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Superconducting topological systems formed by a strong 3D topological insulator (TI) in proximity to a conventional s-wave superconductor (SC) have been intensely studied, as they may host Majorana zero modes. However, there are limited experimental realizations of TI-SC systems in which robust superconducting pairing is induced on the surface states of the TI and a topological superconducting state is established. Here, we fabricate a TI-SC system by depositing, via a focused ion beam, tungsten (W) nanoscale clusters on the surface of TI Bi0.91Sb0.09. We find that the resulting heterostructure supports phase-slip lines (PSLs) that act as effective Josephson junctions (JJs). We probe the response of the system to microwave radiation. We find that for some ac frequencies, and powers, the resulting Shapiro steps' structure of the voltage–current characteristic exhibits a missing first step and an unexpectedly wide second Shapiro step. The theoretical analysis of the measurements shows that the unusual Shapiro response arises from the interplay between a static JJ and a dynamic one and allows us to identify the conditions under which the missing first step can be attributed to the topological nature of the JJs formed by the PSLs. Our results suggest an approach to induce superconductivity in a TI, a route to realizing highly transparent topological JJs, and show how the response of superconducting systems to microwave radiation can be used to infer the dynamics of PSLs. Highly transparent topological junctions are promising candidates to realize vector field sensors with very high sensitivity. In addition, due to the nontrivial Berry phase of the TI's surface states such junctions can be in a topological state which is ideal to create topologically protected qubits. [ABSTRACT FROM AUTHOR]

High‐entropy ceramics (HECs) are solid solutions of inorganic compounds with one or more Wyckoff sites shared by equal or near‐equal atomic ratios of multi‐principal elements. Material design and property tailoring possibilities emerge from this new class of materials. Herein, the discovery of superconductivity 2.35 K and topological properties in the (Ti0.2Zr0.2Nb0.2Hf0.2Ta0.2)C high‐entropy carbide ceramic (HECC) is reported, which is not observed before in any of the investigated HECC. Density functional theory calculations show that six type‐II Dirac points exist in (Ti0.2Zr0.2Nb0.2Hf0.2Ta0.2)C, which mainly contributed from the t2g orbitals of transition metals and the p orbitals of C. Due to the stability of the structure, robust superconductivity (SC) under pressure in this HEC superconductor is also observed. This study expands the physical properties of HECs, which may become a new material platform for SC research, especially for studying the coupling between SC and topological physics. [ABSTRACT FROM AUTHOR]

We present here a theory of Majorana excitons, photo-excited conduction electron-valence band hole pairs, interacting with Majorana Fermions in a Kitaev chain of semiconductor quantum dots embedded in a nanowire. Using analytical tools and exact diagonalization methods, we identify the presence of Majorana zero modes in the nanowire absorption spectra. [ABSTRACT FROM AUTHOR]

Majorana zero modes (MZMs) are exotic excitations (in condensed matter systems) of fundamental scientific interest and hold great promise as the basis of fault-tolerant topological quantum computation. Though MZMs have been predicted in many platforms, their existence is still under debate. In this paper, we review the recent progress in engineering and detecting MZMs in semiconductor-superconductor heterostructures. We also briefly review the protocols for implementing topological quantum computation by hybrid semiconductor-superconductor nanowires. [ABSTRACT FROM AUTHOR]

With a series of recent breakthroughs, iron-based superconductors (FeSC) with a topological Dirac surface state are becoming a promising material platform for hosting Majorana zero modes, which we refer to as the iron-Majorana platform. This platform uniquely combines high-Tc superconductivity, a topological band structure, and electron correlations into a single material, successfully avoiding the difficulties of achieving intrinsic p-wave topological superconductors and superconductor/topological insulator heterojunction systems. The most important advantages of the iron-Majorana platform are its wide topological region and large quasiparticle gap, which provide strong topological protection for pure Majorana zero modes (MZMs). When the superconductor/topological insulator heterojunction systems, e.g., InAs/Al nanowire, have the controversies of being trivial Majorana-like states, the iron-Majorana platform, which possesses well-understood physics and clear experimental evidence of vortex MZMs, is more likely to be a true MZM. However, unlike the nanowire Majorana systems with clear theoretical proposals for braiding schemes, the iron-Majorana system has no concrete method for exchanging the vortex MZMs or constructing a topological qubit. In this article, we propose a roadmap of the future efforts required for more physical exploration and achieving the non-Abelian exchange statistics of MZMs based on the iron-Majorana platform. [ABSTRACT FROM AUTHOR]

Under certain conditions, a fermion in a superconductor can separate in space into two parts known as Majorana zero modes, which are immune to decoherence from local noise sources and are attractive building blocks for quantum computers. Promising experimental progress has been made to demonstrate Majorana zero modes in materials with strong spin-orbit coupling proximity coupled to superconductors. Here we report signatures of Majorana zero modes in a material platform utilizing the surface states of gold. Using scanning tunneling microscope to probe EuS islands grown on top of gold nanowires, we observe two well-separated zero-bias tunneling conductance peaks aligned along the direction of the applied magnetic field, as expected for a pair of Majorana zero modes. This platform has the advantage of having a robust energy scale and the possibility of realizing complex designs using lithographic methods.

Ca1−xLaxFeAs2 (CLFA112) belongs to a new family of Fe-based superconductors (FeSCs) and has a unique crystal structure featuring an arsenic zigzag chain layer, which has been proposed to be a possible two-dimensional topological insulator. This suggests that CLFA112 is a potential topological superconductor—a platform to realize Majorana fermions. Up to now, even a clear superconducting (SC) gap in CLFA112 has never been observed, and the SC properties of CLFA112 remain largely elusive. In this letter, we report the results of an atomic-scale investigation of the electronic structure of CLFA112 crystals using low-temperature scanning tunneling microscopy (STM). We revealed four different types of surfaces exhibiting distinct electronic properties, with all surfaces displaying dominating 2 × 1 surface reconstructions. On a Ca/La layer on top of an FeAs layer, a clear SC gap of ~12 mV was observed only at the crevices (vacancies) where the FeAs layer can be directly accessed. Remarkably, the FeAs termination layer displayed a dispersing nematic modulation both in real and q space. We also present peculiar zero-bias conductance peaks for the very As chain layer that is believed to exhibit a topological edge state as well as the influence of La dopants on the As chain layer. [ABSTRACT FROM AUTHOR]

The search for anyons is a field of immense interest owing to its potential application in the field of quantum information. Quantum critical Kondo impurities constitute one possible platform for their realization and topological Kondo effect (TKE) by virtue of remaining critical in the presence of perturbations, seems to be especially promising in this regard. In this paper we discuss practical steps for a realization of TKE with a relatively high Kondo temperature $T_{\mathrm{K}}$ . Its central feature is the so-called Majorana-Cooper box (MCB) and we argue that a particular type of iron-based topological superconductor is especially suitable for realization of TKE. Once MCB is available one needs to connect it to external metallic leads to produce TKE. A relatively high value of the Kondo temperature $T_{\mathrm{K}}$ is then aided by a large superconducting gap of the iron-based superconductor. We give estimates for $T_{\mathrm{K}}$ , for the cases of both isotropic and anisotropic exchange couplings of MCB with the leads.

Electron teleportation via two separate Majorana bound states(MBSs) is a manifestation of the non-locality of MBSs. A superconductor may host multiple separate or partial overlapping MBSs, and it is difficult to distinguish them. Here, we have studied the electron teleportation between two quantum dots via multiple MBSs in a superconductor island, two of which couple with the quantum dots. We find that in the absence of couplings between MBSs, the electron transferring occurs as the energy level of each quantum dot is aligned with the chemical potential of the superconductor. More importantly, the parity state of the MBSs could be retained or changed after the transport, which depends on the initial state of the MBSs. In the presence of MBSs couplings, this system can yield elastic or inelastic tunnelings, according to which pair of MBSs are coupled together. In addition, different coupling types of the MBSs would give rise to distinct behaviors of electron cotunnelings between the two quantum dots. Our results reveal that the initial state and couplings of multiple MBSs play important roles in the Majorana-aid electron teleportation. These findings could deepen our understanding of the non-locality of MBSs, and can be utilized to distinguish topological MBSs from non-topological bound states. • Electron teleportation manifests the nonlocality of Majorana bound states. • Initial parity state of Majorana bound states is important in teleportation process. • Electron teleportation can be modified by couplings between Majorana bound states. [ABSTRACT FROM AUTHOR]

We present here a theory of Majorana excitons, photo-excited conduction electron-valence band hole pairs, interacting with Majorana Fermions in a Kitaev chain of semiconductor quantum dots embedded in a nanowire. Using analytical tools and exact diagonalization methods, we identify the presence of Majorana zero modes in the nanowire absorption spectra.

The fragile topological phase is an exotic phase with unstable edge states. Its nontrivial band topology can be removed by coupling to additional topologically trivial bands. Here we reveal that the fragile band topology can be realized in the inversion symmetric odd parity superconducting nodal line semimetal (SC-NLSM) materials with and without the spin orbital coupling. Without the spin orbital coupling, both the s -wave SC-NLSM and the p -wave SC-NLSM host a pair of Majorana zero modes on the system surface. For the spinful case, it has fourfold inverted bands and generates fourfold degenerate Majorana surface/edge states. However, we verify that for all of the systems we considered, they belong to the fragile topological superconducting system based on the Wilson loop method. The vortex bound states are studied numerically and the results depend strongly on the systems. [ABSTRACT FROM AUTHOR]

The momentum space of topological insulators and topological superconductors is equipped with a quantum metric defined from the overlap of neighboring valence band states or quasihole states. We investigate the quantum geometrical properties of these materials within the framework of Dirac models and differential geometry. Their momentum space is found to be always a maximally symmetric space with a constant Ricci scalar, and the vacuum Einstein equation is satisfied in 3D with a finite cosmological constant. For linear Dirac models, several geometrical properties are found to be independent of the band gap, including a peculiar straight line geodesic, constant volume of the curved momentum space, and the exponential decay form of the nonlocal topological marker, indicating the peculiar yet universal quantum geometrical properties of these models., (© 2024 IOP Publishing Ltd. All rights, including for text and data mining, AI training, and similar technologies, are reserved.)

My contribution to this collection of articles in honor of David Lee and John Reppy on their 90th birthdays is a reflection on the remarkable phenomenology of the excitation spectra of superfluid 3 He, in particular the B-phase which was identified by NMR and acoustic spectroscopy as the Balian–Werthamer state shown in 1963 to be the ground state of a spin-triplet, p-wave superconductor within weak-coupling BCS theory. The superfluid phases of 3 He provide paradigms for electronic superconductors with broken space-time symmetries and non-trivial ground-state topology. Indeed, broken spin- and orbital rotation symmetries lead to a rich spectrum of collective modes of the order parameter that can be detected using NMR, acoustic and microwave spectroscopies. The topology of the BW state implies its low-temperature, low-energy transport properties are dominated by gapless Majorana modes confined on boundaries or interfaces. Given the central role the BW state played I discuss the acoustic and electromagnetic signatures of the BW state, the latter being relevant if an electronic analog of superfluid 3 He-B is realized. [ABSTRACT FROM AUTHOR]

We study nonequilibrium transport in various open quantum systems whose systems and leads/baths are made of topological superconductors (TSs), semiconductors, and metals. Using quantum Langevin equations and Green's function method, we derive exact expressions for steady-state electrical, thermal, and spin current at the junctions between a system and leads. We validate these current expressions by comparing them with the results from direct time-evolution simulations. We then show how an electrical current injected in TS wires divides into two parts carried by single electronic excitations and Cooper pairs. We further show ballistic thermal transport in an open TS wire in the topological phase under temperature or voltage bias. The thermal current values grow significantly near the topological phase transition, where thermal conductance displays a sharp quantized peak as predicted earlier. We relate the quantized thermal conductance to the zero-frequency thermoelectric transmission coefficient of the open TS wire. We also observe a large thermoelectric current near the topological transition of the TS wires. We introduce a differential spin conductance which displays a quantized zero-bias peak at zero temperature for a spinful TS wire in the topological phase. The role of superconducting baths in transport is demonstrated by thoroughly examining the features of zero-temperature differential electrical conductance and thermal conductance in open systems with TS baths. Our new thermoelectric and spin transport findings in various two-terminal geometries are beneficial to the present challenges in probing the emergence of Majorana quasi-particles in experiments. [ABSTRACT FROM AUTHOR]

Ca1−xLaxFeAs2 (CLFA112) belongs to a new family of Fe-based superconductors (FeSCs) and has a unique crystal structure featuring an arsenic zigzag chain layer, which has been proposed to be a possible two-dimensional topological insulator. This suggests that CLFA112 is a potential topological superconductor—a platform to realize Majorana fermions. Up to now, even a clear superconducting (SC) gap in CLFA112 has never been observed, and the SC properties of CLFA112 remain largely elusive. In this letter, we report the results of an atomic-scale investigation of the electronic structure of CLFA112 crystals using low-temperature scanning tunneling microscopy (STM). We revealed four different types of surfaces exhibiting distinct electronic properties, with all surfaces displaying dominating 2 × 1 surface reconstructions. On a Ca/La layer on top of an FeAs layer, a clear SC gap of ~12 mV was observed only at the crevices (vacancies) where the FeAs layer can be directly accessed. Remarkably, the FeAs termination layer displayed a dispersing nematic modulation both in real and q space. We also present peculiar zero-bias conductance peaks for the very As chain layer that is believed to exhibit a topological edge state as well as the influence of La dopants on the As chain layer.

With the rapid development of topological states in crystals, the study of topological states has been extended to quasicrystals in recent years. In this review, we summarize the recent progress of topological states in quasicrystals, particularly focusing on one-dimensional (1D) and 2D systems. We first give a brief introduction to quasicrystalline structures. Then, we discuss topological phases in 1D quasicrystals where the topological nature is attributed to the synthetic dimensions associated with the quasiperiodic order of quasicrystals. We further present the generalization of various types of crystalline topological states to 2D quasicrystals, where real-space expressions of corresponding topological invariants are introduced due to the lack of translational symmetry in quasicrystals. Finally, since quasicrystals possess forbidden symmetries in crystals such as five-fold and eight-fold rotation, we provide an overview of unique quasicrystalline symmetry-protected topological states without crystalline counterpart. [ABSTRACT FROM AUTHOR]

We study the topological properties of a nodal-line spin gapless semimetal superconductor. In the normal state, the low energy states are completely spin polarized so that the p + i p -wave pairing symmetry is considered. In the superconducting state, the energy bands are fully gapped in both the system bulk and at the system surface. However, when two open boundaries are considered, the gapless states emerge at the system hinges, indicating that the system is actually a higher-order topological superconductor. [ABSTRACT FROM AUTHOR]

One-dimensional (1D) Floquet topological superconductors possess two types of degenerate Majorana edge modes at zero and π quasienergies, leaving more room for the design of boundary time crystals and quantum computing schemes than their static counterparts. In this work, we discover Floquet superconducting phases with large topological invariants and arbitrarily many Majorana edge modes in periodically driven Kitaev chains (KCs). Topological winding numbers defined for the Floquet operator and Floquet entanglement Hamiltonian are found to generate consistent predictions about the phase diagram, bulk-edge correspondence and numbers of zero and π Majorana edge modes of the system under different driving protocols. The bipartite entanglement entropy further shows non-analytic behaviors around the topological transition point between different Floquet superconducting phases. These general features are demonstrated by investigating the KC with periodically kicked pairing or hopping amplitudes. Our discovery reveals the rich topological phases and many Majorana edge modes that could be brought about by periodic driving fields in 1D superconducting systems. It further introduces a unified description for a class of Floquet topological superconductors from their quasienergy bands and entanglement properties.

We employed the self-consistent Bogoliubov-de Gennes equations to explore the states of chiral Majorana mode in quantum anomalous Hall insulators in proximity to a superconductor, leading to the development of an extensive topological phase diagram. Our investigation focused on how an additional potential affects the separation of chiral Majorana modes across different phase conditions. We substantiated our findings by examining the zero-energy Local Density of States spectrum and the probability distribution of the chiral Majorana modes. We established the universality of chiral Majorana mode separation by applying an additional potential. This finding serves as a vital resource for future endeavors aimed at controlling and detecting these particles, thereby contributing to the advancement of quantum computing and condensed matter physics., (© 2024 IOP Publishing Ltd.)

Interfacing the quantum anomalous Hall insulator with a conventional superconductor is known to be a promising manner for realizing a topological superconductor, which has been continuously pursued for years. Such a proximity route depends to a great extent on the control of the delicate interfacial coupling of the two constituents. However, a recent experiment reported the failure to reproduce such a topological superconductor, which is ascribed to the negligence of the electrical short by the superconductor in the theoretical proposal. Here, we reproduce this topological superconductor with attention to the interface control. The resulted conductance matrix under a wide magnetic field range agrees with the fingerprint of this topological superconductor. This allows us to develop a phase diagram that unveils three regions parameterized by various coupling limits, which not only supports the feasibility to fabricate the topological superconductor by proximity but also fully explains the origin of the previous debate. The present work provides a comprehensible guide on fabricating the topological superconductor., (© 2024 IOP Publishing Ltd.)

Resonant exchange of the chiral Majorana fermions (MFs) that is coupled to two parallel Majorana zero modes (MZMs) or two parallel quantum dots (QDs) is investigated. We find that, in the two QDs coupling case, the resonant exchange for the chiral MFs is analogous to that in the MZM coupling case. We further propose a circuit based on topological superconductor, which is formed by the proximity coupling of a quantum anomalous Hall insulator and a s-wave superconductor, to observe the resonant exchange of chiral MFs pairs. The numerical calculations show that the resonant transmission of the chiral MFs can be adjusted by varying the coupling parameters at superconductor phase differenceΔφ=π. It is particularly noteworthy that, by only modulating the coupling strength between the two QDs, the resonant exchange may be switched on or off. By adding another MZM, the non-Abelian braiding like operation can be realized. Therefore, our design scheme may provide another way for non-Abelian braiding operation of MFs and the findings may have potential application value in the realization of topological quantum computers., (© 2024 IOP Publishing Ltd.)

Topological superconductors have attracted increasing attentions for the purpose of the quantum computation, because the character of supporting topological qubits are immune to quantum decoherence and can be manipulated by braiding operation. Since the topological superconducting state is rather rare in the intrinsic topological superconductors, most of experimental efforts focus on inducing topological superconductors by the proximity effect in superconductor (SC)/topological insulator (TI) heterostructures. Fu and Kane have theoretically proposed that the topological superconductivity can be obtained by inducing an s-wave superconducting gap into TIs. After that, a lot of experimental progress has been made in different systems. In the first part of this review, we introduce the heterostructure of threedimensional (3D) TI Bi2Se3 and Bi2Te3 on s-wave SC NbSe2 and d-wave SC Bi2Sr2CaCu2O8+d, topological crystalline insulator Sn1-xPbxTe on Pb, two-dimensional (2D) TI WTe2 on NbSe2, and TiBiSe2 on Pb. In the second part, the TI-based Josephson junctions are reviewed by introducing various experiments, including the Josephson junctions with TI barrier layers based on the Fu-Kane mode, and the superconducting quantum interference devices by TI based Josephson junctions. [ABSTRACT FROM AUTHOR]

We classify gapped topological superconducting (TSC) phases of one-dimensional quantum wires with local magnetic symmetries, in which the time-reversal symmetry |$\mathcal {T}$| is broken, but its combinations with certain crystalline symmetries, such as |$M_x \mathcal {T}$|⁠ , |$C_{2z} \mathcal {T}$|⁠ , |$C_{4z}\mathcal {T}$| and |$C_{6z}\mathcal {T}$|⁠ , are preserved. Our results demonstrate that an equivalent BDI class TSC can be realized in the |$M_x \mathcal {T}$| or |$C_{2z} \mathcal {T}$| 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 |$C_{4z}\mathcal {T}$| and |$C_{6z}\mathcal {T}$| 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 |$C_{6z}\mathcal {T}$| case, the MZMs can be either chiral or helical, and even helical-chiral coexisting. The minimal models preserving |$C_{4z}\mathcal {T}$| or |$C_{6z}\mathcal {T}$| symmetry are presented to illustrate their novel TSC properties and MZMs. [ABSTRACT FROM AUTHOR]

We investigate theoretically the out-of-equilibrium transport properties of a single-level quantum dot coupled to a normal metal electrode and attached to a topological superconductor. Both voltage and thermal bias responses of the system in the nonequilibrium regime are studied. To obtain transport characteristics we used the nonequilibrium Green's function approach. Particularly, we calculated the current and the corresponding differential conductance in two distinct cases. In the former situation, the charge current is induced by applying a bias voltage, whereas in the latter case it is generated by setting a temperature difference between the leads with no bias voltage. Moreover, strong diode effect in thermally generated current is found and non-equilibrium thermopower is analyzed. • Differential conductance reveals zero bias anomaly being a signature of Majorana bound state. • Zero bias anomaly in the differential conductance is robust with respect to changes in the dot's level energy. • Temperature gradient generates a charge current which reveals strong asymmetry under temperature bias reversal. • Under certain conditions strong rectification of thermocurrent is present and the system works as a diode. • Thermocurrent and thermoconductance reveal striking differences in topological and normal phase. [ABSTRACT FROM AUTHOR]

A feasible strategy for realizing the Majorana fermions is searching for a simple compound with both bulk superconductivity and Dirac surface states. In this paper, we perform calculations of electronic band structure, the Fermi surface, and the surface states, and measure the resistivity, magnetization, and specific heat of a TlSb compound with a CsCl-type structure. The band structure calculations show that TlSb is a Dirac semimetal when spin-orbit coupling is considered. TlSb is first determined to be a type-II superconductor with Tc=4.38 K, Hc1(0) = 148 Oe, Hc2(0)=1.12 T, and κGL = 10.6. We also confirm that TlSb is a moderately coupled s-wave superconductor. Although we cannot determine the band near the Fermi level EF that is responsible for superconductivity, its coexistence with topological surface states implies that the TlSb compound may be a simple material platform to realize the fault-tolerant quantum computations. [ABSTRACT FROM AUTHOR]

Based on the stability theory of differential equation, we investigate the robustness of Majorana zero-energy state (MZS) which is an important candidate for the topological quantum computation. In contrast to the prior studies exploring the stability of MZS by the numerical verification which depends on the special perturbations chosen in the simulation, our treatment is suitable for the arbitrary perturbations, so the results we obtained are reliable which is ensured by the exact mathematical theory of stability analysis. As an example, we demonstrate this by the robustness of MZS in the superconductor/topological insulator/ferromagnet insulator junction; the analytical and numerical results indicate that the MZS is unstable in this system. [ABSTRACT FROM AUTHOR]

In this paper, to characterize the effect of topological defects on the entanglement of topological states, we introduce the concept of local entanglement entropy. By using the spinless p x + i p y superconductor with quantized vortices, we numerically calculated the local entanglement entropy. In the topological superconducting (weak-pairing) phase, we found that the local entanglement entropy turns to 0.5 for a Majorana zero mode. For the system with two/four Majorana zero modes, by calculating local entanglement entropy, we found the non-locality effect of Majorana zero modes. In the future, we will use local entanglement entropy to characterize the defect effect on other many-body systems with long entanglement, for example, topological order and quantum spin liquid. [ABSTRACT FROM AUTHOR]

Transmission function of a system composing of a quantum dot (QD) subjected to a photon field and side-coupled to a topological superconductor nanowire hosting a pair of Majorana bound states (MBSs) is calculated by using the non-equilibrium Green's function technique. We find that a series of photon-induced peaks emerge and are split by the coupling between the QD and the MBSs. Moreover, the peaks' height are suppressed to zero because the MBSs absorb (emit) the photon energy. Under this condition, the MBSs may be shifted to the non-zero energy mode, and thus provide another detection scheme for its existence which is quite different from the currently adopted ones depending on the zero-energy mode of the MBSs. In the presence of MBS-MBS overlaping, the central photon-assisted peaks in the transmission function reappear due to the fact that the photon absorbed (emitted) by one mode of the MBSs are subsequently emitted (absorbed) by another MBSs' mode. We also find that the positions of the additional peaks induced by the MBS-MBS overlaping in the presence of the photon field are quite different from the case of zero photon field.

The fragile topological phase is an exotic phase with unstable edge states. Its nontrivial band topology can be removed by coupling to additional topologically trivial bands. Here we reveal that the fragile band topology can be realized in the inversion symmetric odd parity superconducting nodal line semimetal (SC-NLSM) materials with and without the spin orbital coupling. Without the spin orbital coupling, both the s -wave SC-NLSM and the p -wave SC-NLSM host a pair of Majorana zero modes on the system surface. For the spinful case, it has fourfold inverted bands and generates fourfold degenerate Majorana surface/edge states. However, we verify that for all of the systems we considered, they belong to the fragile topological superconducting system based on the Wilson loop method. The vortex bound states are studied numerically and the results depend strongly on the systems.

Two dimensional (2D) topological insulators (TIs) and topological superconductors (TSCs) have been intensively studied for recent years due to their great potential for dissipationless electron transportation and fault-tolerant quantum computing, respectively. Here we focus on stanene, the tin analogue of graphene, to give a brief review of their development as a candidate for both 2D TI and TSC. Stanene is proposed to be a TI with a large gap of 0.3 eV, and its topological properties are sensitive to various factors, e.g., the lattice constants, chemical functionalization and layer thickness, which offer various methods for phase tunning. Experimentally, the inverted gap and edge states are observed recently, which are strong evidences for TI. In addition, stanene is also predicted to be a time reversal invariant TSC by breaking inversion symmetry, supporting helical Majorana edge modes. The layer-dependent superconductivity of stanene is recently confirmed by both transport and scanning tunneling microscopy measurements. This review gives a detailed introduction to stanene and its topological properties and some prospects are also discussed. [ABSTRACT FROM AUTHOR]

Spontaneous symmetry breaking has been a paradigm to describe the phase transitions in condensed matter physics. In addition to the continuous electromagnetic gauge symmetry, an unconventional superconductor can break discrete symmetries simultaneously, such as time reversal and lattice rotational symmetry. In this work we report a characteristic in-plane 2-fold behaviour of the resistive upper critical field and point-contact spectra on the superconducting semimetal PbTaSe 2 with topological nodal-rings, despite its hexagonal lattice symmetry (or D 3 h in bulk while C 3 v on surface, to be precise). The 2-fold behaviour persists up to its surface upper critical field H c 2 R even though bulk superconductivity has been suppressed at its bulk upper critical field H c 2 HC ≪ H c 2 R , signaling its probable surface-only electronic nematicity. In addition, we do not observe any lattice rotational symmetry breaking signal from field-angle-dependent specific heat within the resolution. It is worth noting that such surface-only electronic nematicity is in sharp contrast to the observation in the topological superconductor candidate, Cu x Bi 2 Se 3 , where the nematicity occurs in various bulk measurements. In combination with theory, superconducting nematicity is likely to emerge from the topological surface states of PbTaSe 2 , rather than the proximity effect. The issue of time reversal symmetry breaking is also addressed. Thus, our results on PbTaSe 2 shed new light on possible routes to realize nematic superconductivity with nontrivial topology. [ABSTRACT FROM AUTHOR]

As a promising topological superconductor, Fe(Se0.5Te0.5) recently attracts interest from researchers, while Bi2Se3 has been considered one of the most important topological insulator materials. Therefore the Bi2Se3/FeSe0.5Te0.5 bulk composite could be interesting in demonstrating novel topological properties. We investigated crystalline, morphological, magnetic, and transport properties of Bi2Se3/FeSe0.5Te0.5 composite samples synthesized by the solid-state reaction method, and found that the elevated sintering temperature at the second step during the synthesis improves the crystallinity. All samples exhibit both superconductivity and ferromagnetism. Several indices of superconductivity, such as the transition temperature, change with increasing sintering temperature. [ABSTRACT FROM AUTHOR]

Exploring the topologically nontrivial states in condensed matter, especially their electronic structure and transport property, is very important and timely. Most topics discussed in this paper are relevant to the topological properties of electronic wavefunctions. Besides a brief introduction, the paper contains five sections, including topological quantum phenomena, various topological phases, unusual transport properties of topological quasiparticles, topological collective excitations and coupled excitations, and ongoing research in topological quantum states. These presentations expound a variety of aspects in the studies of topological quantum states. With all these aspects together, a comprehensive understanding of the topological quantum states in condensed matter physics can be achieved. [ABSTRACT FROM AUTHOR]

We report a systematic study on the thickness-dependent superconductivity and transport properties in exfoliated layered topological superconductor β-PdBi2. The superconducting transition temperature Tc is found to decrease with the decreasing thickness. Below a critical thickness of 45 nm, the superconductivity is suppressed, but followed by an abrupt resistance jump near Tc, which is in opposite to the behavior in a superconductor. We attribute suppressed Tc to the enhanced disorder as the thickness decreases. The possible physical mechanisms were discussed for the origination of sharply increased resistance in thinner β-PdBi2 samples.

The usual Berry phase for a Majorana zero-energy state is zero. In this manuscript, we propose a generalized geometric phase for Majorana zero-energy state, which is non-zero for the electron or hole, respectively. We calculate these non-zero geometric phases in a Ferromagnet (FI)/Topological Insulator (TI)/Superconductor (SC) hybrid system, whose magnetization can be manipulated by changing adiabatically the spin degree of freedom. The non-zero geometric phases have potential application on the topological quantum computation treatment of Majorana zero-energy modes. We also discuss the non-adiabatic geometric phase associated with Majorana zero-energy state by the path integral method. [ABSTRACT FROM AUTHOR]

In this paper, we apply the K-theory to classify topological trivial fermionic phases which, nonetheless, host symmetry-protected non-trivial defects. An important implication of our work is that the existence of Majorana zero mode in the vortex core is neither a necessary nor a sufficient condition for the superconductor in question being topologically non-trivial. [ABSTRACT FROM AUTHOR]

As a member of transition metal dichalcogenides, IrTe 2 is expected to exhibit typical layered structure and show large spin-orbital coupling (SOC) due to the large atomic numbers of Ir and Te. We successfully fabricated high quality IrTe 2 crystals, where superconductivity was detected by transport and diamagnetic measurements. Furthermore, linear magnetoresistance was observed over a wide range of magnetic field up to 15 T without any sign of becoming saturated, which might originate from topological nontrivial state. Thus, IrTe 2 could be a candidate of topological superconductors with simple element composition and crystal structure. [ABSTRACT FROM AUTHOR]

We review recent progress in the exploration of topological quantum states of matter in iron-based superconductors. In particular, we focus on the non-trivial topology existing in the band structures and superconducting states of iron's 3d orbitals. The basic concepts, models, materials and experimental results are reviewed. The natural integration between topology and high-temperature superconductivity in iron-based superconductors provides great opportunities to study topological superconductivity and Majorana modes at high temperature. [ABSTRACT FROM AUTHOR]

One-dimensional nanowires with strong spin–orbit coupling and proximity-induced superconductivity are predicted to exhibit topological superconductivity with condensed-matter analogues to Majorana fermions. Here, the nonequilibrium Green’s function approach with the generalized Kadanoff–Baym ansatz is employed to study the electron-correlation effects and their role in the topological superconducting phase in and out of equilibrium. Electron-correlation effects are found to affect the transient signatures regarding the zero-energy Majorana states, when the superconducting nanowire is subjected to external perturbations such as magnetic-field quenching, laser-pulse excitation, and coupling to biased normal-metal leads.

Magnetotransport measurements of PtPb _4 single crystals with the T _c of ∼2.80 K reveal pronounced exotic topological nature. An anomalous negative longitudinal magnetoresistance (MR) related to the chiral anomaly and a linear-like transverse MR examined within the framework of quantum MR theory were studied. Results indicate the presence of chiral-anomaly-induced quantum transport associated with the Weyl states and show a Dirac-cone-like band dispersion in PtPb _4 . This work reveals the drastic impact of the concept that the surface electrons in a Weyl fermion state may dominate the normal-state magnetotransport in PtPb _4 , leading to the conclusion that PtPb _4 can be a Weyl-semimetal superconductor candidate.

We present a detailed analysis about the changes of the orbital electron-correlation effects in one quantum-dot circuit, by considering finite couplings between the quantum dots and Majorana zero modes (MZMs). It is found that the dot-MZM couplings complicate the orbital-Kondo effect, because the orbital correlation occurs between the localized states in the quantum dots and the continuum hybridized states induced by the indirect metal-MZM couplings. When two of such correlation exist in pair, they have an opportunity to induce a long-range RKKY correlation, which is related to the MZMs. Further investigation shows that this RKKY interaction leads to the anomalous fractional Josephson effect. Our work can be helpful in clarifying the influence of MZM on the orbital electron correlation effects.

In this review, we summarize the achievement of the physics of surface Andreev bound states (SABS) up to now. The route of this activity has started from the physics of SABS of unconventional superconductors where the pair potential has a sign change on the Fermi surface. It has been established that SABS can be regarded as a topological edge state with topological invariant defined in the bulk Hamiltonian. On the other hand, SABS accompanies odd-frequency pairing like spin-triplet s-wave or spin-singlet p-wave. In a spin-triplet superconductor junction, induced odd-frequency pairing can penetrate into a diffusive normal metal (DN) attached to the superconductor. It causes so called anomalous proximity effect where the local density of states of quasiparticle in DN has a zero energy peak. When bulk pairing symmetry is spin-triplet px-wave, the anomalous proximity effect becomes prominent and the zero bias voltage conductance is always quantized independent of the resistance in DN and interface. Finally, we show that the present anomalous proximity effect is realized in an artificial topological superconducting system, where a nanowire with spin-orbit coupling and Zeeman field is put on the conventional spin-singlet s-wave superconductor. [ABSTRACT FROM AUTHOR]

The discovery of coexistence of bulk superconductivity and nontrivial topological bands in the compounds, such as LaO and NbC, with a rocksalt structure, motivates us to study the physical properties of TaO with the same structure. Based on preparing successfully TaO polycrystalline sample, we measured its resistivity, magnetization, and specific heat. It was first found that TaO is a type-II superconductor with the critical temperature T c = 6.20 K and the Ginzburg–Landau parameter κ G L (0) = 16.2. The estimated electron–phonon coupling constant λ e p (0.62) and the specific heat jump Δ C e l / γ n T c (0.56) demonstrate that TaO is a weak-coupling phonon-mediated superconductor. The lack of single crystal sample impedes us to study experimentally its properties related to the electronic band topology, which is the subject of endeavor in the future. • TaO polycrystalline sample were first successfully synthesized. • It was first found that TaO is a type-II superconductor. • TaO is a weak-coupling phonon-mediated conventional superconductor. [ABSTRACT FROM AUTHOR]

In topological insulators and topological superconductors, the discrete jump of the topological invariant upon tuning a certain system parameter defines a topological phase transition. A unified framework is employed to address the quantum criticality of the topological phase transitions in one to three spatial dimensions, which simultaneously incorporates the symmetry classification, order of band crossing, m -fold rotational symmetry, correlation functions, critical exponents, scaling laws, and renormalization group approach. We first classify higher-order Dirac models according to the time-reversal, particle-hole, and chiral symmetries, and determine the even–oddness of the order of band crossing in each symmetry class. The even–oddness further constrains the rotational symmetry m permitted in a symmetry class. Expressing the topological invariant in terms of a momentum space integration over a curvature function, the order of band crossing determines the critical exponent of the curvature function, as well as that of the Wannier state correlation function introduced through the Fourier transform of the curvature function. The conservation of topological invariant further yields a scaling law between critical exponents. In addition, a renormalization group approach based on deforming the curvature function is demonstrated for all dimensions and symmetry classes. Through clarification of how the critical quantities, including the jump of the topological invariant and critical exponents, depend on the nonspatial and the rotational symmetry, our work introduces the notion of universality class into the description of topological phase transitions.

We study time-resolved charge transport in a superconducting nanowire using time-dependent Landauer–Büttiker theory. We find that the steady-state Majorana zero-bias conductance peak emerges transiently accompanied by characteristic oscillations after a bias-voltage quench. These oscillations are suppressed for trivial impurity states (IS) that otherwise show a similar steady-state signal as the Majorana zero mode (MZM). In addition, we find that Andreev bound states or quasi-Majorana states (QMS) in the topologically trivial bulk phase can give rise to a zero-bias conductance peak, also retaining the transient properties of the MZM. Our results imply that (1) time-resolved transport may be used as a probe to distinguish between the topological MZM and trivial IS; and (2) the QMS mimic the transient signatures of the topological MZMs.

Chiral and helical Majorana fermions are two archetypal edge excitations in two-dimensional topological superconductors. They emerge from systems of different Altland–Zirnbauer symmetries and characterized by ${\mathbb{Z}}$ and ${{\mathbb{Z}}}_{2}$ topological invariants respectively. It seems improbable to tune a pair of co-propagating chiral edge modes to counter-propagate in a single system without symmetry breaking. Here, we explore the peculiar behaviors of Majorana edge modes in topological superconductors with an additional ‘mirror’ symmetry which changes the bulk topological invariant to ${\mathbb{Z}}\oplus {\mathbb{Z}}$ type. A theoretical toy model describing the proximity structure of a Chern insulator and a p _x -wave superconductor is proposed and solved analytically to illustrate a direct transition between two topologically nontrivial phases. The weak pairing phase has two chiral Majorana edge modes, while the strong pairing phase is characterized by mirror-graded Chern number and hosts a pair of counter-propagating Majorana fermions protected by the mirror symmetry. The edge theory is worked out in detail, and implications to braiding of Majorana fermions are discussed.