arXiv:1105.2353v2 [cond-mat.str-el] 27 May 2011 Quantum Hall effects in a Weyl Semi-Metal: possible application in pyrochlore Iridates Kai-Yu Yang, 1 Yuan-Ming Lu, 1 and Ying Ran 1 1 Department of Physics, Boston College, Chestnut Hill, MA 02467 (Dated: May 30, 2011) There have been lots of interest in pyrochlore Iridates A2Ir2O7 where both strong spin-orbital coupling and strong correlation are present. A recent LDA calculation 1 suggests that the system is likely in a novel three dimensional topological semi-metallic phase: a Weyl semi-metal. Such a system has zero carrier density and arrives at the quantum limit even in a weak magnetic field. In this paper we discuss two novel quantum effects of this system in a magnetic field: a pressure-induced anomalous Hall effect and a magnetic field induced charge density wave at the pinned wavevector connecting Weyl nodes with opposite chiralities. A general formula of the anomalous hall coefficients in a Weyl semi-metal is also given. Both proposed effects can be probed by experiments in the near future, and can be used to detect the Weyl semi-metal phase. I. INTRODUCTION Experimental realizations of two-dimensional massless Dirac electrons in condensed matter systems have gener- ated a lot of interest. These include the intrinsic two- dimensional graphene system 2 , as well as the surface of a three-dimensional topological insulator 3–5 . One of the many exciting phenomena for these systems is their anomalous response to an external magnetic field. For example, the room-temperature integer quantum hall effect 6,7 has been observed in graphene system. A minimal model for a two dimensional Dirac elec- tronic system is H = v F (p x σ x + p y σ y ), where  p is mo- mentum and σ are Pauli matrices. Clearly a mass term mσ z will generate an energy gap for the electronic struc- ture. One can ask whether mass terms will appear in the experimental systems mentioned above, in which case the linear dispersive band touching points, the Dirac nodes, will be destroyed. In these systems, it turns out that the Dirac nodes are protected by extra physical symmetries apart from the lattice translational symmetry. For ex- ample, in the case of the surface states of a topological insulator, it is protected by time-reversal symmetry. Recently a remarkable theoretical work 1 indicates that, a novel three-dimensional relativistic electronic struc- ture, the Weyl semi-metal phase, is likely to be realized in pyrochlore Iridates A 2 Ir 2 O 7 where A=Yttrium, or a Lanthanide element. On one hand, similar to graphene, the electronic dispersion of a Weyl semi-metal is charac- terized by a set of linear-dispersive band-touching points of two adjacent bands, the Weyl nodes. On the other hand, there are important differences between the 3D Weyl nodes and the 2D Dirac nodes in graphene, be- cause the Weyl nodes are protected by the topology of the band structure. One direct way to see this is to write down the effective hamiltonian in the neighborhood of a Weyl node: H = v F (p x σ x +p y σ y +p z σ z ). The three Pauli matrices are used up and there is simply no local mass term. Consequently, as long as there is no translational- symmetry-breaking inter-valley mixings between differ- ent Weyl nodes, the Weyl semi-metal phase is robust for arbitrary perturbation. The concept of Weyl fermions was firstly introduced in high energy physics and has been used to describe neutri- nos. The possible realizations of Weyl electronic struc- tures in condensed matter systems and their supercon- ducting analogs were discussed by various authors 8–10 . In fact, the original attempt to realize Weyl fermions in 3D lattice systems results in the famous fermion- doubling theorem, which dictates the total number of Weyl nodes must be even 11 . This is related to another famous phenomena, the Adler-Bell-Jackiw anomaly (or chiral anomaly) 8 . Weyl Fermions have its handiness or chirality. Chiral anomaly states that a quantized space- time electromagnetic field event would pump quantized electric charge from a node with positive chirality to one with negative chirality. Thus the number nodes of pos- itive chirality must equal those with negative chirality; totally one must have even number of nodes. Because the Dirac spectrum is known to have anoma- lous response to a magnetic field, a natural question to ask is: what is the response of a Weyl semi-metal in a magnetic field? Motivated by the fact that Weyl semi- metal is a novel phase of matter whose experimental sig- natures are of fundamental interest, and also by the re- cent experiment efforts on the pyrochlore Iridates, we study the effects of an external magnetic field on a Weyl semi-metal. Let us state the main results of this work. We find two novel quantum effects of a Weyl semi-metal in a magnetic field: a pressure-induced anomalous Hall ef- fect and a magnetic field induced charge density wave at the pinned wavevector that connects nodes with oppo- site chiralities. A general formula of the anomalous hall conductivity in a Weyl semi-metal is also given. We also apply these results to the proposed Weyl phase in py- rochlore Iridates and address the experimental relevant questions in these specific systems. Pyrochlore Iridates A 2 Ir 2 O 7 have attracted a lot of attentions both experimentally and theoretically 1,12–18 . Because of the feature of the Ir 4+ ion, these systems are in a novel regime where strong spin-orbital coupling,