Weyl and Dirac semimetals are three-dimensional phases of matter with gapless electronic excitations that are protected by topology and symmetry. As three-dimensional analogs of graphene, they have generated much recent interest. Deep connections exist with particle physics models of relativistic chiral fermions, and, despite their gaplessness, to solid-state topological and Chern insulators. Their characteristic electronic properties lead to protected surface states and novel responses to applied electric and magnetic fields. The theoretical foundations of these phases, their proposed realizations in solid-state systems, and recent experiments on candidate materials as well as their relation to other states of matter are reviewed.

Weyl and Dirac semimetals in three-dimensional solids

Three-dimensional (3D) topological Dirac semimetals (TDSs) represent an unusual state of quantum matter that can be viewed as “3D graphene.” In contrast to 2D Dirac fermions in graphene or on the surface of 3D topological insulators, TDSs possess 3D Dirac fermions in the bulk. By investigating the electronic structure of Na 3 Bi with angle-resolved photoemission spectroscopy, we detected 3D Dirac fermions with linear dispersions along all momentum directions. Furthermore, we demonstrated the robustness of 3D Dirac fermions in Na 3 Bi against in situ surface doping. Our results establish Na 3 Bi as a model system for 3D TDSs, which can serve as an ideal platform for the systematic study of quantum phase transitions between rich topological quantum states.

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Discovery of a Three-Dimensional Topological Dirac Semimetal, Na3Bi

We propose a simple realization of the three-dimensional (3D) Weyl semimetal phase, utilizing a multilayer structure, composed of identical thin films of a magnetically doped 3D topological insulator, separated by ordinary-insulator spacer layers. We show that the phase diagram of this system contains a Weyl semimetal phase of the simplest possible kind, with only two Dirac nodes of opposite chirality, separated in momentum space, in its band structure. This Weyl semimetal has a finite anomalous Hall conductivity and chiral edge states and occurs as an intermediate phase between an ordinary insulator and a 3D quantum anomalous Hall insulator. We find that the Weyl semimetal has a nonzero dc conductivity at zero temperature, but Drude weight vanishing as ${T}^{2}$, and is thus an unusual metallic phase, characterized by a finite anomalous Hall conductivity and topologically protected edge states.

Weyl semimetal in a topological insulator multilayer.

Weyl semimetals (WSMs) are topological quantum states wherein the electronic bands disperse linearly around pairs of nodes with fixed chirality, the Weyl points In WSMs, nonorthogonal electric and magnetic fields induce an exotic phenomenon known as the chiral anomaly, resulting in an unconventional negative longitudinal magnetoresistance, the chiral-magnetic effect However, it remains an open question to which extent this effect survives when chirality is not well-defined Here, we establish the detailed Fermi-surface topology of the recently identified WSM TaP via combined angle-resolved quantum-oscillation spectra and band-structure calculations The Fermi surface forms banana-shaped electron and hole pockets surrounding pairs of Weyl points Although this means that chirality is ill-defined in TaP, we observe a large negative longitudinal magnetoresistance We show that the magnetoresistance can be affected by a magnetic field-induced inhomogeneous current distribution inside the sample

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Negative magnetoresistance without well-defined chirality in the Weyl semimetal TaP

Topological insulators and topological semimetals are both new classes of quantum materials, which are characterized by surface states induced by the topology of the bulk band structure. Topological Dirac or Weyl semimetals show linear dispersion around nodes, termed the Dirac or Weyl points, as the three-dimensional analog of graphene. We review the basic concepts and compare these topological states of matter from the materials perspective with a special focus on Weyl semimetals. The TaAs family is the ideal materials class to introduce the signatures of Weyl points in a pedagogical way, from Fermi arcs to the chiral magnetotransport properties, followed by hunting for the type-II Weyl semimetals in WTe2, MoTe2, and related compounds. Many materials are members of big families, and topological properties can be tuned. As one example, we introduce the multifunctional topological materials, Heusler compounds, in which both topological insulators and magnetic Weyl semimetals can be found. Instead of a comp...

Topological Materials: Weyl Semimetals

There has been much interest in pyrochlore iridates A${}_{2}$Ir${}_{2}$O${}_{7}$ where both strong spin-orbital coupling and strong correlation are present. A recent local density approximation calculation [X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Phys. Rev. B 83, 205101 (2011)] suggests that the system is likely in a three-dimensional topological semimetallic phase: a Weyl semimetal. Such a system has zero carrier density and arrives at the quantum limit even in a weak magnetic field. In this paper, we discuss two quantum effects of this system in a magnetic field: a pressure-induced anomalous Hall effect and a magnetic-field-induced charge density wave at the pinned wave vector connecting Weyl nodes with opposite chiralities. A general formula of the anomalous Hall coefficients in a Weyl semimetal is also given. Both proposed effects can be probed by experiments in the near future and can be used to detect the Weyl semimetal phase.

Quantum Hall effects in a Weyl semimetal: Possible application in pyrochlore iridates

We use the variational mean-field approach to systematically study the phase diagram of a bilayer heterostructure of the correlated transition-metal oxide LaNiO${}_{3}$, grown along the (111) direction. The Ni${}^{3+}$ ions with a ${d}^{7}$ (or ${e}_{g}^{1}$) configuration form a buckled honeycomb lattice. We show that, as a function of the strength of the on-site interactions, various topological phases emerge. In the presence of a reasonable size of the Hund's coupling, as the correlation is tuned from intermediate to strong, the following sequence of phases is found: (1) a Dirac half-semimetal phase, (2) a quantum anomalous Hall insulator (QAHI) phase with Chern number one, and (3) a ferromagnetic nematic phase breaking the lattice point-group symmetry. The spin-orbit couplings and magnetism are both dynamically generated in the QAHI phase.

Possible interaction driven topological phases in (111) bilayers of LaNiO3

We investigate the two-dimensional cooperon-fermion model in the correlated regime with a continuous-time diagrammatic determinant quantum Monte Carlo algorithm. We estimate the transition temperature ${T}_{c}$, examine the effectively reduced band gap and cooperon mass, and find that delocalization of the cooperons enhances the diamagnetism. When applied to diamagnetism of the pseudogap phase in high-${T}_{c}$ cuprates, we obtain results in qualitative agreement with recent torque magnetization measurements.

Kai-Yu Yang

Papers

Quantum Hall effects in a Weyl semimetal: Possible application in pyrochlore iridates

Possible interaction driven topological phases in (111) bilayers of LaNiO3

Diagrammatic quantum Monte Carlo solution of the two-dimensional cooperon-fermion model