Fractional Quantum Hall Effect in Weyl Semimetals

Abstract

Weyl semimetal may be thought of as a gapless topological phase protected by the chiral anomaly, where the symmetries involved in the anomaly are the U (1) charge conservation and the crystal translational symmetry. The absence of a band gap in a weakly-interacting Weyl semimetal is mandated by the electronic structure topology and is guaranteed as long as the symmetries and the anomaly are intact. The nontrivial topology also manifests in the Fermi arc surface states and topological response, in particular taking the form of an anomalous Hall effect in magnetic Weyl semimetals, whose magnitude is only determined by the location of the Weyl nodes in the Brillouin zone. Here we consider the situation when the interactions are not weak and ask whether it is possible to open a gap in a magnetic Weyl semimetal while preserving its nontrivial electronic structure topology along with the translational and the charge conservation symmetries. Surprisingly, the answer turns out to be yes. The resulting topologically ordered state provides a nontrivial realization of the fractional quantum Hall effect in three spatial dimensions in the absence of an external magnetic field, which cannot be viewed as a stack of two dimensional states. Our state contains loop excitations with nontrivial braiding statistics when linked with lattice dislocations.Weyl semimetal is the first example of a bulk gapless topological phase [1][2][3][4]. The gaplessness of the bulk electronic structure in Weyl semimetals is mandated by topology: there exist closed surfaces in momentum space, which carry nonzero Chern numbers (flux of Berry curvature through the surface), which makes the presence of a band-touching point inside the Brillouin zone (BZ) volume, enclosed by the surface, inevitable. This picture, however, relies on separation between the individual Weyl nodes in momentum space, which involves symmetry considerations. In particular, either inversion or time reversal (TR) symmetry need to be violated in order for the Weyl nodes to be separated. In addition, crystal translational symmetry needs to be present, since otherwise even separated Weyl nodes may be hybridized and gapped out.A very useful viewpoint on topology-mandated gaplessness is provided by the concept of quantum anomalies. The best known example of this is the gapless surface states of three dimensional (3D) TR-invariant topological insulator (TI). The relevant anomaly in this case is the parity anomaly: the θ-term topological response of the bulk 3D TI [5] violates TR (and parity) when evaluated in a sample with a boundary. This anomaly of the bulk response must be cancelled by the corresponding anomaly of the gapless surface state [6], which is simply the parity anomaly of the massless 2D Dirac fermion [7][8][9].Analogously, the gaplessness of the bulk spectrum in Weyl semimetals may be related to the chiral anomaly [10,11]. Suppose we have a magnetic Weyl semimetal with two band-touching nodes, located at k ± = ±Q = ±Qẑ. Crystal translations in the z-direction act on the low-...

Torsional Landau levels and geometric anomalies in condensed matter Weyl systems

“…More importantly, the axial anomaly can then arise from the momentum space structure itself since one has to shift by → e ∓ip W •x W to obtain the low-energy Weyl excitation W . While this gives almost the same Hamiltonian and spectrum as for of U(1) fields, the momentum dependence is crucial in the case of the protected Fermi arcs [19], anomalous quantum Hall effect [47,56], as well as the torsional anomaly with the tetrads analyzed in the main text.…”

“…In the presence of elastic deformations corresponding to torsion, i.e., phonons, the anomalous chiral symmetry corresponding to translations is manifested as the nonconservation of (lattice) node momentum flux k W e j μ 5 between the Weyl fermions and the background phonons [38,47], as found in superfluid 3 He-A for the (p + ip)-wave paired Fermi liquid [3]. See also [16,76,78,109].…”

“…In three spatial dimensions Weyl fermions experience chiral anomalies [46]. Here, the anomalous chiral symmetries leading to axial anomalies in the system [38,47] correspond to translational symmetries in momentum space. For relevant condensed matter considerations of the chiral anomaly, see, e.g., [3,9,11,[48][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63].…”

Weyl fermions, Fermi arcs, and minority-spin carriers in ferromagnetic CoS 2

“…Here, we reveal that CoS 2 , a material that has been long studied because of its itinerant ferromagnetism and potential for half-metallicity, actually hosts Weyl fermions and Fermi arc surface states in its band structure close to the Fermi level E F , as well as topological nodal lines below the Fermi level. CoS 2 is, therefore, a rare example of the recently discovered class of experimentally verified magnetic topological metals (8)(9)(10), which have been proposed to realize new mechanisms of spin-to-charge conversion (11), and are of broader interest for fundamental science, for instance, as a platform to realize axion insulators (7), the intrinsic anomalous Hall effect (12), or the anomalous fractional quantum Hall effect (13).…”

Chiral anomalies induced transport in Weyl metals in quantizing magnetic field

“…Weyl metals (WMs) host massless chiral relativistic quasiparticles which show very interesting and novel phenomena [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. One of the most interesting aspects of massless relativistic chiral fluids, in quantum field theory is the breakdown of the chiral gauge symmetry in the presence of an external electromagnetic field [16][17][18].…”