Dirac and Weyl superconductors in three dimensions

Topological Degeneracy Induced by Twisting

In recent years, twisting has emerged as a new degree of freedom that plays an increasingly important role in Bloch bands of various physical systems. However, there is currently a lack of reports on the nontrivial physics of topological degeneracy in twisted systems. In this Letter, we investigated the intrinsic physical correlation between twisting and topological degeneracy. We found that twisting not only breaks the symmetry of the system but also introduces topological degeneracy that does not exist under the original symmetric system without twisting. Furthermore, the topological degeneracy can be easily tuned through twisting. This new twist-induced topological degeneracy gives rise to a unique polarization-degenerate birefringent medium, wherein the twist angle acts as a novel degree of freedom for dispersion and polarization management of interface states. Exhibiting fascinating properties and experimental feasibilities, our Letter points to new possibilities in the research of various topological physics in twisted photonics.

Dual Dirac Nodal Line in Nearly Freestanding Electronic Structure of β-Sn Monolayer

Two-dimensional topological insulators (2D TIs) have distinct electronic properties that make them attractive for various applications, especially in spintronics. The conductive edge states in 2D TIs are protected from disorder and perturbations and are spin-polarized, which restrict current flow to a single spin orientation. In contrast, topological nodal line semimetals (TNLSM) are distinct from TIs because of the presence of a 1D ring of degeneracy formed from two bands that cross each other along a line in the Brillouin zone. These nodal lines are protected by topology and can be destroyed only by breaking certain symmetry conditions, making them highly resilient to disorder and defects. However, 2D TNLSMs do not possess protected boundary modes, which makes their investigation challenging. There have been several theoretical predictions of 2D TNLSMs, however, experimental realizations are rare. β-Sn, a metallic allotrope of tin with a superconducting temperature of 3.72 K, may be a candidate for a topological superconductor that can host Majorana Fermions for quantum computing. In this work, single layers of α-Sn and β-Sn on a Cu(111) substrate are successfully prepared and studied using scanning tunneling microscopy, angle-resolved photoemission spectroscopy, and density functional theory calculations. The lattice and electronic structure undergo a topological transition from 2D topological insulator α-Sn to 2D TNLSM β-Sn, with two types of nodal lines coexisting in monolayer β-Sn. Such a realization of two types of nodal lines in one 2D material has not been reported to date. Moreover, we also observed an unexpected phenomenon of freestanding-like electronic structures of β-Sn/Cu(111), highlighting the potential of ultrathin β-Sn films as a platform for exploring the electronic properties of 2D TNLSM and topological superconductors, such as few-layer superconducting β-Sn in lateral contact with topological nodal line single-layer β-Sn.

Revealing the spatial nature of sublattice symmetry

The sublattice symmetry on a bipartite lattice is commonly regarded as the chiral symmetry in the AIII class of the tenfold Altland-Zirnbauer classification. Here, we reveal the spatial nature of sublattice symmetry and show that this assertion holds only if the periodicity of primitive unit cells agrees with that of the sublattice labeling. In cases where the periodicity does not agree, sublattice symmetry is represented as a glide reflection in energy-momentum space, which inverts energy and simultaneously translates some k by π, leading to substantially different physics. Particularly, it introduces novel constraints on zero modes in semimetals and completely alters the classification table of topological insulators compared to class AIII. Notably, the dimensions corresponding to trivial and nontrivial classifications are switched, and the nontrivial classification becomesZ 2 instead of Z . We have applied these results to several models, including the Hofstadter model both with and without dimerization.

Direction-dependent conductivity in planar Hall set-ups with tilted Weyl/multi-Weyl semimetals

We compute the magnetoelectric conductivity tensors in planar Hall set-ups, which are built with tilted Weyl semimetals (WSMs) and multi-Weyl semimetals (mWSMs), considering all possible relative orientations of the electromagnetic fields (EandB) and the direction of the tilt. The non-Drude part of the response arises from a nonzero Berry curvature in the vicinity of the WSM/mWSM node under consideration. Only in the presence of a nonzero tilt do we find linear-in-|B|terms in set-ups where the tilt-axis is not perpendicular to the plane spanned byEandB. The advantage of the emergence of the linear-in-|B|terms is that, unlike the various|B|2-dependent terms that can contribute to experimental observations, they have purely a topological origin, and they dominate the overall response-characteristics in the realistic parameter regimes. The important signatures of these terms are that they (1) change the periodicity of the response fromπto 2π, when we consider their dependence on the angleθbetweenEandB; and (2) lead to an overall change in sign of the conductivity depending onθ, when measured with respect to theB=0case.

Hyperbolic metamaterial empowered controllable photonic Weyl nodal line semimetals

Motivated by unique topological semimetals in condensed matter physics, we propose an effective Hamiltonian with four degrees of freedom to describe evolutions of photonic double Weyl nodal line semimetals in one-dimensional hyper-crystals, which supports the energy bands translating or rotating independently in the form of Weyl quasiparticles. Especially, owing to the unit cells without inversion symmetry, a pair of reflection-phase singularities carrying opposite topological charges emerge near each nodal line, and result in a unique bilateral drumhead surface state. After reducing radiation leakages and absorption losses, these two singularities gather together gradually, and form a quasi-bound state in the continuum (quasi-BIC) ring at the nodal line ultimately. Our work not only reports the first realization of controllable photonics Weyl nodal line semimetals, establishes a bridge between two independent topological concepts-BICs and Weyl semimetals, but also heralds new possibilities for unconventional device applications, such as dual-mode schemes for highly sensitive sensing and switching.

Monolayer Cu2Se: a topological catalysis in CO2electroreduction

This investigation provides a comprehensive exploration into the intricate interplay between topological surface states (TSS) and catalytic performance in two-dimensional (2D) materials, with specific emphasis on monolayer Cu2Se. Leveraging the unique characteristics of nodal loop semimetals (NLSMs), we delve into the precise influence of TSS on catalytic activity, particularly in the domain of CO2electrochemical reduction. Our findings illuminate the central role played by these TSS, arising from the underlying NLSM framework, in sculpting catalytic efficiency. The length of these surface states emerges as a critical determinant of surface density of states (DOSs), a fundamental factor governing catalytic behavior. Extension of these surface states correlates with heightened surface DOSs, yielding lower Gibbs free energies and consequently enhancing catalytic performance, particularly in the electrochemical reduction of CO2. Moreover, we underscore the profound importance of preserving symmetries that protect the nodal loop. The disruption of these symmetries is found to result in a significant degradation of catalytic efficacy, underscoring the paramount significance of topological features in facilitating catalytic processes. Therefore, this study not only elucidates the fundamental role of TSS in dictating the catalytic performance of topological 2D materials but also paves the way for harnessing these unique attributes to drive sustainable and highly efficient catalysis across a diverse spectrum of chemical processes.

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.

Existence of nodal-arc and its evolution into Weyl-nodes in the presence of spin-orbit coupling in TaAs & TaP

In this work, we report the existence ofnodal-arc, which acts as the building block of all the nodal-rings in TaAs & TaP. Thisnodal-arcis found to be capable of generating all the nodal-rings in these materials upon the application of space-group symmetry operations including time-reversal symmetry. The arcs are obtained to be dispersive with the energy spread of ∼109 (∼204) meV in TaAs (TaP). Also, the orbitals leading to bands-inversion and thus the formation ofnodal-arcsare found to be Ta-5d& As-4p(P-3p) in TaAs (TaP). The area of nodal-rings is found to be highly sensitive to the change in hybridization-strength, where the increase in hybridization-strength leads to the decrease in the area of nodal-rings. In the presence of spin-orbit coupling (SOC), all the points on these arcs get gaped-up and two pairs of Weyl-nodes are found to evolve from them. Out of the two pair, one is found to be situated close to the joining point of the two arcs forming a ring. This causes the evolution of each nodal-ring into three pairs of Weyl-nodes. The coordinates of these Weyl-nodes are found to be robust to the increase in SOC-strength from ∼0.7-3.5 eV. All the results are obtained at thefirst-principlelevel. This work provides a clear picture of the existence ofnodal-arcdue to accidental degeneracy and its evolution into Weyl-nodes under the effect of SOC.

Nontrivial Topological Properties and Synthesis of Sn2CoS with L21 Structure

We synthesize Sn₂CoS in experiment and study its topological properties in theory. By first-principles calculations, we study the band structure and surface state of Sn₂CoS with L2₁ structure. It is found that the material has type-II nodal line in the Brillouin zone and clear drumhead-like surface state when the spin-orbit coupling is not considered. In the case of spin-orbit coupling, the nodal line will open gap, leaving the Dirac points. To check the stability of the material in nature, we synthesize Sn₂CoS nanowires with L2₁ structure in an anodic aluminum oxide (AAO) template directly by the electrochemical deposition (ECD) method with direct current (DC). Additionally, the diameter of the typical Sn₂CoS nanowires is about 70 nm, with a length of about 70 μm. The Sn₂CoS nanowires are single crystals with an axis direction of [100], and the lattice constant determined by XRD and TEM is 6.0 Å. Overall, our work provides realistic material to study the nodal line and Dirac fermions.

Two-dimensional half-metallicity and fully spin-polarized topological fermions in monolayer EuOBr

Two-dimensional (2D) half-metal and topological states have been the current research focus in condensed matter physics. Herein, we report a novel 2D material named EuOBr monolayer, which can simultaneously show 2D half-metal and topological fermions. This material shows a metallic state in the spin-up channel but a large insulating gap of 4.38 eV in the spin-down channel. In the conducting spin channel, the EuOBr monolayer shows the coexistence of Weyl points and nodal-lines near the Fermi level. These nodal-lines are classified by type-I, hybrid, closed, and open nodal-lines. The symmetry analysis suggests these nodal-lines are protected by the mirror symmetry, which cannot be broken even spin-orbit coupling is included because the ground magnetization direction in the material is out-of-plane [001]. The topological fermions in the EuOBr monolayer are fully spin-polarized, which can be meaningful for future applications in topological spintronic nano-devices.

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

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

Topological Hall Effect Anisotropy in Kagome Bilayer Metal Fe_{3}Sn_{2}

Kagome lattice materials have attracted growing interest for their topological properties and flatbands in electronic structure. We present a comprehensive study on the anisotropy and out-of-plane electric transport in Fe_{3}Sn_{2}, a metal with bilayer of Fe kagome planes and with massive Dirac fermions that features high-temperature noncollinear magnetic structure and magnetic skyrmions. For the electrical current path along the c axis, in micron-size crystals, we found a large topological Hall effect over a wide temperature range down to spin-glass state. Twofold and fourfold angular magnetoresistance are observed for different magnetic phases, reflecting the competition of magnetic interactions and magnetic anisotropy in kagome lattice that preserve robust topological Hall effect for inter-kagome bilayer currents. This provides new insight into the anisotropy in Fe_{3}Sn_{2}, of interest in skyrmionic-bubble application-related micron-size devices.

Acoustic Möbius Insulators from Projective Symmetry

In the presence of gauge symmetry, common but not limited to artificial crystals, the algebraic structure of crystalline symmetries needs to be projectively represented, giving rise to unprecedented topological physics. Here, we demonstrate this novel idea by exploiting a projective translation symmetry and constructing a variety of Möbius-twisted topological phases. Experimentally, we realize two Möbius insulators in acoustic crystals for the first time: a two-dimensional one of first-order band topology and a three-dimensional one of higher-order band topology. We observe unambiguously the peculiar Möbius edge and hinge states via real-space visualization of their localiztions, momentum-space spectroscopy of their 4π periodicity, and phase-space winding of their projective translation eigenvalues. Not only does our work open a new avenue for artificial systems under the interplay between gauge and crystalline symmetries, but it also initializes a new framework for topological physics from projective symmetry.

Experimental Demonstration of Bulk-Hinge Correspondence in a Three-Dimensional Topological Dirac Acoustic Crystal

Three-dimensional topological Dirac semimetal (DSM) is a vital state to explore topological phases and phase transitions. However, its bulk-boundary correspondence is elusive. Here, we experimentally investigate the higher-order hinge states in an acoustic DSM. Not only removable trivial surface states but also robust nontrivial hinge arcs are observed, attributed to the direct correspondence between bulk polarization and hinge charge. We further reveal that a pair of zigzag and bearded hinges possess arcs located in complementary momentum regions. Our work provides solid proof of the bulk-hinge correspondence in DSM and sheds light on the study of topological hierarchy across dimensions.

Theoretical realization of hybrid Weyl state and associated high catalytic performance for hydrogen evolution in NiSi

For electrochemical hydrogen evolution reaction (HER), developing high-performance catalysts without containing precious metals have been a major research focus in the present. Herein, we show the feasibility of HER catalytic enhancement in Ni-based materials based on topological engineering from hybrid Weyl states. Via a high-throughput computational screening from ∼140,000 materials, we identify that a chiral compound NiSi is a hybrid Weyl semimetal (WSM) showing bulk type-I and type-II Weyl nodes and long surface Fermi arcs near the Fermi level. Sufficient evidences verify that topological charge carriers participate in the HER process, and make the certain surface of NiSi highly active with the Gibbs free energy nearly zero (0.07 eV), which is even lower than Pt and locates on the top of the volcano plots. This work opens up a new routine to develop no-precious-metal-containing HER catalysts via topological engineering, rather than traditional defect engineering, doping engineering, or strain engineering.

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A new designation to catalytic enhancement via topological engineering is constructed

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An ideal hybrid Weyl catalyst NiSi is identified by a high-throughput material screening

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Sufficient evidences verify topological charge participates in catalytic enhancement

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NiSi has a higher theoretical hydrogen evolution activity than precious-metal Pt

Second-Order Real Nodal-Line Semimetal in Three-Dimensional Graphdiyne

Real topological phases featuring real Chern numbers and second-order boundary modes have been a focus of current research, but finding their material realization remains a challenge. Here, based on first-principles calculations and theoretical analysis, we reveal the already experimentally synthesized three-dimensional (3D) graphdiyne as the first realistic example of the recently proposed second-order real nodal-line semimetal. We show that the material hosts a pair of real nodal rings, each protected by two topological charges: a real Chern number and a 1D winding number. The two charges generate distinct topological boundary modes at distinct boundaries. The real Chern number leads to a pair of hinge Fermi arcs, whereas the winding number protects a double drumhead surface bands. We develop a low-energy model for 3D graphdiyne which captures the essential topological physics. Experimental aspects and possible topological transition to a 3D real Chern insulator phase are discussed.

Fully spin-polarized Weyl fermions and in/out-of-plane quantum anomalous Hall effects in a two-dimensional d0 ferromagnet

The quantum anomalous Hall effect (QAHE) in intrinsic ferromagnets has attracted considerable attention recently. Previously, studies of the QAHE have mostly focused on the default assumption of out-of-plane magnetization. In fact, the QAHE can also be achieved via in-plane magnetization, but such candidate materials are very scarce. Here, we find that two-dimensional (2D) YN₂ not only possesses the previously reported out-of-plane QAHE, but it also possesses a tunable in-plane QAHE. More importantly, unlike the previously reported in-plane QAHE in d/f-type ferromagnets, here we report the effect in a 2D d⁰ ferromagnet, namely YN₂, for the first time. In the ground state, a YN₂ monolayer has a half-metal band structure, and manifests six pairs of fully spin-polarized Weyl points at the Fermi level. When spin-orbit coupling is included, the YN₂ monolayer can realize multiple topological phases, determined based on the magnetization direction. Under in-plane magnetization, the YN₂ monolayer shows either the Weyl state or in-plane QAHE state. Remarkably, the Chern number (±1) and the propagating direction of QAHE edge channels can be continuously switched via shifting the direction of the in-plane magnetic field. When magnetization is applied out-of-plane, the YN₂ monolayer realizes an out-of-plane QAHE phase with a high Chern number of 3. The nontrivial edge states for all the topological phases in the YN₂ monolayer have been clearly identified. This work suggests that 2D YN₂ is an excellent candidate for investigating in-plane QAHE phases in d⁰ ferromagnets.

Boundary Criticality of PT-Invariant Topology and Second-Order Nodal-Line Semimetals

For conventional topological phases, the boundary gapless modes are determined by bulk topological invariants. Based on developing an analytic method to solve higher-order boundary modes, we present PT-invariant 2D topological insulators and 3D topological semimetals that go beyond this bulk-boundary correspondence framework. With unchanged bulk topological invariants, their first-order boundaries undergo transitions separating different phases with second-order boundary zero modes. For the 2D topological insulator, the helical edge modes appear at the transition point for two second-order topological insulator phases with diagonal and off-diagonal corner zero modes, respectively. Accordingly, for the 3D topological semimetal, the criticality corresponds to surface helical Fermi arcs of a Dirac semimetal phase. Interestingly, we find that the 3D system generically belongs to a novel second-order nodal-line semimetal phase, possessing gapped surfaces but a pair of diagonal or off-diagonal hinge Fermi arcs.

Interacting Dirac materials

We investigate the extent to which the class of Dirac materials in two-dimensions provides general statements about the behavior of both fermionic and bosonic Dirac quasiparticles in the interacting regime. For both quasiparticle types, we find common features for the interaction induced renormalization of the conical Dirac spectrum. We perform the perturbative renormalization analysis and compute the self-energy for both quasiparticle types with different interactions and collate previous results from the literature whenever necessary. Guided by the systematic presentation of our results in table1, we conclude that long-range interactions generically lead to an increase of the slope of the single-particle Dirac cone, whereas short-range interactions lead to a decrease. The quasiparticle statistics does not qualitatively impact the self-energy correction for long-range repulsion but does affect the behavior of short-range coupled systems, giving rise to different thermal power-law contributions. The possibility of a universal description of the Dirac materials based on these features is also mentioned.

Refined symmetry indicators for topological superconductors in all space groups

Topological superconductors are exotic phases of matter featuring robust surface states that could be leveraged for topological quantum computation. A useful guiding principle for the search of topological superconductors is to relate the topological invariants with the behavior of the pairing order parameter on the normal-state Fermi surfaces. The existing formulas, however, become inadequate for the prediction of the recently proposed classes of topological crystalline superconductors. In this work, we advance the theory of symmetry indicators for topological (crystalline) superconductors to cover all space groups. Our main result is the exhaustive computation of the indicator groups for superconductors under a variety of symmetry settings. We further illustrate the power of this approach by analyzing fourfold symmetric superconductors with or without inversion symmetry and show that the indicators can diagnose topological superconductors with surface states of different dimensionalities or dictate gaplessness in the bulk excitation spectrum.

Electronic structure, doping effect and topological signature in realistic intermetallics Li3-xNaxM (x = 3, 2, 1, 0; M = N, P, As, Sb, Bi)

Topological aspects of electronic structures have received intensive research interest in recent years. Here, we systematically investigate the electronic structure, doping effect and topological signature in a family of realistic compounds Li_3-xNa_xM (x = 3, 2, 1, 0; M = N, P, As, Sb, Bi). Without considering SOC, their electronic band structures show a doubly degenerate nodal line (NL) near the Fermi level in the Γ-A path. In addition, some compounds including Li₂NaN, LiNa₂N, Na₃N and Na₃Bi also exhibit one (or two) pair(s) of triply-degenerate nodal points (TDNPs) in the Γ-A path, locating at both sides of the Γ point. When SOC is taken into account, the band degeneracy of the NLs splits, and the scale of band splitting follows a positive correlation with the atomic weight of the M elements. Due to the band splitting by SOC, most of the Li_3-xNa_xM compounds show a pair of Dirac points (DPs) near the Fermi level. Very interestingly, we find that these DPs possess different types of band dispersions, namely type-I, type-II and the critical-type. The Fermi arcs from the DPs are identified. Our results indicate that Li_3-xNa_xM compounds are good candidates to study the novel properties of NLs, TDNPs, and DPs with different slopes of band dispersions.

Palladium oxide: an excellent topological electronic material with 0-D and 1-D band crossings and definite nontrivial surface states

PdO is an excellent topological semimetal with coexisting 0-D and 1-D band crossings and clear surface states.

Two-dimensional Tunable Dirac/Weyl Semimetal in Non-Abelian Gauge Field

Three-dimensional(3D) Weyl semimetal(WSM) with linear energy spectra has attracted significant interest. Especially they have been observed experimentally in several solid materials with the breaking of inversion symmetry. Here we predict a new family of particle-hole([Formula: see text]) invariant 2D WSMs in the non-Abelian gauge field, which can emerge in the low energy bands being close to Fermi energy (dubbed Weyl-I) and the high energy bands being away from Fermi energy (dubbed Weyl-II), only when the time-reversal symmetry([Formula: see text]) of the 2D Dirac semimetal is broken in the presence of in-plane Zeeman fields. Moreover, a 2D Dirac node can split into a pair of Weyl nodes showing the same Berry phase, and the 2D WSM, being protected by [Formula: see text] symmetry, exhibits four Weyl-I nodes, whose energies are invariant with the variation of the magnetic field. The corresponding Fermi velocity and Berry connection have been calculated. Based on the 2D WSMs, we also examine inhomogeneous pairings of attractive Fermi gases and find a new kind of the LO states with the beat frequency. This 2D WSM provides a realistic and promising platform for exploring and manipulating exotic Weyl physics, which may increase the experimental feasibility in the context of ultracold atoms.

Topological crystalline superconductors in two dimensions

We propose a general framework to classify the topological edge modes in two dimensional topological crystalline superconductors with time-reversal symmetry, which are protected by different crystalline symmetries. The realization of topological crystalline phases is investigated in the superconductors and a criterion for distinguishing topological phases in crystalline superconductor both with mirror and mirror-inversion symmetry is founded. In these topological phases, Majorana edge modes can be identified along the mirror invariant line, while topological edge modes in the AIII class exist along the mirror-inversion invariant line. Furthermore, for the time-reversal invariant superconductor when only the mirror-inversion symmetry is preserving, topological edge modes in the AIII class with nearly flat band emerge in the mirror-inversion parity subspace.

Screening topological materials with a CsCl-type structure in crystallographic databases

CsCl-type materials have many outstanding characteristics, i.e. simple in structure, ease of synthesis and good stability at room temperature, thus are an excellent choice for designing functional materials. Using high-throughput first-principles calculations, a large number of topological semimetals/metals (TMs) were designed from CsCl-type materials found in crystallographic databases and their crystal and electronic structures have been studied. The CsCl-type TMs in this work show rich topological character, ranging from triple nodal points, type-I nodal lines and critical-type nodal lines, to hybrid nodal lines. The TMs identified show clean topological band structures near the Fermi level, which are suitable for experimental investigations and future applications. This work provides a rich data set of TMs with a CsCl-type structure.

Evidence for singular-phonon-induced nematic superconductivity in a topological superconductor candidate Sr0.1Bi2Se3

Superconductivity mediated by phonons is typically conventional, exhibiting a momentum-independent s-wave pairing function, due to the isotropic interactions between electrons and phonons along different crystalline directions. Here, by performing inelastic neutron scattering measurements on a superconducting single crystal of Sr0.1Bi2Se3, a prime candidate for realizing topological superconductivity by doping the topological insulator Bi2Se3, we find that there exist highly anisotropic phonons, with the linewidths of the acoustic phonons increasing substantially at long wavelengths, but only for those along the [001] direction. This observation indicates a large and singular electron-phonon coupling at small momenta, which we propose to give rise to the exotic p-wave nematic superconducting pairing in the MxBi2Se3 (M = Cu, Sr, Nb) superconductor family. Therefore, we show these superconductors to be example systems where electron-phonon interaction can induce more exotic superconducting pairing than the s-wave, consistent with the topological superconductivity.

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.

Majorana Doublets, Flat Bands, and Dirac Nodes in s-Wave Superfluids

Topological superfluids protected by mirror and time-reversal symmetries are exotic states of matter possessing Majorana Kramers pairs (MKPs), yet their realizations have long been hindered by the requirement of unconventional pairing. We propose to realize such a topological superfluid by utilizing s-wave pairing and emergent mirror and time-reversal symmetries in two coupled 1D ultracold atomic Fermi gases with spin-orbit coupling. By stacking such systems into 2D, we discover topological and Dirac-nodal superfluids hosting distinct MKP flat bands. We show that the emergent symmetries make the MKPs and their flat bands stable against pairing fluctuations that otherwise annihilate Majorana pairs. Exploiting new experimental developments, our scheme provides a unique platform for exploring MKPs and their applications in quantum computation.

Ideal Inner Nodal Chain Semimetals in Li2XY (X = Ca, Ba; Y = Si, Ge) Materials

The chain-type nodal loops in the reciprocal space can generate exotic nodal chain fermions. Here, we report that Li₂XY (X = Ca, Ba; Y = Si, Ge) compounds are ideal inner nodal chain semimetals. Their band structures are composed of two connecting nodal loops with either hybrid or type-I band dispersion. The signatures of the nodal chain, such as the nontrivial surface states, are quite pronounced in these Li₂XY compounds, since there is only a single inner nodal chain without other extraneous bands near the Fermi level. These compounds are existing materials and ambient-stable, which is available to realize the experimental detection of inner nodal chain fermions or further the practical applications.

Multifaceted properties of Andreev bound states: interplay of symmetry and topology

Andreev bound states (ABSs) ubiquitously emerge as a consequence of non-trivial topological structures of the order parameter of superfluids and superconductors and significantly contribute to thermodynamics and low-energy quantum transport phenomena. We here share the current status of our knowledge on their multifaceted properties such as Majorana fermions and odd-frequency pairing. A unified concept behind ABSs originates from a soliton state in the one-dimensional Dirac equation with mass domain wall and interplay of ABSs with symmetry and topology enrich their physical characteristics. We make an overview of ABSs with a special focus on superfluid 3He. The quantum liquid confined to restricted geometries serves as a rich repository of noteworthy quantum phenomena, such as the mass acquisition of Majorana fermions driven by spontaneous symmetry breaking, topological quantum criticality, Weyl superfluidity and the anomalous magnetic response. The marriage of the superfluid 3He and nano-fabrication techniques will take one to a new horizon of topological quantum phenomena associated with ABSs.This article is part of the theme issue 'Andreev bound states'.

Realistic Floquet Semimetal with Exotic Topological Linkages between Arbitrarily Many Nodal Loops

Valence and conduction bands in nodal loop semimetals (NLSMs) touch along closed loops in momentum space. If such loops can proliferate and link intricately, NLSMs become exotic topological phases, which require nonlocal hopping and are therefore unrealistic in conventional quantum materials or cold atom systems alike. In this Letter, we show how this hurdle can be surmounted through an experimentally feasible periodic driving scheme. In particular, by tuning the period of a two-step periodic driving or certain experimentally accessible parameters, we can generate arbitrarily many nodal loops that are linked with various levels of complexity. Furthermore, we propose to use both a Berry-phase related winding number and the Alexander polynomial topological invariant to characterize the fascinating linkages among the nodal loops. This Letter thus presents a class of exotic Floquet topological phases that has hitherto not been proposed in any realistic setup. Possible experimental confirmation of such exotic topological phases is also discussed.

Topological Triply Degenerate Points Induced by Spin-Tensor-Momentum Couplings

The recent discovery of triply degenerate points (TDPs) in topological materials has opened a new perspective toward the realization of novel quasiparticles without counterparts in quantum field theory. The emergence of such protected nodes is often attributed to spin-vector-momentum couplings. We show that the interplay between spin-tensor- and spin-vector-momentum couplings can induce three types of TDPs, classified by different monopole charges (C=±2, ±1, 0). A Zeeman field can lift them into Weyl points with distinct numbers and charges. Different TDPs of the same type are connected by intriguing Fermi arcs at surfaces, and transitions between different types are accompanied by level crossings along high-symmetry lines. We further propose an experimental scheme to realize such TDPs in cold-atom optical lattices. Our results provide a framework for studying spin-tensor-momentum coupling-induced TDPs and other exotic quasiparticles.

Topological Nodal Cooper Pairing in Doped Weyl Metals

We generalize the concept of Berry connection of the single-electron band structure to that of a two-particle Cooper pairing state between two Fermi surfaces with opposite Chern numbers. Because of underlying Fermi surface topology, the pairing Berry phase acquires nontrivial monopole structure. Consequently, pairing gap functions have topologically protected nodal structure as vortices in the momentum space with the total vorticity solely determined by the pair monopole charge q_{p}. The nodes of gap function behave as the Weyl-Majorana points of the Bogoliubov-de Gennes pairing Hamiltonian. Their relation with the connection patterns of the surface modes from the Weyl band structure and the Majorana surface modes inside the pairing gap is also discussed. Under the approximation of spherical Fermi surfaces, the pairing symmetry are represented by monopole harmonic functions. The lowest possible pairing channel carries angular momentum number j=|q_{p}|, and the corresponding gap functions are holomorphic or antiholomorphic functions on Fermi surfaces. After projected on the Fermi surfaces with nontrivial topology, all the partial-wave channels of pairing interactions acquire the monopole charge q_{p} independent of concrete pairing mechanism.

Hourglass Dirac chain metal in rhenium dioxide

Nonsymmorphic symmetries, which involve fractional lattice translations, can generate exotic types of fermionic excitations in crystalline materials. Here we propose a topological phase arising from nonsymmorphic symmetries-the hourglass Dirac chain metal, and predict its realization in the rhenium dioxide. We show that ReO2 features hourglass-type dispersion in the bulk electronic structure dictated by its nonsymmorphic space group. Due to time reversal and inversion symmetries, each band has an additional two-fold degeneracy, making the neck crossing-point of the hourglass four-fold degenerate. Remarkably, close to the Fermi level, the neck crossing-point traces out a Dirac chain-a chain of connected four-fold-degenerate Dirac loops-in the momentum space. The symmetry protection, the transformation under symmetry-breaking, and the associated topological surface states of the Dirac chain are revealed. Our results open the door to an unknown class of topological matters, and provide a platform to explore their intriguing physics.

Topological States Characterized by Mirror Winding Numbers in Graphene with Bond Modulation

Localized electrons appear at the zigzag-shaped edge of graphene due to quantum interference. Here we propose a way for harnessing the edge electronic states to make them mobile, by incorporating a topological view point. The manipulation required is to introduce a pattern of strong-weak bonds between neighboring carbon atoms, and to put side by side two graphene sheets with strong-weak alternation conjugating to each other. The electrons with up and down pseudospins propagate in opposite directions at the interface, similar to the prominent quantum spin Hall effect. The system is characterized by a topological index, the mirror winding number, with its root lying in the Su-Schrieffer-Heeger model for polymer. Taking this point of view, one is rewarded by several ways for decorating graphene edge which result in similar mobile electronic states with topological protection. This work demonstrates that celebrated nanotechnology can be used to derive topological states.

The prediction of a family group of two-dimensional node-line semimetals

Using first-principles calculations, we predict a family group of two-dimensional semimetals MX (M = Pd, Pt; X = S, Se, Te), which has a zig-zag type mono-layer structure in the Pmma (no. 41) layer group. Band structure analysis reveals that node-line features are caused by band inversion and the inversion exists even in the absence of spin-orbital-coupling. First-principles calculations show the robust lattice stability of these predicted materials. This work provides the possibility of making a group of novel two-dimensional materials with semimetal features.

Topological superconductors: a review

This review elaborates pedagogically on the fundamental concept, basic theory, expected properties, and materials realizations of topological superconductors. The relation between topological superconductivity and Majorana fermions are explained, and the difference between dispersive Majorana fermions and a localized Majorana zero mode is emphasized. A variety of routes to topological superconductivity are explained with an emphasis on the roles of spin-orbit coupling. Present experimental situations and possible signatures of topological superconductivity are summarized with an emphasis on intrinsic topological superconductors.

Three-dimensional Pentagon Carbon with a genesis of emergent fermions

Carbon, the basic building block of our universe, enjoys a vast number of allotropic structures. Owing to its bonding characteristic, most carbon allotropes possess the motif of hexagonal rings. Here, with first-principles calculations, we discover a new metastable three-dimensional carbon allotrope entirely composed of pentagon rings. The unique structure of this Pentagon Carbon leads to extraordinary electronic properties, making it a cornucopia of emergent topological fermions. Under lattice strain, Pentagon Carbon exhibits topological phase transitions, generating a series of novel quasiparticles, from isospin-1 triplet fermions to triply degenerate fermions and further to Hopf-link Weyl-loop fermions. Its Landau level spectrum also exhibits distinct features, including a huge number of almost degenerate chiral Landau bands, implying pronounced magneto-transport signals. Our work not only discovers a remarkable carbon allotrope with highly rare structural motifs, it also reveals a fascinating hierarchical particle genesis with novel topological fermions beyond the Dirac and Weyl paradigm.

Weyl Exceptional Rings in a Three-Dimensional Dissipative Cold Atomic Gas

Three-dimensional topological Weyl semimetals can generally support a zero-dimensional Weyl point characterized by a quantized Chern number or a one-dimensional Weyl nodal ring characterized by a quantized Berry phase in the momentum space. Here, in a dissipative system with particle gain and loss, we discover a new type of topological ring, dubbed a Weyl exceptional ring consisting of exceptional points at which two eigenstates coalesce. Such a Weyl exceptional ring is characterized by both a quantized Chern number and a quantized Berry phase, which are defined via the Riemann surface. We propose an experimental scheme to realize and measure the Weyl exceptional ring in a dissipative cold atomic gas trapped in an optical lattice.

Three-dimensional Majorana fermions in chiral superconductors

Using a systematic symmetry and topology analysis, we establish that three-dimensional chiral superconductors with strong spin-orbit coupling and odd-parity pairing generically host low-energy nodal quasiparticles that are spin-nondegenerate and realize Majorana fermions in three dimensions. By examining all types of chiral Cooper pairs with total angular momentum J formed by Bloch electrons with angular momentum j in crystals, we obtain a comprehensive classification of gapless Majorana quasiparticles in terms of energy-momentum relation and location on the Fermi surface. We show that the existence of bulk Majorana fermions in the vicinity of spin-selective point nodes is rooted in the nonunitary nature of chiral pairing in spin-orbit-coupled superconductors. We address experimental signatures of Majorana fermions and find that the nuclear magnetic resonance spin relaxation rate is significantly suppressed for nuclear spins polarized along the nodal direction as a consequence of the spin-selective Majorana nature of nodal quasiparticles. Furthermore, Majorana nodes in the bulk have nontrivial topology and imply the presence of Majorana bound states on the surface, which form arcs in momentum space. We conclude by proposing the heavy fermion superconductor PrOs4Sb12 and related materials as promising candidates for nonunitary chiral superconductors hosting three-dimensional Majorana fermions.

Predicted Unusual Magnetoresponse in Type-II Weyl Semimetals

We show several distinct signatures in the magnetoresponse of type-II Weyl semimetals. The energy tilt tends to squeeze the Landau levels (LLs), and, for a type-II Weyl node, there always exists a critical angle between the B field and the tilt, at which the LL spectrum collapses, regardless of the field strength. Before the collapse, signatures also appear in the magneto-optical spectrum, including the invariable presence of intraband peaks, the absence of absorption tails, and the special anisotropic field dependence.

Chiral Spin-Orbital Liquids with Nodal Lines

Strongly correlated materials with strong spin-orbit coupling hold promise for realizing topological phases with fractionalized excitations. Here, we propose a chiral spin-orbital liquid as a stable phase of a realistic model for heavy-element double perovskites. This spin liquid state has Majorana fermion excitations with a gapless spectrum characterized by nodal lines along the edges of the Brillouin zone. We show that the nodal lines are topological defects of a non-Abelian Berry connection and that the system exhibits dispersing surface states. We discuss some experimental signatures of this state and compare them with properties of the spin liquid candidate Ba_{2}YMoO_{6}.

Unified Theory of PT and CP Invariant Topological Metals and Nodal Superconductors

As PT and CP symmetries are fundamental in physics, we establish a unified topological theory of PT and CP invariant metals and nodal superconductors, based on the mathematically rigorous KO theory. Representative models are constructed for all nontrivial topological cases in dimensions d=1, 2, and 3, with their exotic physical meanings being elucidated in detail. Intriguingly, it is found that the topological charges of Fermi surfaces in the bulk determine an exotic direction-dependent distribution of topological subgap modes on the boundaries. Furthermore, by constructing an exact bulk-boundary correspondence, we show that the topological Fermi points of the PT and CP invariant classes can appear as gapless modes on the boundary of topological insulators with a certain type of anisotropic crystalline symmetry.

Weak Topological Insulators and Composite Weyl Semimetals: β-Bi4X4 (X=Br, I)

While strong topological insulators (STIs) were experimentally realized soon after they were theoretically predicted, a weak topological insulator (WTI) has yet to be unambiguously confirmed. A major obstacle is the lack of distinct natural cleavage surfaces to test the surface selective hallmark of a WTI. With a new scheme, we discover that β-Bi4X4 (X=Br, I), dynamically stable or synthesized before, can be a prototype WTI with two natural cleavage surfaces, where two anisotropic Dirac cones stabilize and annihilate, respectively. We further find four surface-state Lifshitz transitions under charge doping and two bulk topological phase transitions under uniaxial strain. Near the WTI-STI transition, there emerges a novel Weyl semimetal phase, in which the Fermi arcs generically appear at both cleavage surfaces whereas the Fermi circle only appears at one selected surface.

Topological nodal-line fermions in spin-orbit metal PbTaSe2

Topological semimetals can support one-dimensional Fermi lines or zero-dimensional Weyl points in momentum space, where the valence and conduction bands touch. While the degeneracy points in Weyl semimetals are robust against any perturbation that preserves translational symmetry, nodal lines require protection by additional crystalline symmetries such as mirror reflection. Here we report, based on a systematic theoretical study and a detailed experimental characterization, the existence of topological nodal-line states in the non-centrosymmetric compound PbTaSe₂ with strong spin-orbit coupling. Remarkably, the spin-orbit nodal lines in PbTaSe₂ are not only protected by the reflection symmetry but also characterized by an integer topological invariant. Our detailed angle-resolved photoemission measurements, first-principles simulations and theoretical topological analysis illustrate the physical mechanism underlying the formation of the topological nodal-line states and associated surface states for the first time, thus paving the way towards exploring the exotic properties of the topological nodal-line fermions in condensed matter systems.

Nodal-line shaped bands appearing near the Fermi level host unique properties in topological matter, which has yet to be confirmed in real materials. Here, the authors report the existence of topological nodal-line states in the non-centrosymmetric single-crystalline spin-orbit semimetal PbTaSe₂.

Structured Weyl Points in Spin-Orbit Coupled Fermionic Superfluids

We demonstrate that a Weyl point, widely examined in 3D Weyl semimetals and superfluids, can develop a pair of nondegenerate gapless spheres. Such a bouquet of two spheres is characterized by three distinct topological invariants of manifolds with full energy gaps, i.e., the Chern number of a 0D point inside one developed sphere, the winding number of a 1D loop around the original Weyl point, and the Chern number of a 2D surface enclosing the whole bouquet. We show that such structured Weyl points can be realized in the superfluid quasiparticle spectrum of a 3D degenerate Fermi gas subject to spin-orbit couplings and Zeeman fields, which supports Fulde-Ferrell superfluids as the ground state.

Topological Superconductivity in Dirac Semimetals

Dirac semimetals host bulk band-touching Dirac points and a surface Fermi loop. We develop a theory of superconducting Dirac semimetals. Establishing a relation between the Dirac points and the surface Fermi loop, we clarify how the nontrivial topology of Dirac semimetals affects their superconducting state. We note that the unique orbital texture of Dirac points and a structural phase transition of the crystal favor symmetry-protected topological superconductivity with a quartet of surface Majorana fermions. We suggest the possible application of our theory to recently discovered superconducting states in Cd_{3}As_{2}.

Fermi points and topological quantum phase transitions in a multi-band superconductor

The importance of models with an exact solution for the study of materials with non-trivial topological properties has been extensively demonstrated. The Kitaev model plays a guiding role in the search for Majorana modes in condensed matter systems. Also, the sp-chain with an anti-symmetric mixing among the s and p bands is a paradigmatic example of a topological insulator with well understood properties. Interestingly, these models share the same universality class for their topological quantum phase transitions. In this work we study a two-band model of spinless fermions with attractive inter-band interactions. We obtain its zero temperature phase diagram, which presents a rich variety of phases including a Weyl superconductor and a topological insulator. The transition from the topological to the trivial superconducting phase has critical exponents different from those of Kitaev's model.

Nanostructured Carbon Allotropes with Weyl-like Loops and Points

Carbon allotropes are subject of intense investigations for their superb structural, electronic, and chemical properties, but not for topological band properties because of the lack of strong spin-orbit coupling (SOC). Here, we show that conjugated p-orbital interactions, common to most carbon allotropes, can in principle produce a new type of topological band structure, forming the so-called Weyl-like semimetal in the absence of SOC. Taking a structurally stable interpenetrated graphene network (IGN) as example, we show, by first-principles calculations and tight-binding modeling, that its Fermi surface is made of two symmetry-protected Weyl-like loops with linear dispersion along perpendicular directions. These loops are reduced to Weyl-like points upon breaking of the inversion symmetry. Because of the topological properties of these band-structure anomalies, remarkably, at a surface terminated by vacuum there emerges a flat band in the loop case and two Fermi arcs in the point case. These topological carbon materials may also find applications in the fields of catalysts.

Topological Imbert-Fedorov Shift in Weyl Semimetals

The Goos-Hänchen (GH) shift and the Imbert-Fedorov (IF) shift are optical phenomena which describe the longitudinal and transverse lateral shifts at the reflection interface, respectively. Here, we predict the GH and IF shifts in Weyl semimetals (WSMs)-a promising material harboring low energy Weyl fermions, a massless fermionic cousin of photons. Our results show that the GH shift in WSMs is valley independent, which is analogous to that discovered in a 2D relativistic material-graphene. However, the IF shift has never been explored in nonoptical systems, and here we show that it is valley dependent. Furthermore, we find that the IF shift actually originates from the topological effect of the system. Experimentally, the topological IF shift can be utilized to characterize the Weyl semimetals, design valleytronic devices of high efficiency, and measure the Berry curvature.