Locality of the anomalous Hall conductivity

Abstract

The geometrical intrinsic contribution to the anomalous Hall conductivity (AHC) of a metal is commonly expressed as a reciprocal-space integral: as such, it only addresses unbounded and macroscopically homogeneous samples. Here we show that the geometrical AHC has an equivalent expression as a local property. We define a "geometrical marker" which actually probes the AHC in inhomogeneous systems (e.g. heterojunctions), as well as in bounded samples. The marker may even include extrinsic contributions of geometrical nature.PACS numbers: 72.15.Eb,72.20.My The Hall conductivity is "anomalous" whenever it is nonzero in absence of an applied magnetic field. The phenomenon requires the absence of time-reversal symmetry: it was discovered by Hall himself in 1881 in ferromagnetic metals. The possibility of observing anomalous Hall conductivity (AHC) in insulators was pointed out in 1988 by Haldane, who proposed a model Hamiltonian where the AHC is nonzero and quantized [1]; the AHC value is determined by the topology of the electronic ground state. In metals extrinsic mechanisms are essential to make the longitudinal dc conductivity finite. In absence of timereversal symmetry extrinsic mechanisms contribute to the AHC as well: these go under the name of side-jump and skew-scattering [2]. Since the early 2000s [3,4] it became clear that an intrinsic effect, only dependent on the ground wavefunction of the pristine crystal, provides an important additional contribution to the AHC. The latter contribution is geometrical in nature; its expression is the nonquantized version of the corresponding formula for insulators. In this work we only address the geometrical/topological AHC in metals/insulators, which is customarily expressed as the Fermi-volume integral of the Berry curvature, Eq. (4) below. The standard approach requires an unbounded crystalline sample where the orbitals have the Bloch form; this was extended in Ref. [5] to "dirty" metals in a supercell framework, where the geometric contribution includes some extrinsic effects.Recent work has demonstrated the locality of AHC, although in the insulating case only [6]. One does not require lattice periodicity and reciprocal-space paraphernalia: the AHC can be defined and computed for bounded samples and/or for macroscopically inhomogeneous systems (e.g. heterojunctions). The extension to the metallic case is not obvious, since one of the reasons for the locality of the (quantized) AHC is the r-space exponential decay of the one-body density matrix in insulators ("nearsightedness" [7]). In metals instead such decay is only power-law, which hints to a possibly different behavior. Furthermore the Hall current in insulators may only flow in the edge region, while in metals the current flows through the bulk of the sample as well. Our major result is that even in metals the AHC is a local property: for a bounded sample it can be expressed in terms of the one-body density matrix, evaluated in the sample bulk. We show this by means of simulations on model two-dimensional b...

Itinerant ferromagnetism and intrinsic anomalous Hall effect in amorphous iron-germanium

“…While the intrinsic AHC is generally expressed as the integral of the Berry curvature taken over all occupied bands, this result independently verifies the outcome of our scaling argument and unambiguously shows that the intrinsic mechanism persists in systems whose band structure is ill-defined, further validating the real-space formulation of the AHC in Ref. 15. Our spin-orbit coupled DFT calculations provide computational foundations and a theoretical paradigm shift for understanding the intrinsic anomalous Hall conductivity in an amorphous material.…”

Surfaces of axion insulators

“…For the calculation of the surface anomalous Hall conductivity (AHC), we implement a recently proposed approach 14 based in part on previous developments 5, 20,21 showing that the Chern-Simons (CS) contribution to the AHC can be expressed as a local, real space property. In this framework, one defines a local Chern-number density C(r) = −2π Im r|P rQ × QrP |r (8) from which the local CS contribution to the AHC can be obtained via σ CS (r) = (e 2 /h)C(r).…”

Geometric and nongeometric contributions to the surface anomalous Hall conductivity

“…The nongeometric part of the local AHC was overlooked in some previous studies [7,16], where the local AHC was formulated as a spatially-resolved Berry curvature. As for the geometric part, the expression in Eq.…”

“…where |v and |c denote occupied and empty energy eigenstates, respectively, and E cv = E c − E v . Equations (15) and (16) give the full local AHC; below, we separate it into geometric and nongeometric parts.…”