Threes company

Abstract

Enabled by recent advances in symmetry and electronic structure, researchers have observed signatures of unconventional threefold degeneracies in tungsten carbide, challenging a longstanding paradigm in nodal semimetals.Take a single layer of graphite and you get graphene, a material whose structural and electronic properties allow diverse applications ranging from biosensing to electrical engineering. Try and explain graphene's properties using solid state physics, and you get an equation similar to one otherwise seen in discussions of cosmology and colliders: the Dirac Equation. In the decade following graphene's discovery, all materials of this kind, known as 'nodal semimetals', were named and categorized assuming a one-to-one correspondence between the low-energy electronic behavior of crystalline solids and the equations of high-energy particle physics, leading to the classification of a wide variety of chemical compounds as Dirac or Weyl semimetals. Now, writing in Nature Physics, Jun-Zhang Ma and colleagues [1] present experimental evidence in tungsten carbide of a semimetal that escapes this paradigm.This assumption of one-to-one correspondence was first upended in early 2016 by a pair of papers [2,3] noting that, in fact, plenty of nodal semimetals are instead governed by equations decidedly unlike those in high energy physics. These 'unconventional' semimetals, as characterized in a flood of subsequent theoretical proposals, feature electronic states twisted into loops, chains, and hourglasses, or meeting in unexpected multiples of three. Materials candidates accompanying these proposals were identified so readily that one might even question how exotic unconventional semimetals really are. Rather, it was possible that we had all just been a bit spoiled by the natural simplicity of graphene. By observing electronically relevant threefold nodal degeneracies and surface Fermi arcs in tungsten carbide [1], Ma et al. provide some of the first experimental support for the growing body of imaginative proposals for unconventional semimetals with topological electronic character.In both particle physics and nodal semimetals, the possible energies of a particle or quasiparticle are determined by its momentum. This is known as a dispersion relation. When a particle has no mass, the solutions of its dispersion relation come together and meet in a degenerate nodal point. In high-energy physics, the structure of this dispersion relation and the degree of its degeneracy are determined by the fundamental symmetries of nature, such as charge conjugation, parity inversion, and time reversal. Conversely, as recognized since the 1970's [4], the dispersion relation of a crystalline solid is instead more accurately described by the mathematical representations of its spatial symmetries. With the advent of computers powerful enough to perform large-scale calculations of the electronic structures of real materials, researchers have only recently begun to link this abstract mathematical characterization to nodal points in kno...

Topological Semimetals from First Principles

“…A symmetry-protected crossing of three energy bands can also occur at a (generic) point on a high-symmetry line, provided the little group of the high-symmetry line admits both one-and two-dimensional (double) group representations. This is true for threefold rotation axes with little group C 3v and threefold band crossings of this kind have been identified as "triple-point" semimetals (14,15,16,17).…”

“…Another possible distinction can be made based on the origin of the crossing, that is to say, whether it is symmetry-enforced or arises as a result of a band inversion. These attributes, in combination with their topological characteristics, have led to the identification of a growing number of different TSM families, which include Dirac and Weyl semimetals (4,5,6,7), nodal line semimetals (8,9,10), type-I and type-II semimetals (11), multifold fermion semimetals (12,13), and "triple-point" semimetals (14,15,16,17), among others.…”

Transport of Topological Semimetals

“…Here we take a particular type of three-band crossing as an example (33)(34)(35)(36)(168)(169)(170). In such a triple-point semimetal, three singly degenerate bands cross at discrete points, the triple points (Figure 1d).…”

All Topological Bands of All Stoichiometric Materials

“…Typically, the bulk Fermi pockets of ES materials are characterized by fourfold-degenerate, linearly dispersing Dirac fermions [6,156,197,198]. However, it is also possible for ES materials to host chiral fermions with larger Chern numbers [199][200][201][202][203], threefold-degenerate "nexus" fermions [204][205][206][207][208][209], or more exotic mixtures of chiral and achiral fermions [210]. Almost all of the materials shown in Figs.…”