Class of distorted Landau levels and Hall phases in a two-dimensional electron gas subject to an inhomogeneous magnetic field

Abstract

An analytic closed form solution is derived for the bound states of a two-dimensional electron gas subject to a static, inhomogeneous ($1/r$ in plane decaying) magnetic field, including the Zeeman interaction. The solution provides access to many-body properties of a two-dimensional, noninteracting, electron gas in the thermodynamic limit. Radially distorted Landau levels can be identified as well as magnetic field induced density and current oscillations close to the magnetic impurity. These radially localized oscillations depend strongly on the coupling of the spin to the magnetic field, which gives rise to nontrivial spin currents. Moreover, the Zeeman interaction introduces a unique flat band, i.e., infinitely degenerate energy level in the ground state, assuming a spin ${g}_{s}$-factor of two. Surprisingly, the charge and current densities can be computed analytically for this fully filled flat band in the thermodynamic limit. Numerical calculations show that the total magnetic response of the electron gas remains diamagnetic (similar to Landau levels) independent of the Fermi energy. However, the contribution of certain, infinitely degenerate energy levels may become paramagnetic. Furthermore, numerical computations of the Hall conductivity reveal asymptotic properties of the electron gas, which are driven by the anisotropy of the vector potential instead of the magnetic field, i.e., become independent of spin. Eventually, the distorted Landau levels give rise to negative and positive Hall conductivity phases, with sharp transitions at specific Fermi energies. Overall, our work merges ``impurity'' with Landau-level physics, which provides novel physical insights, not only locally, but also in the asymptotic limit. This paves the way for a large number of future theoretical as well as experimental investigations, e.g., to include electronic correlations and to investigate two-dimensional systems such as graphene or transition metal dichalcogenides under the influence of inhomogeneous magnetic fields.