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

Observations of topological surface states provide strong evidence that MoTe2 is a type-II Weyl semimetal, hosting Weyl fermions that have no counterpart in high-energy physics.

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Experimental observation of topological Fermi arcs in type-II Weyl semimetal MoTe2

Materials harbouring exotic quasiparticles, such as massless Dirac and Weyl fermions, have garnered much attention from physics and material science communities due to their exceptional physical properties such as ultra-high mobility and extremely large magnetoresistances. Here, we show that the highly stable, non-toxic and earth-abundant material, ZrSiS, has an electronic band structure that hosts several Dirac cones that form a Fermi surface with a diamond-shaped line of Dirac nodes. We also show that the square Si lattice in ZrSiS is an excellent template for realizing new types of two-dimensional Dirac cones recently predicted by Young and Kane. Finally, we find that the energy range of the linearly dispersed bands is as high as 2 eV above and below the Fermi level; much larger than of other known Dirac materials. This makes ZrSiS a very promising candidate to study Dirac electrons, as well as the properties of lines of Dirac nodes.

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Dirac cone protected by non-symmorphic symmetry and three-dimensional Dirac line node in ZrSiS.

A Dirac nodal-line semimetal phase, which represents a new quantum state of topological materials, has been experimentally realized only in a few systems, including PbTaSe_{2}, PtSn_{4}, and ZrSiS. In this Letter, we report evidence of nodal-line fermions in ZrSiSe and ZrSiTe probed in de Haas-van Alphen quantum oscillations. Although ZrSiSe and ZrSiTe share a similar layered structure with ZrSiS, our studies show the Fermi surface (FS) enclosing a Dirac nodal line has a 2D character in ZrSiTe, in contrast with 3D-like FS in ZrSiSe and ZrSiS. Another important property revealed in our experiment is that the nodal-line fermion density in this family of materials (∼10^{20}  cm^{-3}) is much higher than the Dirac fermion density of other topological materials with discrete nodes. In addition, we have demonstrated ZrSiSe and ZrSiTe single crystals can be thinned down to 2D atomic thin layers through microexfoliation, which offers the first platform to explore exotic properties of topological nodal-line fermions in low dimensions.

Evidence of Topological Nodal-Line Fermions in ZrSiSe and ZrSiTe.

Topological Weyl semimetal (TWS), a new state of quantum matter, has sparked enormous research interest recently. Possessing unique Weyl fermions in the bulk and Fermi arcs on the surface, TWSs offer a rare platform for realizing many exotic physical phenomena. TWSs can be classified into type-I that respect Lorentz symmetry and type-II that do not. Here, we directly visualize the electronic structure of MoTe2, a recently proposed type-II TWS. Using angle-resolved photoemission spectroscopy (ARPES), we unravel the unique surface Fermi arcs, in good agreement with our ab initio calculations that have nontrivial topological nature. Our work not only leads to new understandings of the unusual properties discovered in this family of compounds, but also allows for the further exploration of exotic properties and practical applications of type-II TWSs, as well as the interplay between superconductivity (MoTe2 was discovered to be superconducting recently) and their topological order.

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Signature of type-II Weyl semimetal phase in MoTe2

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.

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Topological nodal-line fermions in spin-orbit metal PbTaSe2.

A Weyl semimetal is a new state of matter that hosts Weyl fermions as emergent quasiparticles. The Weyl fermions correspond to isolated points of bulk band degeneracy, Weyl nodes, which are connected only through the crystal's boundary by exotic Fermi arcs. The length of the Fermi arc gives a measure of the topological strength, because the only way to destroy the Weyl nodes is to annihilate them in pairs in the reciprocal space. To date, Weyl semimetals are only realized in the TaAs class. Here, we propose a tunable Weyl state in Mo(x)W(1-x)Te2 where Weyl nodes are formed by touching points between metallic pockets. We show that the Fermi arc length can be changed as a function of Mo concentration, thus tuning the topological strength. Our results provide an experimentally feasible route to realizing Weyl physics in the layered compound Mo(x)W(1-x)Te2, where non-saturating magneto-resistance and pressure-driven superconductivity have been observed.

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Bao Kai Wang

Papers

Topological nodal-line fermions in spin-orbit metal PbTaSe2.

Prediction of an arc-tunable Weyl Fermion metallic state in MoxW1−xTe2