Landau levels in a gravitational field: The Schwarzschild spacetime case

Abstract

We investigate the gravitational effect on Landau levels. We show that the familiar infinite Landau degeneracy of the energy levels of a quantum particle moving inside a uniform and constant magnetic field is removed by the interaction of the particle with a gravitational field. First, two independent approaches are used to solve the relevant Schrödinger equation. It is found that both approaches yield qualitatively similar results within their respective approximations. Then, with the goal of clarifying some confusing results found in the literature concerning the use of a third independent approach for extracting the quantization condition based on a similar differential equation, we show that such an approach cannot yield a general and yet consistent result. We point out to the more accurate, and yet impractical, way to use such an approach, a way which does in principle yield a consistent quantization condition.Finally, we show how our results could be used to contribute in a novel way to the existing methods for testing gravity at the tabletop experiments level as well as at the astrophysical observational level.

A fresh look at the influence of gravity on the quantum Hall effect

“…The solution to this equation without the gravitational potential V g (r) has been given in Refs. [17,18] with all the steps outlined in great detail. The result is a function R(r) ∼ r e − eB 4 r 2 , with the proportionality factor being a hypergeometric function.…”

“…In order to solve this equation and extract the quantization condition on the energy E, we proceed along the same steps used in Ref. [17] to solve the simpler equation (III.1) without the last term. Here, we are going to solve Eq.…”

“…For that purpose, one needs to gravitationally affect the quantized Landau levels themselves. In fact, it was recently shown that the Landau levels around a massive sphere and around a cylinder are indeed affected by the gravitational field of those massive objects [17,18]. However, the way the levels were found to be modified there was only through the removal of their degeneracy in the orbital quantum number.…”

“…In Refs. [17,18], it was found that the gravitational field due to a (static and/or rotating) sphere as well as that due to a long cylinder both remove the degeneracy of the Landau levels by splitting the orbital energies of a charged particle going around such massive objects. As we shall see, using, instead, two massive hemispheres separated by a small distance gives rise in the motion of the free 2DEG inside a conductor sandwiched between the two hemispheres to a gravitationally induced harmonic oscillator.…”

Prospects for testing the inverse-square law and gravitomagnetism using quantum interference

“…We refer, however, to the recent proposal in Ref. [21] for a new way of putting into evidence deviations from the ISL based on the effect of gravity on the quantized Landau levels.…”