The notion of topological phases has been extended to higher-order and has been generalized to different dimensions. As a paradigm, Cd3As2 is predicted to be a higher-order topological semimetal, possessing three-dimensional (3D) bulk Dirac fermions, two-dimensional (2D) Fermi arcs, and one-dimensional (1D) hinge states. These topological states have different characteristic length scales in electronic transport, allowing to distinguish their properties when changing sample size. Here, we report an anomalous dimensional reduction of supercurrent transport by increasing the size of Dirac semimetal Cd3As2-based Josephson junctions. An evolution of the supercurrent quantum interferences from a standard Fraunhofer pattern to a superconducting quantum interference device (SQUID)-like one is observed when the junction channel length is increased. The SQUID-like interference pattern indicates the supercurrent flowing through the 1D hinges. The identification of 1D hinge states should be valuable for deeper understanding the higher-order topological phase in a 3D Dirac semimetal.
We theoretically predict proximity-induced odd-frequency (odd-$ω$) pairing on the surface of a Weyl nodal loop semimetal, characterized by a nodal loop Fermi surface and drumhead-like surface states (DSSs), attached to conventional spin-singlet $s$-wave superconducting leads. Due to the complete spin polarization of the DSS, odd-$ω$ equal-spin triplet pairing is present, and we show that it gives rise to a finite Josephson current. Placing an additional ferromagnet in the junction can also generate odd-$ω$ mixed-spin triplet pairing, but the pairing and current is not affected if the magnetiza…
Dirac semimetals (DSMs) are an important class of topological states of matter. Here, focusing on DSMs of band inversion type, we investigate their boundary modes from the effective model perspective. We show that in order to properly capture the boundary modes, $k$-cubic terms must be included in the effective model, which would drive an evolution of surface degeneracy manifold from a nodal line to a nodal point. Using first-principles calculations, we demonstrate that this feature and the topological hinge modes can be clearly exhibited in $β$-CuI. We further extend the discussion to magneti…
The existence of $1/2$ modes in the entanglement spectrum (ES) has been shown to be a powerful quantum-informative signature of boundary states of gapped topological phases of matter, e.g., topological insulators and topological superconductors, where the finite bulk gap allows us to establish a crystal-clear correspondence between $1/2$ modes and boundary states. Here we investigate the recently proposed higher-order Weyl semimetals (HOWSM), where bulk supports gapless higher-order Weyl nodes and boundary supports hinge arcs and Fermi arcs. We find that the aim of unambiguously identifying hi…
One-dimensional (1D) gapless hinge states are predicated in the three-dimensional (3D) higher-order topological insulators and topological semimetals, because of the higher-order bulk-boundary correspondence. Nevertheless, the topologically protected property of the hinge states is still not demonstrated so far, because it is not accessible by conventional methods, such as spectroscopy experiments and quantum oscillations. Here, we reveal the topological nature of hinge states in the higher-order topological semimetal Cd3As2 nanoplate through spin potentiometric measurements. The results of cu…
Transition metal dichalcogenides XTe$_{2}$ (X=Mo,W) have been shown to be second-order topological insulators based on first-principles calculations, while topological hinge states have been shown to emerge based on the associated tight-binding model. The model is equivalent to the one constructed from a loop-nodal semimetal by adding mass terms and spin-orbit interactions. We propose to study a chiral-symmetric model obtained from the original Hamiltonian by simplifying it but keeping almost identical band structures and topological hinge states. A merit is that we are able to derive various …
We investigate higher-order Weyl semimetals (HOWSMs) having bulk Weyl nodes attached to both surface and hinge Fermi arcs. We identify a new type of Weyl node, that we dub a $2nd$ order Weyl node, that can be identified as a transition in momentum space in which both the Chern number and a higher order topological invariant change. As a proof of concept we use a model of stacked higher order quadrupole insulators to identify three types of WSM phases: $1st$-order, $2nd$-order, and hybrid-order. The model can also realize type-II and hybrid-tilt WSMs with various surface and hinge arcs. Moreove…
Higher-order topology yields intriguing multidimensional topological phenomena, while Weyl semimetals have unconventional properties such as chiral anomaly. However, so far, Weyl physics remain disconnected with higher-order topology. Here, we report the theoretical discovery of higher-order Weyl points and thereby the establishment of such an important connection. We demonstrate that higher-order Weyl points can emerge in chiral materials such as chiral tetragonal crystals as the intermediate phase between the conventional Weyl semimetal and 3D higher-order topological phases. Higher-order We…
Weyl semimetals are extraordinary systems where exotic phenomena such as Fermi arcs, pseudo-gauge fields and quantum anomalies arise from topological band degeneracy in crystalline solids for electrons and metamaterials for photons and phonons. On the other hand, higher-order topological insulators unveil intriguing multidimensional topological physics beyond the conventional bulk-edge correspondences. However, it is unclear whether higher-order topology can emerge in Weyl semimetals. Here, we report the experimental discovery of higher-order Weyl semimetals in its phononic analog which exhibi…
Bulk-boundary correspondence has achieved a great success in the identification of topological states. However, this elegant strategy doesn't apply to the Dirac semimetals (DSMs). Here, we propose that kagome lattices Pd$_3$Pb$_2$X$_2$ (X = S, Se) are unique type-I DSMs without surface Fermi arc states, which are different from the previous well-known DSMs, such as Na$_3$Bi and Cd$_3$As$_2$. Pd$_3$Pb$_2$X$_2$ are characterized by nontrivial topological invariant Z$_3$, guaranteeing a higher-order bulk-hinge correspondence and the existence of higher-order Fermi arcs, as well as fractional corn…
In this article we study 3D non-Hermitian higher-order Dirac semimetals (NHHODSMs). Our focus is on $C_4$-symmetric non-Hermitian systems where we investigate inversion ($\mathcal{I}$) or time-reversal ($\mathcal{T}$) symmetric models of NHHODSMs having real bulk spectra. We show that they exhibit the striking property that the bulk and surfaces are anti-PT and PT symmetric, respectively, and so belong to two different topological classes realizing a novel non-Hermitian topological phase which we call a \emph{hybrid-PT topological phases}. Interestingly, while the bulk spectrum is still fully …
We study the transport properties of a Josephson junction consisting of two identical $s$-wave superconductors separated by an even-layer ${\mathrm{MnBi}}_{2}{\mathrm{Te}}_{4}$ (MBT). Using recursive Green's function method, we calculate the supercurrent in the presence of a perpendicular magnetic field and find that its quantum interference exhibits distinct patterns when the MBT is in different magnetic states. In the antiferromagnetic state, the MBT is an axion insulator supporting an extended "hinge" supercurrent, which leads to a sinusoidal interference pattern decaying with the field str…
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 ge…
Three-dimensional quantum anomalous Hall effect in Weyl semimetals
Fudan University（复旦大学）；Soochow University（苏州大学）；Nankai University（南开大学）；Beijing Academy of Quantum Information Sciences（北京量子信息科学研究院）；Peking University（北京大学）
The real Chern insulator state, featuring nontrivial real Chern number and second-order boundary modes, has been revealed in a few two-dimensional systems. The concept can be extended to three dimensions (3D), but a proper material realization is still lacking. Here, based on first-principles calculations and theoretical analysis, we identify the recently synthesized bulk $γ$-graphyne as a 3D real Chern insulator. Its nontrivial bulk topology leads to topological hinge modes spreading across the 1D edge Brillouin zone. Under compression of the interlayer distance, the system can undergo a topo…
Much of the dramatic growth in research on topological materials has focused on topologically protected surface states. While the domain walls of topological materials such as Weyl semimetals with broken inversion or time-reversal symmetry can provide a hunting ground for exploring topological interfacial states, such investigations have received little attention to date. Here, utilizing in-situ cryogenic transmission electron microscopy combined with first-principles calculations, we discover intriguing domain-wall structures in MoTe2, both between polar variants of the low-temperature(T) Wey…
Most topological insulators discovered today in spinful systems can be transformed from topological semimetals (TSMs) with vanishing bulk gap via introducing the spin-orbit coupling (SOC), which manifests the intrinsic links between the gapped TI phases and the gapless TSMs. Recently, we have proposed a new family of TSMs in time-reversal invariant {\it spinless} systems, which host butterfly-like nodal-lines (NLs) consisting of a pair of identical concentric intersecting coplanar ellipses (CICE). In this Communication, we unveil the intrinsic link between this exotic class of nodal-line semim…
Although the richness of spatial symmetries has led to a rapidly expanding inventory of possible topological crystalline (TC) phases of electrons, physical realizations have been slow to materialize due to the practical difficulty to ascertaining band topology in realistic calculations. Here, we integrate the recently established theory of symmetry indicators of band topology into first-principle band-structure calculations, and test it on a databases of previously synthesized crystals. The combined algorithm is found to efficiently unearth topological materials and predict topological propert…
Weyl semimetals for their exotic topological properties have drawn considerable attention in many research fields. When in combination with s-wave superconductors, the supercurrent can be carried by their topological surface channels, forming junctions mimic the behavior of Majorana bound states. Here, we present a transmon-like superconducting quantum intereference device (SQUID) consists of lateral junctions made of Weyl semimetal Td-MoTe2 and superconducting leads niobium nitride (NbN). The SQUID is coupled to a readout cavity made of molybdenum rhenium (MoRe), whose response at high power …
We study non-Hermitian higher-order Weyl semimetals (NHHOWSMs) possessing real spectra and having inversion $\mathcal{I}$ ($\mathcal{I}$-NHHOWSM) or time-reversal symmetry $\mathcal{T}$ ($\mathcal{T}$-NHHOWSM). When the reality of bulk spectra is lost, the NHHOWSMs exhibit various configurations of surface Fermi Arcs (FAs) and Exceptional Fermi Rings (EFRs), providing a setup to investigate them on an equal footing. The EFRs only appear in the region between 2nd-order WNs. We also discover Weyl nodes originating from non-Hermicity, called non-Hermitian Weyl nodes (NHWNs). Remarkably, we find T…
We demonstrate a topological classification of vortices in three dimensional time-reversal invariant topological superconductors based on superconducting Dirac semimetals with an s-wave superconducting order parameter by means of a pair of numbers $(N_Φ,N)$, accounting how many units $N_Φ$ of magnetic fluxes $hc/4e$ and how many $N$ chiral Majorana modes the vortex carries. From these quantities, we introduce a topological invariant which further classifies the properties of such vortices under linking processes. While such processes are known to be related to instanton processes in a field th…
Recently, the notion of topological phases of matter has been extended to higher-order incarnations, supporting gapless modes on even lower dimensional boundaries, such as corners and hinges. We here identify a collection of cubic spin-3/2 fermions with biquadratic touching of Kramers degenerate valence and conduction bands as a platform to strain-engineer higher-order topological (HOT) phases: external uniaxial strain gives birth to a HOT Dirac semimetal or an insulator, depending on its sign, featuring topological \emph{hinge} modes in the strain direction. The insulator in fact exhibits \em…
In recent years, transition metal dichalcogenides (TMDs) have garnered great interest as topological materials -- monolayers of centrosymmetric $β$-phase TMDs have been identified as 2D topological insulators (TIs), and bulk crystals of noncentrosymmetric $γ$-phase MoTe$_2$ and WTe$_2$ have been identified as type-II Weyl semimetals. However, ARPES and STM probes of these TMDs have revealed huge, "arc-like" surface states that overwhelm, and are sometimes mistaken for, the much smaller topological surface Fermi arcs of bulk type-II Weyl points. In this letter, we use first-principles calculati…
The mathematical field of topology has become a framework to describe the low-energy electronic structure of crystalline solids. A typical feature of a bulk insulating three-dimensional topological crystal are conducting two-dimensional surface states. This constitutes the topological bulk-boundary correspondence. Here, we establish that the electronic structure of bismuth, an element consistently described as bulk topologically trivial, is in fact topological and follows a generalized bulk-boundary correspondence of higher-order: not the surfaces of the crystal, but its hinges host topologica…
Triple nodal points are degeneracies of energy bands in momentum space at which three Hamiltonian eigenstates coalesce at a single eigenenergy. For spinless particles, the stability of a triple nodal point requires two ingredients: rotation symmetry of order three, four or six; combined with mirror or space-time-inversion symmetry. However, despite ample studies of their classification, robust boundary signatures of triple nodal points have until now remained elusive. In this work, we first show that pairs of triple nodal points in semimetals and metals can be characterized by Stiefel-Whitney …
