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-...

We have investigated the interplay between band inversion and size quantization in spherically shaped nanoparticles made from topological-insulator (TI) materials. A general theoretical framework is developed based on a versatile continuum-model description of the TI bulk band structure and the assumption of a hard-wall mass confinement. Analytical results are obtained for the wave functions of single-electron energy eigenstates and the matrix elements for optical transitions between them. As expected from spherical symmetry, quantized levels in TI nanoparticles can be labeled by quantum numbers j and m = −j, −j + 1, . . . , j for total angular momentum and its projection on an arbitrary axis. The fact that TIs are narrow-gap materials, where the charge-carrier dynamics is described by a type of two-flavor Dirac model, requires j to assume half-integer values and also causes a doubling of energy-level degeneracy where two different classes of states are distinguished by being parity eigenstates with eigenvalues (−1) j∓1/2 . The existence of energy eigenstates having the same j but opposite parity enables optical transitions where j is conserved, in addition to those adhering to the familiar selection rule where j changes by ±1. All optical transitions satisfy the usual selection rule ∆m = 0, ±1. We treat intra-and inter-band optical transitions on the same footing and establish ways for observing unusual quantum-size effects in TI nanoparticles, including oscillatory dependences of the band gap and of transition amplitudes on the nanoparticle radius. Our theory also provides a unified perspective on multi-band models for charge carriers in semiconductors and Dirac fermions from elementary-particle physics.

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