We propose a simple realization of the three-dimensional (3D) Weyl semimetal phase, utilizing a multilayer structure, composed of identical thin films of a magnetically-doped 3D topological insulator (TI), separated by ordinary-insulator spacer layers. We show that the phase diagram of this system contains a Weyl semimetal phase of the simplest possible kind, with only two Dirac nodes of opposite chirality, separated in momentum space, in its bandstructure. This Weyl semimetal has a finite anomalous Hall conductivity, chiral edge states, and occurs as an intermediate phase between an ordinary insulator and a 3D quantum anomalous Hall insulator. We find that the Weyl semimetal has a nonzero DC conductivity at zero temperature, but Drude weight vanishing as T 2 , and is thus an unusual metallic phase, characterized by a finite anomalous Hall conductivity and topologically-protected edge states.

We present a study of "nodal semimetal" phases, in which non-degenerate conduction and valence bands touch at points (the "Weyl semimetal") or lines (the "line node semimetal") in threedimensional momentum space. We discuss a general approach to such states by perturbation of the critical point between a normal insulator (NI) and a topological insulator (TI), breaking either time reversal (TR) or inversion symmetry. We give an explicit model realization of both types of states in a NI-TI superlattice structure with broken TR symmetry. Both the Weyl and the line-node semimetals are characterized by topologically-protected surface states, although in the line-node case some additional symmetries must be imposed to retain this topological protection. The edge states have the form of "Fermi arcs" in the case of the Weyl semimetal: these are chiral gapless edge states, which exist in a finite region in momentum space, determined by the momentum-space separation of the bulk Weyl nodes. The chiral character of the edge states leads to a finite Hall conductivity. In contrast, the edge states of the line-node semimetal are "flat bands": these states are approximately dispersionless in a subset of the two-dimensional edge Brillouin zone, given by the projection of the line node onto the plane of the edge. We discuss unusual transport properties of the nodal semimetals, and in particular point out quantum critical-like scaling of the DC and optical conductivity of the Weyl semimetal, and similarities to the conductivity of graphene in the line node case.

We demonstrate that topological transport phenomena, characteristic of Weyl semimetals, namely the semi-quantized anomalous Hall effect and the chiral magnetic effect (equilibrium magnetic-field-driven current), may be thought of as two distinct manifestations of the same underlying phenomenon, the chiral anomaly. We show that the topological response in Weyl semimetals is fully described by a $\theta$-term in the action for the electromagnetic field, where $\theta$ is not a constant parameter, like e.g. in topological insulators, but is a field, which has a linear dependence on the space-time coordinates. We also show that the $\theta$-term and the corresponding topological response survive for sufficiently weak translational symmetry breaking perturbations, which open a gap in the spectrum of the Weyl semimetal, eliminating the Weyl nodes.Comment: 9 pages, 1 figure, published versio

Topological semimetals and metals have emerged as a new frontier in the field of quantum materials. Novel macroscopic quantum phenomena they exhibit are not only of fundamental interest, but may hold some potential for technological applications.The study of the electronic structure topology of crystalline materials has emerged in the last decade as a major new theme in the modern condensed matter physics. The starting impetus came from the remarkable discovery of topological insulators, 1,2 but the focus has recently shifted towards topological semimetals and even metals. While the idea that metals can have a topologically nontrivial electronic structure is not entirely new and some of the recent developments were anticipated in earlier work, 3-5 this shift was precipitated by the theoretical discovery of Weyl 6-12 and later Dirac semimetals. [13][14][15] The experimental realization of both Weyl and Dirac semimetals [16][17][18][19][20] within the last couple of years has brought the field to the forefront of quantum condensed matter research.

We present a theory of the anomalous Hall effect (AHE) in a doped Weyl semimetal, or Weyl metal, including both intrinsic and extrinsic (impurity scattering) contributions. We demonstrate that a Weyl metal is distinguished from an ordinary ferromagnetic metal by the absence of the extrinsic and the Fermi surface part of the intrinsic contributions to the AHE, as long as the Fermi energy is sufficiently close to the Weyl nodes. The AHE in a Weyl metal is thus shown to be a purely intrinsic, universal property, fully determined by the location of the Weyl nodes in the first Brillouin zone.An exciting recent development in condensed matter physics is the emerging extension of the concepts of nontrivial electronic structure topology, which have long been confined exclusively to insulators, to gapless metallic states. These ideas, pioneered some time ago by Volovik [1], have recently been brought to the forefront of condensed matter research, with specific solid-state realizations of the first topologically-nontrivial metallic state, a Weyl semimetal, proposed [2][3][4][5]. The recent observation of the closely related Dirac semimetals [6][7][8][9][10][11] paves the way for the realization of Weyl semimetals in the near future.The electronic structure of a Weyl semimetal contains points in momentum space, at which two nondegenerate bands touch at the Fermi energy. Such points, called Weyl nodes, can occur generically (but not necessarily at the Fermi energy) in three-dimensional (3D) band structures, as long as either time-reversal (TR) or inversion (I) symmetries are violated, which is needed to create nondegenerate bands, otherwise prohibited by the Kramers theorem. These points are topologically-nontrivial objects, characterized by an integer topological charge, and are monopole sources of the Berry curvature, momentumspace dual of the magnetic field in real space.Apart from the appearance of the Weyl nodes themselves, which is generically possible in 3D, Weyl semimetal requires the Fermi energy to be aligned with the nodes. This situation is not generic, but is a special case of a Weyl metal: a metal is which the Fermi surface is broken up into disjoint pieces, each surrounding, in the simplest case, a single Weyl node (we will assume such individual sheets of the Fermi surface may be characterized by a Chern number, which is equal to the topological charge, enclosed by the Fermi surface sheet). One may then ask the following question: what are, if any, observable consequences of such a topologically-nontrivial character of the Fermi surface of a Weyl metal?The purpose of this paper is to describe one such phenomenon, which is characteristic of a specific subclass of Weyl metals, namely the ferromagnetic (FM) Weyl metals, in which the nodes owe their existence to broken TR [12]. Any FM metal exhibits anomalous Hall effect (AHE), i.e. an antisymmetric contribution to the offdiagonal resistivity, "proportional" to the magnetization rather than to the applied magnetic field. As has been clearly demonstrated in recen...

Weyl semimetal is a new topological state of matter, characterized by the presence of nondegenerate band-touching nodes, separated in momentum space, in its bandstructure. Here we discuss a particular realization of a Weyl semimetal: a superlattice heterostructure, made of alternating layers of topological insulator (TI) and normal insulator (NI) material, introduced by one of us before. The Weyl node splitting is achieved most easily in this system by breaking time-reversal (TR) symmetry, for example by magnetic doping. If, however, spatial inversion (I) symmetry remains, the Weyl nodes will occur at the same energy, making it possible to align the Fermi energy simultaneously with both nodes. The goal of this work is to explore the consequences of breaking the I symmetry in this system. We demonstrate that, while this generally moves the Weyl nodes to different energies, thus eliminating nodal semimetal and producing a state with electron and hole Fermi surfaces, the topological properties of the Weyl semimetal state, i.e. the chiral edge states and the corresponding Hall conductivity, survive for moderate I symmetry breaking. Moreover, we demonstrate that a new topological phenomenon arises in this case, if an external magnetic field along the growth direction of the heterostructure is applied. Namely, this leads to an equilibrium dissipationless current, flowing along the direction of the field, whose magnitude is proportional to the energy difference between the Weyl nodes and to the magnetic field, with a universal coefficient, given by a combination of fundamental constants.Comment: 9 pages, 5 figures; minor corrections and extensions, published versio

We describe the localization transition of superfluids on two-dimensional lattices into commensurate Mott insulators with average particle density p/q (p, q relatively prime integers) per lattice site. For bosons on the square lattice, we argue that the superfluid has at least q degenerate species of vortices which transform under a projective representation of the square lattice space group (a PSG). The formation of a single vortex condensate produces the Mott insulator, which is required by the PSG to have density wave order at wavelengths of q/n lattice sites (n integer) along the principle axes; such a second-order transition is forbidden in the Landau-Ginzburg-Wilson framework. We also discuss the superfluid-insulator transition in the direct boson representation, and find that an interpretation of the quantum criticality in terms of deconfined fractionalized bosons is only permitted at special values of q for which a permutative representation of the PSG exists. We argue (and demonstrate in detail in a companion paper: L. Balents et al., cond-mat/0409470) that our results apply essentially unchanged to electronic systems with short-range pairing, with the PSG determined by the particle density of Cooper pairs. We also describe the effect of static impurities in the superfluid: the impurities locally break the degeneracy between the q vortex species, and this induces density wave order near each vortex. We suggest that such a theory offers an appealing rationale for the local density of states modulations observed by Hoffman et al. (cond-mat/0201348) in STM studies of the vortex lattice of BSCCO, and allows a unified description of the nucleation of density wave order in zero and finite magnetic fields. We note signatures of our theory that may be tested by future STM experiments.Comment: 35 pages, 16 figures; (v2) part II is cond-mat/0409470; (v3) added new appendix and clarifying remarks; (v4) corrected typo

We use microscopic linear response theory to derive a set of equations that provide a complete description of coupled spin and charge diffusive transport in a two-dimensional electron gas (2DEG) with the Rashba spin-orbit (SO) interaction. These equations capture a number of interrelated effects including spin accumulation and diffusion, Dyakonov-Perel spin relaxation, magnetoelectric, and spin-galvanic effects. They can be used under very general circumstances to model transport experiments in 2DEG systems that involve either electrical or optical spin injection. We comment on the relationship between these equations and the exact spin and charge density operator equations of motion. As an example of the application of our equations, we consider a simple electrical spin injection experiment and show that a voltage will develop between two ferromagnetic contacts if a spin-polarized current is injected into a 2DEG, that depends on the relative magnetization orientation of the contacts. This voltage is present even when the separation between the contacts is larger than the spin diffusion length.