Quantized Anomalous Hall Effect in Two-Dimensional Ferromagnets: Quantum Hall Effect in Metals

Abstract

We study the effect of disorder on the anomalous Hall effect (AHE) in two-dimensional ferromagnets. The topological nature of AHE leads to the integer quantum Hall effect from a metal, i.e., the quantization of σxy induced by the localization except for the few extended states carrying Chern number. Extensive numerical study on a model reveals that Pruisken's two-parameter scaling theory holds even when the system has no gap with the overlapping multibands and without the uniform magnetic field. Therefore the condition for the quantized AHE is given only by the Hall conductivity σxy without the quantum correction, i.e., |σxy| > e 2 /(2h).PACS numbers: 72.15. Rn, The origin of the anomalous Hall effect (AHE) has been a subject of extensive controversy for a long term. One is based on the band picture with the relativistic spin-orbit interaction [1], while the other is due to the impurity scatterings [2]. Most of the succeeding theories follows the idea that the AHE occurs due to the scattering events modified by the spin-orbit interaction, i.e., the skew scattering or the side jump mechanism [3]. Recently several authors recognized the topological nature of the AHE discussed in Refs. [4,5,6]. In this formalism, the Hall conductivity σ xy is given by the Berry phase curvature in the momentum ( k-) space integrated over the occupied states [7]. Also there appeared some experimental evidences supporting it [8]. Therefore it is very important to study the effect of the scatterings by disorder, which makes k ill-defined, to see the topological stability of this mechanism for AHE.This issue is closely related to the integer quantum Hall effect (IQHE) [9] but there are several essential differences. Usually the topological stability which guarantees the quantization of some physical quantity, e.g., σ xy , has been discussed in the context of the adiabatic continuation [9]. Therefore it appears that the gaps between Landau levels in pure system are needed to start with even though the disorder potential eventually buries it. In the IQHE system without disorder, the periodic potential is irrelevant because the carrier concentration is much smaller than unity per atom. Although numerical simulations [10] use lattice models, the main concern is put on the limit of dispersionless Landau levels separated by the gaps. In the present case, i.e., in ferromagnetic metals, there are multiple bands overlapping without the gaps in the density of states. The periodicity of the lattice remains unchanged, which prohibits the uniform magnetic field and also gives a large energy dispersion. In * Electronic address: m.onoda@aist.go.jp † Electronic address: nagaosa@appi.t.u-tokyo.ac.jp the language of the effective magnetic field for electrons, it reaches a huge value of the order of ∼ 10 4 Tesla, i.e., the magnetic cyclotron length is of the order of the lattice constant, but the net flux is zero when averaged over the unit cell. Therefore these two cases belong to quite different limits although the symmetries of the systems are common, ...

Electrically tunable multiple Dirac cones in thin films of the (LaO)2(SbSe2)2 family of materials

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