence on the chemical potential. With the increase of the Zeeman field, the MBSs are almost fully spin-polarized and do not localize at the wire ends gradually. We also extend our discussion to the property of Majorana modes in an inhomogeneous nanowire, in which a local gate voltage is applied to one end of the nanowire. It is found that the local potential barrier or well could modulate the Majorana energy splitting periodically. The leakage of MBSs to the potential region is exponentially suppressed for the barrier case. A potential well could induce near-zero-energy bound states and these states merge with MBSs, leading to the delocalization of MBSs. In the potential well region, both the spin-up and spin-down components of the trivial states account for a significant proportion, which can be detected experimentally.

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.

Block-spiral magnetism: An exotic type of frustrated order

Competing interactions in quantum materials induce exotic states of matter such as frustrated magnets, an extensive field of research from both the theoretical and experimental perspectives. Here, we show that competing energy scales present in the low-dimensional orbital-selective Mott phase (OSMP) induce an exotic magnetic order, never reported before. Earlier neutron-scattering experiments on iron-based 123 ladder materials, where OSMP is relevant, already confirmed our previous theoretical prediction of block magnetism (magnetic order of the form [Formula: see text]). Now we argue that another phase can be stabilized in multiorbital Hubbard models, the block-spiral state. In this state, the magnetic islands form a spiral propagating through the chain but with the blocks maintaining their identity, namely rigidly rotating. The block-spiral state is stabilized without any apparent frustration, the common avenue to generate spiral arrangements in multiferroics. By examining the behavior of the electronic degrees of freedom, parity-breaking quasiparticles are revealed. Finally, a simple phenomenological model that accurately captures the macroscopic spin spiral arrangement is also introduced, and fingerprints for the neutron-scattering experimental detection are provided.

First- and Second-Order Topological Superconductivity and Temperature-Driven Topological Phase Transitions in the Extended Hubbard Model with Spin-Orbit Coupling

The combination of spin-orbit coupling with interactions results in many exotic phases of matter. In this Letter, we investigate the superconducting pairing instability of the two-dimensional extended Hubbard model with both Rashba and Dresselhaus spin-orbit coupling within the mean-field level at both zero and finite temperature. We find that both first- and second-order time-reversal symmetry breaking topological gapped phases can be achieved under appropriate parameters and temperature regimes due to the presence of a favored even-parity s+id-wave pairing even in the absence of an external magnetic field or intrinsic magnetism. This results in two branches of chiral Majorana edge states on each edge or a single zero-energy Majorana corner state at each corner of the sample. Interestingly, we also find that not only does tuning the doping level lead to a direct topological phase transition between these two distinct topological gapped phases, but also using the temperature as a highly controllable and reversible tuning knob leads to different direct temperature-driven topological phase transitions between gapped and gapless topological superconducting phases. Our findings suggest new possibilities in interacting spin-orbit coupled systems by unifying both first- and higher-order topological superconductors in a simple but realistic microscopic model.

Evidence for dispersing 1D Majorana channels in an iron-based superconductor

The possible realization of Majorana fermions as quasiparticle excitations in condensed-matter physics has created much excitement. Most studies have focused on Majorana bound states; however, propagating Majorana states with linear dispersion have also been predicted. Here, we report scanning tunneling spectroscopic measurements of crystalline domain walls (DWs) in FeSe0.45Te0.55. We located DWs across which the lattice structure shifts by half a unit cell. These DWs have a finite, flat density of states inside the superconducting gap, which is a hallmark of linearly dispersing modes in one dimension. This signature is absent in DWs in the related superconductor, FeSe, which is not in the topological phase. Our combined data are consistent with the observation of dispersing Majorana states at a π-phase shift DW in a proximitized topological material.

Higher-Order Topological Odd-Parity Superconductors

The topological property of a gapped odd-parity superconductor is jointly determined by its pairing nodes and Fermi surfaces in the normal state. We reveal that the contractibility of Fermi surfaces without crossing any time-reversal invariant momentum and the presence of nontrivial Berry phase on Fermi surfaces are two key conditions for the realization of higher-order topological odd-parity superconductors. When the normal state is a normal metal, we reveal the necessity of removable Dirac pairing nodes and provide a general and simple principle to realize higher-order topological odd-parity superconductors. Our findings can be applied to design new platforms of higher-order topological superconductors, as well as higher-order topological insulators owing to their direct analogy in Hamiltonian description.

Higher-Order Topology and Nodal Topological Superconductivity in Fe(Se,Te) Heterostructures

We show, theoretically, that a heterostructure of monolayer FeTe_{1-x}Se_{x}-a superconducting quantum spin Hall material-with a monolayer of FeTe-a bicollinear antiferromagnet-realizes a higher order topological superconductor phase characterized by emergent Majorana zero modes pinned to the sample corners. We provide a minimal effective model for this system, analyze the origin of higher order topology, and fully characterize the topological phase diagram. Despite the conventional s-wave pairing, we find a rather surprising emergence of a novel topological nodal superconductor in the phase diagram. Featured by edge-dependent Majorana flat bands, the topological nodal phase is protected by an antiferromagnetic chiral symmetry. We also discuss the experimental feasibility, the estimation of realistic model parameters, and the robustness of the Majorana corner modes against magnetic and potential disorder. Our work provides a new experimentally feasible high-temperature platform for both higher order topology and non-Abelian Majorana physics.

Lattice-Symmetry-Assisted Second-Order Topological Superconductors and Majorana Patterns

We propose a realization of the lattice-symmetry-assisted second-order topological superconductors with corner Majorana zero modes (MZM) based on two-dimensional topological insulators (2DTI). The lattice symmetry can naturally lead to the anisotropic coupling of edge states along different directions to the in-plane magnetic field and conventional s-wave pairings, thus leading to a single MZM located at the corners for various lattice patterns. In particular, we focus on the 2DTI with D_{3d} lattice symmetry and found different types of gap opening for the edge states along the armchair and zigzag edges in a broad range of parameters. As a consequence, a single MZM exists at the corner between the zigzag and armchair edges, and is robust against weakly broken lattice symmetry. We propose to realize such corner MZMs in a variety of polygon patterns, such as triangles and quadrilaterals. We further show their potentials in building the Majorana network through constructing the Majorana Y junction under an in-plane magnetic field.

Evidence for Helical Hinge Zero Modes in an Fe-Based Superconductor

Combining topology and superconductivity provides a powerful tool for investigating fundamental physics as well as a route to fault-tolerant quantum computing. There is mounting evidence that the Fe-based superconductor FeTe_0.55Se_0.45 (FTS) may also be topologically nontrivial. Should the superconducting order be s^±, then FTS could be a higher order topological superconductor with helical hinge zero modes (HHZMs). To test the presence of these modes, we have fabricated normal-metal/superconductor junctions on different surfaces via 2D atomic crystal heterostructures. As expected, junctions in contact with the hinge reveal a sharp zero bias anomaly that is absent when tunneling purely into the c-axis. Additionally, the shape and suppression with temperature are consistent with highly coherent modes along the hinge and are incongruous with other origins of zero bias anomalies. Additional measurements with soft-point contacts in bulk samples with various Fe interstitial contents demonstrate the intrinsic nature of the observed mode. Thus, we provide evidence that FTS is indeed a higher order topological superconductor.

Second-Order Topological Superconductors with Mixed Pairing

We show that a two-dimensional semiconductor with Rashba spin-orbit coupling could be driven into the second-order topological superconducting phase when a mixed-pairing state is introduced. The superconducting order we consider involves only even-parity components and meanwhile breaks time-reversal symmetry. As a result, each corner of a square-shaped Rashba semiconductor would host one single Majorana zero mode in the second-order nontrivial phase. Starting from edge physics, we are able to determine the phase boundaries accurately. A simple criterion for the second-order phase is further established, which concerns the relative position between Fermi surfaces and nodal points of the superconducting order parameter. In the end, we propose two setups that may bring this mixed-pairing state into the Rashba semiconductor, followed by a brief discussion on the experimental feasibility of the two platforms.

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.