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

Landau Levels in a Gravitational Field: The Schwarzschild Spacetime Case

“…Therefore, the correction term in Equation ( 24) is indeed smaller than GMm/ρ 0 , which, as we already argued below Equation ( 16), is nothing but a small perturbation compared to the Landau energy represented by the first term. The same remark is also valid for the general splitting formula (23). In the latter, the smallness of the correction term compared to the first is less transparent but can, nevertheless, still be inferred from the less trivial expressions (A10) and (A17) of M n and P n , respectively.…”

“…Indeed, around the poles of the star the cylindrical coordinate ρ is such that ρ = r sin θ r. The results obtained in Section 3.1 can then easily be adapted to such a case (as done in Ref. [23] for the inside of a sphere) by expanding here, instead, the non-relativistic Newtonian potential around the poles of the star into that of a simple harmonic oscillator. Denoting now the radius of the star by R, we have at the poles, r = R 2 + ρ 2 , with ρ R. Therefore, we have, up to a constant, −GM/r ∼ 1 2 GMρ 2 /R 3 .…”

“…As shown in Ref. [23], the gravitational influence on the quantum Hall effect is due to the splitting of Landau levels caused by the potential gradient of gravity in the case of Earth's field and to the quadratic gravitational potential in the case of two weakly separated massive hemispheres.…”

“…Indeed, we restrict here the particle's motion to a plane in view of the many applications such a configuration gives rise to. Such applications range from the two-dimensional motion of the electron gas in the quantum Hall effect [23] to the study of the equation of state of the dense matter confined to the thin envelop or crust of magnetized astrophysical objects, as will be discussed in Section 4.2. In fact, for a particle which is instead free to explore the three-dimensional space the crucial Landau levels do not arise, for then the particle's wavefunction would be described by spherical harmonics governed by a central gravitational potential inside a strong magnetic field, much like in the case of atoms inside large magnetic fields (see the discussion below Equation ( 16))).…”

“…With such a potential, Equation ( 16) can easily be solved to yield the following counterpart of Equation ( 24) for the splitting of the Landau levels (see Equation (12) of Ref. [23], where the role of the constant K there is here played by GM/R 3 ):…”