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We introduce higher-order topological Dirac superconductor (HOTDSC) as a new gapless topological phase of matter in three dimensions, which extends the notion of Dirac phase to a higher-order topological version. Topologically distinct from the traditional topological superconductors and known Dirac superconductors, a HOTDSC features helical Majorana hinge modes between adjacent surfaces, which are direct consequences of the symmetry-protected higher-order band topology manifesting in the system. Specifically, we show that rotational, spatial inversion, and time-reversal symmetries together protect the coexistence of bulk Dirac nodes and hinge Majorana modes in a seamless way. We define a set of topological indices that fully characterizes the HOTDSC. We further show that a practical way to realize the HOTDSC phase is to introduce unconventional odd-parity pairing to a three-dimensional Dirac semimetal while preserving the necessary symmetries. As a concrete demonstration of our idea, we construct a corresponding minimal lattice model for HOTDSC obeying the symmetry constraints. Our model exhibits the expected topological invariants in the bulk and the defining spectroscopic features on an open geometry, as we explicitly verify both analytically and numerically. Remarkably, the HOTDSC phase offers an example of a higher-order topological quantum critical point, which enables realizations of various higher-order topological phases under different symmetry-breaking patterns. In particular, by breaking the inversion symmetry of a HOTDSC, we arrive at a higher-order Weyl superconductor, which is yet another new gapless topological state that exhibits hybrid higher-order topology.
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Weyl superconductors feature Weyl points at zero energy in the three-dimensional (3D) Brillouin zone and arc states that connect the projections of these Weyl points on the surface. We report that higher-order Weyl superconductors can be realized in odd-parity topological superconductors with time-reversal symmetry being broken by periodic driving. Different from conventional Weyl points, the higher-order Weyl points in the bulk separate 2D first- and second-order topological phases, while on the surface, their projections are connected not only by conventional surface Majorana arcs, but also by hinge Majorana arcs. We show that the Weyl-point connectivity via Majorana arcs is largely enriched by the underlying higher-order topology and becomes anisotropic with respect to surface orientations. We identify the anisotropic Weyl-point connectivity as a characteristic feature of higher-order Weyl materials. As each 2D subsystem can be singled out by fixing the periodic driving, we propose how the Majorana zero modes in the 2D higher-order topological phases can be detected and manipulated in experiments.
We identify three-dimensional higher-order superconductors characterized by the coexistence of one-dimensional Majorana hinge states and gapless surface sates. We show how such superconductors can be obtained starting from the model of a spinful quadrupolar semimetal with two orbitals and adding an s-wave superconducting pairing term. By considering all the possible s-wave pairings satisfying Fermi-Dirac statistics we obtain six different superconducting models. We find that for two of these models a flat-band of hinge Majorana states coexist with surface states, and that these models have a non-vanishing quadrupole-like topological invariant. Two of the other models, in the presence of a Zeeman term, exhibit helical and dispersive hinge states localized only at two of the four hinges. We find that these states are protected by combinations of rotation and mirror symmetries, and that the pair of corners exhibiting hinge states switches upon changing the sign of the Zeeman term. Furthermore, we show that these states can be localized to a single hinge with suitable perturbations. The remaining two models retain gapless bulk and surface states that spectroscopically obscure any possible hinge states.
Dirac and Weyl semimetals both exhibit arc-like surface states. However, whereas the surface Fermi arcs in Weyl semimetals are topological consequences of the Weyl points themselves, the surface Fermi arcs in Dirac semimetals are not directly related to the bulk Dirac points, raising the question of whether there exists a topological bulk-boundary correspondence for Dirac semimetals. In this work, we discover that strong and fragile topological Dirac semimetals exhibit 1D higher-order hinge Fermi arcs (HOFAs) as universal, direct consequences of their bulk 3D Dirac points. To predict HOFAs coexisting with topological surface states in solid-state Dirac semimetals, we introduce and layer a spinful model of an $s-d$-hybridized quadrupole insulator (QI). We develop a rigorous nested Jackiw-Rebbi formulation of QIs and HOFA states. Employing $ab initio$ calculations, we demonstrate HOFAs in both the room- ($alpha$) and intermediate-temperature ($alpha$) phases of Cd$_{3}$As$_2$, KMgBi, and rutile-structure ($beta$-) PtO$_2$.
Three-dimensional topological (crystalline) insulators are materials with an insulating bulk, but conducting surface states which are topologically protected by time-reversal (or spatial) symmetries. Here, we extend the notion of three-dimensional topological insulators to systems that host no gapless surface states, but exhibit topologically protected gapless hinge states. Their topological character is protected by spatio-temporal symmetries, of which we present two cases: (1) Chiral higher-order topological insulators protected by the combination of time-reversal and a four-fold rotation symmetry. Their hinge states are chiral modes and the bulk topology is $mathbb{Z}_2$-classified. (2) Helical higher-order topological insulators protected by time-reversal and mirror symmetries. Their hinge states come in Kramers pairs and the bulk topology is $mathbb{Z}$-classified. We provide the topological invariants for both cases. Furthermore we show that SnTe as well as surface-modified Bi$_2$TeI, BiSe, and BiTe are helical higher-order topological insulators and propose a realistic experimental setup to detect the hinge states.
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 real, we find that exceptional Fermi-rings (EFRs) appear connecting the two Dirac nodes on the surface. This provides a route to probe and utilize both the bulk Dirac physics and exceptional rings/points on equal footing. Moreover, particularly for $mathcal{T}$-NHHODSMs, we also find real hinge-arcs connecting the surface EFRs. We show that this higher-order topology can be characterized using a biorthogonal real-space formula of the quadrupole moment. Furthermore, by applying Hermitian $C_4$-symmetric perturbations, we discover various novel phases, particularly: (i) an intrinsic $mathcal{I}$-NHHODSM having hinge arcs and gapped surfaces, and (ii) a novel $mathcal{T}$-symmetric skin-topological HODSM which possesses both topological and skin hinge modes. The interplay between non-Hermition and higher-order topology in this work paves the way toward uncovering rich phenomena and hybrid functionality that can be readily realized in experiment.
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