Topological superconductors with spin-triplet pairings and Majorana Fermi arcs

We construct a three-dimensional topological superconductor Bogoliubov–de Gennes(BdG)Hamiltonian with the normal state being a three-dimensional topological insulator.By introducing inter-orbital spin-triplet pairings term△3,there are topological Majorana nodes in the bulk and they are connected by Majorana Fermi arcs on the surface,similar to the case of Weyl semimetal.Furthermore,by adding an inversion-breaking term to the normal state,momentum-independent pairing terms with different parities can coexist in the Bd G Hamiltonian,which creates more Majorana modes similar to Andreev bound states and a richer phase diagram.

Optical study on topological superconductor candidate Sr-doped Bi_(2)Se_(3)

Utilizing infrared spectroscopy,we study the charge dynamics of the topological superconductor candidate Sr_(x)Bi_(2)Se_(3).The frequency-dependent reflectivity R(ω)demonstrates metallic feature and the scattering rate of the free carriers decreases with temperature decreasing.The plasma edge shows a slight blue shift upon cooling,similar to the behavior of Cu_(x)Bi_(2)Se_(3).As the carrier concentration n obtained by Hall resistivity increases slightly with the decreasing temperature,the effective mass is proved to increase as well,which is in contrast with that of Cu_(x)Bi_(2)Se_(3).We also perform the ultrafast pump-probe study on the Sr_(0.2)Bi_(2)Se_(3)compounds.Resembling its parent compound Bi_(2)Se_(3),three distinct relaxation processes are found to contribute to the transient reflectivity.However,the deduced relaxation times are quite different.In addition,the electron-optical-phonon coupling constant is identified to beλ=0.88.

Phase Diagram and Edge States of Surface States of Topological Superconductors

Majorana fermions in two-dimensional systems satisfy non-Abelian statistics. They are possible to exist in topological superconductors as quasi particles, which is of great significance for topological quantum computing. In this paper, we study a new promising system of superconducting topological surface state topological insulator thin films. We also study the phase diagrams of the model by plotting the Majorana edge states and the density of states in different regions of the phase diagram. Due to the mirror symmetry of the topological surface states, the Hamiltonian can be block diagonalized into two spin-triplet p-wave superconductors, which are also confirmed by the phase diagrams. The chiral Majorana edge modes may provide a new route for realizing topological quantum computation.

Two-Dimensional Topological Superconductors with Spin-Orbit Coupling and Zeeman Term

Topological superconductors have attracted much attention for their potential applications in realizing topological quantum computing. In this paper, we show a system with 8 × 8 Bogoliubov-de Gennes Hamiltonian. The system has particle-hole symmetry. By adding a Zeeman term to the model, we discuss this system from two situations. In the first case of breaking the inversion symmetry while preserving the mirror symmetry, it is obtained from the topological phase diagram of the system that it is a topological superconductor. In another case where the mirror symmetry is broken while preserving the inversion symmetry, the system has two nodes connected by a flat band of zero-energy Andreev edge states and the topological number is non-zero . It can be concluded that the system is a topological nodal superconductor.

Customizing topological phases in the twisted bilayer superconductors with even-parity pairings

We investigate the topological properties of twisted bilayer superconductors with different even-parity pairings in each layer.In the presence of spin-orbit coupling,the Hamiltonian is mapped into an effective odd-parity superconductor.Based on this,we deduce the topological properties by examining the relative configuration between Fermi surface and Dirac pairing node.We show that mixed Rashba and Dresselhaus spin-orbit coupling and anisotropic hopping terms,which break the C_(4)symmetry of the Fermi surface,can induce first-order topological superconductors with non-zero bulk Chern number.This provides a versatile way to control the topological phases of bilayer superconductors by adjusting the twisted angle and chemical potential.We demonstrate our results using a typical twisted angle of 53.13°,at which the translation symmetry is restored and the Chern number and edge state are calculated using the Moir′e momentum.

Stanene: A good platform for topological insulator and topological superconductor

Two dimenslonal(2D)topological insulators(TIs)and topological superonductors(TSC&)have been intensively studied for recent yars due to their great poterntial for dissipat iouless eloctron transporta-tion and fault-tolerant quantum computing,respectively.Here we focus on stanene,the tin analogue of grapbene,to give a brief review of their development a a candidate for both 2D TI and TSC.Stanene is proposed to bea TI with a large gap of 0.3 eW,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 syumetry,supporting belical Majorana edge modes.The layer dependent superconduetivity of stanene is receatly confirmed by both transport and scauning tumeling microsoopy measurenents.This review givs a detailod introduction to stanene and its topological properties and some proepects are also discussed.

A brief review on Majorana bound states in topological superconductors

Topological superconductivity has drawn much attention recently, and most interests are focused on the Majorana bound states existing at the edges of one-dimensional topological superconductors. These Majorana bound states are ideal platform for studying non-Abelian statistics. Meanwhile, they are proposed to be useful in quantum computation. In this review, we introduce the basic concepts and models in this area. We begin from the Kitaev model, which is the most concise model for one-dimensional topological superconductivity. Then, we discuss how to realize this model with spin-orbit coupling in realistic materials. Finally, we show some simple methods to detect the Majorana bound states and study their novel properties with the help of adjacent quantum dots.

Majorana zero mode in the vortex of an artificial topological superconductor

Majorana fermion （MF）, an exotic particle that is identical to its own antiparticle, was recently found in solid matter as a quasiparticle excitation, the Majorana zero mode （MZM）, in the vortex of an artificial topological superconductor （TSC）. This artificial TSC, first proposed by Fu and Kane in 2008, is a heterostructure made of a topological insulator BiETe3 and an s-wave superconductor NbSe2. This paper will briefly review the experimental progresses based on the Bi2Te3/NbSe2 heterostructure. All evidences are self-consistent and reveal that the MZM exists in the center of vortex. Those experimental results are also supported by theory. This finding is a milestone in the research ofMajorana fermions in solid state physics and a starting point of MZM＇s application in topological quantum computation.

Identification of Topological Surface State in PdTe2 Superconductor by Angle-Resolved Photoemission Spectroscopy

High-resolution angle-resolved photoemission measurements are carried out on transition metal dichalcogenide PdTe2 that is a superconductor with a Tc at 1.7K. Combined with theoretical calculations, we discover for the first time the existence of topologically nontrivial surface state with Dirac cone in PbTe2 superconductor. It is located at the Brillouin zone center and possesses helical spin texture. Distinct from the usual three-dimensional topological insulators where the Dirac cone of the surface state lies at the Fermi level, the Dirac point of the surface state in PdTe2 lies deeply below the Fermi level at - 1.75 eV binding energy and is well separated from the bulk states. The identification of topological surface state in PdTe2 superconductor deeply below the Fermi level provides a unique system to explore new phenomena and properties and opens a door for finding new topological materials in transition metal ehalcogenides.

Chiral symmetry protected topological nodal superconducting phase and Majorana Fermi arc

With an external in-plane magnetic field, we show the emergence of a topological nodal superconducting phase of the two-dimensional topological surface states. This nodal superconducting phase is protected by the chiral symmetry with a non-zero magnetic field, and there are corresponding Majorana Fermi arcs(also known as flat band Andreev bound states) connecting the two Majorana nodes along the edges, similar to the case of Weyl semimetal. The topological nodal superconductor is an intermediate phase between two different chiral superconductors, and is stable against the effects of substrates. The two-dimensional effective theory of the nodal superconducting phase also captures the low energy behavior of a three-dimensional lattice model which describes the iron-based superconductor with a thin film geometry. The localizations of the Majorana nodes can be manipulated through external in-plane magnetic fields, which may introduce a non-trivial topological Berry phase between them.

Topological phase boundary in a generalized Kitaev model

We study the effects of the next-nearest-neighbor hopping and nearest-neighbor interactions on topological phases in a one-dimensional generalized Kitaev model. In the noninteracting case, we define a topological number and calculate exactly the phase diagram of the system. With addition of the next-nearest-neighbor hopping, the change of phase boundary between the topological and trivial regions can be described by an effective shift of the chemical potential. In the interacting case, we obtain the entanglement spectrum, the degeneracies of which correspond to the topological edge modes, by using the infinite time-evolving block decimation method. The results show that the interactions change the phase boundary as adding an effective chemical potential which can be explained by the change of the average number of particles.

Topological hierarchy matters — topological matters with superlattices of defects

Topological insulators/superconductors are new states of quantum matter with metallic edge/surface states.In this paper,we review the defects effect in these topological states and study new types of topological matters — topological hierarchy matters.We find that both topological defects（quantized vortices） and non topological defects（vacancies） can induce topological mid-gap states in the topological hierarchy matters after considering the superlattice of defects.These topological mid-gap states have nontrivial topological properties,including the nonzero Chern number and the gapless edge states.Effective tight-binding models are obtained to describe the topological mid-gap states in the topological hierarchy matters.

Majorana zero modes induced by skyrmion lattice

One-dimensional s-wave superconductor with spin-orbit coupling is a platform for the realization of Majorana zero modes. The spin-exchange with the magnetic skyrmion lattice can induce spin-orbit coupling in a s-wave superconductor system and the effects are different from the constant spin–orbit coupling. The strength of the effective spin–orbit coupling as well as the rich topoloigcal phase diagram are directly connected to the radius of the skyrmion lattice R. We obtain the rich topological phase diagram of this system with different skyrmion lattice radii by numerically evaluating the spectrum of the system under the periodic boundary condition, and we also find the Majorana zero modes under the open boundary condition to verify the bulk–edge correspondence.

A brief review on one-dimensional topological insulators and superconductors

Recently one-dimensional topological phases are gaining increasing attentions. Like two- and three-dimensional ones, Onedimensional systems are important in a complete understanding of the topological properties. One-dimensional topological phases have been realized using current experimental setups. Specially the signatures of Majorana fermions have been observed in onedimensional topological superconductors engineered with Rashiba nanowires. From the many studies, the paper reviews typical theoretical models of one-dimensional topological insulators and superconductors. For one-dimensional topological insulators, we introduce the Su-Schrieffer-Heeger, superlattices and Creutz models, while for topological superconductors the Kitaev model and Rashiba nanowire are introduced. These models not only provide an overview of one-dimensional topological phases, but also are the starting points for further studies.

Recent progress on Majorana in semiconductor-superconductor heterostructures—engineering and detection

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 briefiy review the protocols for implementing topological quantum computation by hybrid semiconductor-superconductor nanowires.

Roadmap of the iron-based superconductor Majorana platform

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-Tcsuperconductivity,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.,In As/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.

具有高阶范霍夫奇点的自选轨道耦合体系中的鲁棒拓扑超导

Van Hove singularities in proximity to the Fermi level promote electronic interactions and generate diverse competing instabilities.It is also known that a nontrivial Berry phase derived from spin–orbit coupling can introduce an intriguing decoration into the interactions and thus alter correlated phenomena.However,it is unclear how and what type of new physics can emerge in a system featured by the interplay between van Hove singularities(VHSs)and the Berry phase.Here,based on a general Rashba model on the square lattice,we comprehensively explore such an interplay and its significant influence on the competing electronic instabilities by performing a parquet renormalization group analysis.Despite the existence of a variety of comparable fluctuations in the particle–particle and particle-hole channels associated with higher-order VHSs,we find that the chiral p±ip pairings emerge as two stable fixed trajectories within the generic interaction parameter space,namely the system becomes a robust topological superconductor.The chiral pairings stem from the hopping interaction induced by the nontrivial Berry phase.The possible experimental realization and implications are discussed.Our work sheds new light on the correlated states in quantum materials with strong spin–orbit coupling(SOC)and offers fresh insights into the exploration of topological superconductivity.

Topological states in quasicrystals

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.

Evidence for nematic superconductivity of topological surface states in PbTaSe2

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 PbTaSe2 with topological nodal-rings,despite its hexagonal lattice symmetry(or D3 h in bulk while C3 v on surface,to be precise).The 2-fold behaviour persists up to its surface upper critical field Hc2R even though bulk superconductivity has been suppressed at its bulk upper critical field Hc2HC<<Hc2R,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,CuxBi2Se3,where the nematicity occurs in various bulk measurements.In combination with theory,superconducting nematicity is likely to emerge from the topological surface states of PbTaSe2,rather than the proximity effect.The issue of time reversal symmetry breaking is also addressed.Thus,our results on PbTaSe2 shed new light on possible routes to realize nematic superconductivity with nontrivial topology.

Reveal the nematic topological superconductivity from the anisotropic unconventional Josephson effect:Theory and application to doped Bi_(2)Se_(3)

We propose a scheme to reveal the possible nematic superconducting order parameter in the doped Bi2Se3 by observing the anisotropic unconventional Josephson effect without an external magnetic field.We find the presence of an intrinsicπ-phase in the spin-triplet channel of Andreev reflection.Its competition with the odd-parity superconducting gap phase can lead to unconventional Josephson effect in the Josephson junction,whose normal region is connected to the same side of the superconductor,called the U-shaped junction according to its geometry.For Josephson junctions with the interfaces perpendicular to the nematic direction,the competition will lead to a Josephsonπ-junctions.In the case where the interface is parallel to the nematic direction,it will lead to a Josephson 0-junction.Thus,this can directly reflect the nematic superconductivity.It is worth noting that Josephson coupling with the 4πperiod appears only in the normal injected channels.Interestingly,if the Josephson junction adopts a conventional geometry,it always exhibits a normal Josephson 0-junction regardless of the gap function taken by the doped Bi2Se3 and therefore cannot distinguish the pairing symmetry.We thus propose a superconducting quantum interference device containing a U-shaped Josephson junction to detect nematic superconductivity.This proposal not only can be applied to detect nematic superconductivity but also provides a feasible platform for topological quantum computation.