Vanishing of power-law corrections to Kubo's formula for the Hall current at incommensurate magnetic fields

TL;DR

This paper demonstrates that in a two-dimensional electron gas under a perpendicular magnetic field, the Hall current response is linear in the electric field without power-law corrections, even with incommensurate magnetic flux, extending previous results.

Contribution

It generalizes the analysis of Hall response to incommensurate magnetic flux and lattice-periodic perturbations, showing the absence of power-law corrections in the linear response.

Findings

Hall current is linear in electric field strength with no power-law corrections.

Hall conductivity can be expressed via the equilibrium Fermi projection.

Results extend previous work to incommensurate magnetic flux settings.

Abstract

We consider a non-interacting electron gas confined to a two-dimensional crystal by the action of a perpendicular magnetic field; in the one-particle approximation, the dynamics of the system is modelled by a spectrally gapped Bloch-Landau Hamiltonian. No commensurability condition is assumed between the magnetic flux per unit cell and the quantum of magnetic flux. We construct a non-equilibrium almost-stationary state (NEASS) which "dresses" the equilibrium Fermi projection on states below the spectral gap, and models the state of the system after the addition of a weak external electric field of strength $\varepsilon \ll 1$. Having in mind applications to the integer quantum Hall effect, we probe the response of a current operator in the direction transverse to that of the applied electric field, and show that the resulting current density in the NEASS is linear in $\varepsilon$, with no power-law corrections. The linear response coefficient, namely the Hall conductivity, is computed in terms of the equilibrium Fermi projection via the double-commutator formula, in accordance with the prediction from Kubo's linear response theory. Our results generalize the methods and findings of [Lett. Math. Phys. 112 (2022), 91] to the setting of uniform magnetic fields with incommensurate magnetic flux per unit cell, and to lattice-periodic perturbation of such magnetic fields.