Hall Conductivity for Two Dimensional Magnetic Systems

TL;DR

This paper develops a formalism based on the Kubo approach to calculate the dynamical conductivities of electrons in two-dimensional magnetic systems, including effects of local magnetic fields, spin coupling, and disorder.

Contribution

It introduces a novel Kubo-inspired method for computing conductivities in 2D magnetic systems, incorporating spin effects and disorder perturbatively.

Findings

Homogeneous magnetic field Hall conductivity rederived.

Vortex at the origin exhibits diverging transverse conductivity at low frequency.

Perturbative analysis shows oscillations in Hall conductivity near classical values.

Abstract

A Kubo inspired formalism is proposed to compute the longitudinal and transverse dynamical conductivities of an electron in a plane (or a gas of electrons at zero temperature) coupled to the potential vector of an external local magnetic field, with the additional coupling of the spin degree of freedom of the electron to the local magnetic field (Pauli Hamiltonian). As an example, the homogeneous magnetic field Hall conductivity is rederived. The case of the vortex at the origin is worked out in detail. This system happens to display a transverse Hall conductivity ($P$ breaking effect) which is subleading in volume compared to the homogeneous field case, but diverging at small frequency like $1/\omega^2$. A perturbative analysis is proposed for the conductivity in the random magnetic impurity problem (Poissonian vortices in the plane). At first order in perturbation theory, the Hall conductivity displays oscillations close to the classical straight line conductivity of the mean magnetic field.