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Dirac semimetals lack a simple bulk-boundary correspondence. Recently, Dirac materials with four-fold rotation symmetry have been shown to exhibit a higher order bulk-hinge correspondence: they display higher order Fermi arcs, which are localized on hinges where two surfaces meet and connect the projections of the bulk Dirac points. In this paper, we classify higher order Fermi arcs for Dirac semimetals protected by a rotation symmetry and the product of time-reversal and inversion. Such Dirac points can be either linear in all directions or linear along the rotation axis and quadratic in other directions. By computing the filling anomaly for momentum-space planes on either side of the Dirac point, we find that all linear Dirac points exhibit higher order Fermi arcs terminating at the projection of the Dirac point, while the Dirac points that are quadratic in two directions lack such higher order Fermi arcs. When higher order Fermi arcs do exist, they obey either a $mathbb{Z}_2$ (four-fold rotation axis) or $mathbb{Z}_3$ (three- or six-fold rotation axis) group structure. Finally, we build two models with six-fold symmetry to illustrate the cases with and without higher order Fermi arcs. We predict higher order Fermi arcs in Na$_3$Bi.
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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$_2$.
We theoretically study the topological robustness of the surface physics induced by Weyl Fermi-arc surface states in the presence of short-ranged quenched disorder and surface-bulk hybridization. This is investigated with numerically exact calculations on a lattice model exhibiting Weyl Fermi-arcs. We find that the Fermi-arc surface states, in addition to having a finite lifetime from disorder broadening, hybridize with nonperturbative bulk rare states making them no longer bound to the surface (i.e. they lose their purely surface spectral character). Thus, we provide strong numerical evidence that the Weyl Fermi-arcs are not topologically protected from disorder. Nonetheless, the surface chiral velocity is robust and survives in the presence of strong disorder, persisting all the way to the Anderson-localized phase by forming localized current loops that live within the localization length of the surface. Thus, the Weyl semimetal is not topologically robust to the presence of disorder, but the surface chiral velocity is.
Van Hove points are special points in the energy dispersion, where the density of states exhibits analytic singularities. When a Van Hove point is close to the Fermi level, tendencies towards density wave orders, Pomeranchuk orders, and superconductivity can all be enhanced, often in more than one channel, leading to a competition between different orders and unconventional ground states. Here we consider the effects from higher-order Van Hove points, around which the dispersion is flatter than near a conventional Van Hove point, and the density of states has a power-law divergence. We argue that such points are present in intercalated graphene and other materials. We use an effective low-energy model for electrons near higher-order Van Hove points and analyze the competition between different ordering tendencies using an unbiased renormalization group approach. For purely repulsive interactions, we find that two key competitors are ferromagnetism and chiral superconductivity. For a small attractive exchange interaction, we find a new type of spin Pomeranchuk order, in which the spin order parameter winds around the Fermi surface. The supermetal state, predicted for a single higher-order Van Hove point, is an unstable fixed point in our case.
We present an exactly solvable spin-3/2 model defined on a pentacoordinated three-dimensional graphite lattice, which realizes a novel quantum spin liquid with second-order topology. The exact solutions are described by Majorana fermions coupled to a background $mathbb{Z}_2$ gauge field, whose ground-state flux configuration gives rise to an emergent off-centered spacetime inversion symmetry. The symmetry protects topologically nontrivial band structures for the Majorana fermions, particularly including nodal-line semimetal phases with twofold topological charges: the second Stiefel-Whitney number and the quantized Berry phase. The former leads to rich topological phenomena on the system boundaries. There are two nodal-line semimetal phases hosting hinge Fermi arcs located on different hinges, and they are separated by a critical Dirac semimetal state with surface helical Fermi arcs. In addition, we show that rich symmetry/topology can be explored in our model by simply varying the lattice or interaction arrangement. As an example, we discuss how to achieve a topological gapped phase with surface Dirac points.
We use tunable, vacuum ultraviolet laser-based angle-resolved photoemission spectroscopy and density functional theory calculations to study the electronic properties of Dirac semimetal candidate cubic PtBi${}_{2}$. In addition to bulk electronic states we also find surface states in PtBi${}_{2}$ which is expected as PtBi${}_{2}$ was theoretical predicated to be a candidate Dirac semimetal. The surface states are also well reproduced from DFT band calculations. Interestingly, the topological surface states form Fermi contours rather than double Fermi arcs that were observed in Na$_3$Bi. The surface bands forming the Fermi contours merge with bulk bands in proximity of the Dirac points projections, as expected. Our data confirms existence of Dirac states in PtBi${}_{2}$ and reveals the fragility of the Fermi arcs in Dirac semimetals. Because the Fermi arcs are not topologically protected in general, they can be deformed into Fermi contours, as proposed by [Kargarian {it et al.}, PNAS textbf{113}, 8648 (2016)]. Our results demonstrate validity of this theory in PtBi${}_{2}$.
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