Discovery of Weyl Fermions, Topological Fermi Arcs and Topological Nodal-Line States of Matter
Author: M. Zahid Hasan
Affiliation: Department of Physics, Princeton University, USA
Link to Hasan Research Group: Laboratory for Topological Quantum Matter & Advanced Spectroscopy >>
The eponymous Dirac equation describes the first synthesis of quantum mechanics and special relativity in describing the nature of electron. Its solutions suggest three distinct forms of relativistic particles - the Dirac, Majorana and Weyl fermions [1-3]. In 1929, Hermann Weyl proposed the simplest version of the equation, whose solution predicted massless fermions with a definite chirality or handedness . Weyl’s equation was intended as a model of elementary articles, but in nearly 86 years, no candidate Weyl fermions have ever been established in high-energy experiments. Neutrinos were once thought to be such particles but later found to possess a small mass. Recently, analogs of the fermion particles have been discovered in certain electronic materials that exhibit strong spin-orbit coupling and topological behavior. Just as Dirac fermions emerge as signatures of topological insulators , researchers have shown that electronic excitations in semimetals such as tantalum or niobium arsenides (TaAs and NbAs) behave like Weyl fermions [5-7]. And such a behavior is consistent with their topological semimetal bandstructures [8,9].
In 1937 physicist Conyers Herring considered under what conditions electronic bands in solids have the same energy by accident in crystals that lack certain symmetries . Near these accidental band touching points, the low-energy excitations, or electronic quasiparticles can be described by an equation that is essentially identical to the 1929 Weyl equation. In recent times, these touching points have been theoretically studied in the context of topological materials and are referred to as Weyl points and the quasiparticles near them are the emergent Weyl fermions . In these solids, the electrons’ quantum-mechanical wave functions acquire a phase, as though they were moving in a superficial magnetic field that is defined in momentum space. In contrast to a real magnetic field, this fictional field (known as a Berry curvature) admits excitations that behave like magnetic monopoles. These monopoles are topological defects or singularities that locate at the Weyl points. So the real space Weyl points are associated with chiral fermions and in momentum space they behave like magnetic monopoles [11-17]. The fact that Weyl nodes are related to magnetic monopoles suggests they will be found in topological materials that are in the vicinity of a topological phase transition [14,15]. The surface of a topological insulator has a Fermi surface that forms a closed loop in momentum space; in a Weyl semimetal, these loops become non-closed arcs as some symmetry is lifted [11,12]. These Fermi arcs terminate at the location of the bulk Weyl points ensuring their topological nature . Theory had suggested that Weyl semimetals should occur in proximity to topological insulators in which inversion or time-reversal symmetry was broken [12,14,16].
Building on these ideas, researchers, including the Princeton University group, used ab initio calculations to predict candidate materials [8,9] and perform angle-resolved photoemission spectroscopy to detect the Fermi arcs, characteristic of Weyl nodes, on the surface of TaAs and NbAs [5-7]. ARPES is an ideal tool for studying such a topological material as known from the extensive body of works on topological insulators . The ARPES technique involves shooting high-energy photons on a material and measuring the energy, momentum and spin of the ejected electrons both from the surface and the bulk. This allows for the explicit determination of both bulk Weyl nodes and the Fermi-arc surface states (Figure 1).
Figure 1: (click on the image to view with high resolution) Weyl fermion and Fermi arcs (a) Schematic of the band structure of a Weyl fermion semimetal. (b) Correspondence of the bulk Weyl fermions to surface Fermi arc states. (c) ARPES mapping of TaAs Fermi surface. (d) Fermi arc surface states and Weyl nodes on the (001) surface of TaAs. (e) Linear dispersion of Weyl quasi-particles in TaAs. (Adapted form Ref. )
In the absence of spin-orbit coupling, the tantalum arsenide material is a nodal-line semimetal in which the bulk Fermi surface is a closed loop in momentum space [8,17,18]. With spin-orbit coupling turned on, the loop-shaped nodal line condenses into discrete Weyl points in momentum space . In this sense the topological nodal-line semimetal can be thought of as a state where the Weyl semimetals originate from by further symmetry breaking (Figure 2). Such a state has been considered in theory previously  but it lacked concrete experimental realizations. Very recently, the first example of a topological nodal-line semimetal in the lead tantalum selenide (PbTaSe2) materials has been reported experimentally . Even though many predictions existed, no concrete experimentally realizable material was found. These findings suggest that Weyl semimetals [5-7] and nodal-line semimetals [17-18] are the first two examples of topological materials that are intrinsically gapless in contrast to topological insulators .
Figure 2: (click on the image to view with high resolution) Topological nodal-line semimetals (a) Schematic of a Weyl semimetal and a topological nodal-line semimetal. (b) ARPES mapping and theoretical simulation of (001)-surface band structure of PbTaSe2 showing the loop-shaped bulk Fermi surface. (c) ARPES spectrum and theoretical band structure along some momentum space directions. (e) Calculated iso-energy band contour showing the nodal line and topological surface states. (Adapted from Ref. )
In the 1980s, Nielsen and Ninomiya suggested that exotic effects, like the ABJ (Adler-Bell-Jackiw) chiral anomaly—in which the combination of an applied electric and magnetic fields generates an excess of quasiparticles with a particular chirality—were associated with Weyl fermions and could be observable in 3D crystals . A further correspondence has been established more recently with the increased understanding of materials with band structures that are topologically protected [11-17]. Unusual transport properties that are associated with Weyl fermions, such as a reduction of the electrical resistance in the presence of an applied magnetic field, have already been reported in the TaAs class of materials [19,20] (Figure 3). Weyl materials can also act as a novel platform for topological superconductivity leading to the realization of Weyl-Majorana modes potentially opening a new pathway for investigating qubit possibilities . Weyl particles have also been observed in photonic (bosonic) crystals. In these systems the number of optical modes has an unusual scaling with the volume of the photonic crystal, which may allow for the construction of large-volume single-mode lasers . Development in the last few months seems to suggest that Weyl particles are indeed associated with a number of unexpected quantum phenomena and these findings may lead to applications in next-generation photonics and electronics.
Figure 3: (click on the image to view with high resolution) Signature of the chiral anomaly in the Weyl fermion semimetal TaAs. (a) Magneto-resistance (MR) data of the Weyl semimetal TaAs in the presence of parallel electric and magnetic fields at T = 2 K. The MR decreases as one increases the magnetic field. (b) MR as a function of angle between the electric and the magnetic fields. The negative magneto-resistance is quickly suppressed as one varies the direction of the magnetic ~B field away from that of the electric ~E field. The observed negative MR and its angular dependence serve as the key signature of the chiral anomaly. (c,d) Landau energy spectra of the left- and right-handed Weyl fermions in the presence of parallel electric and magnetic fields. (Adapted from Ref. )
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