Vanishing of power-law corrections to Kubo's formula for the Hall current at incommensurate magnetic fields
Gabriele Mazzini, Domenico Monaco
TL;DR
This work proves that the Hall response in a two-dimensional crystal under a uniform magnetic field remains linear with respect to a weak electric perturbation even when the magnetic flux per unit cell is incommensurate with the quantum flux. By constructing a magnetic non-equilibrium almost-stationary state (NEASS) through space-adiabatic perturbation theory, the authors express the Hall conductivity entirely in terms of the equilibrium Fermi projection via the double-commutator formula $\sigma_{\mathrm{Hall}} = i\mathcal{T}(\Pi_0\,[ [\Pi_0,X_1],[\Pi_0,X_2] ]\Pi_0)$. They show the current density satisfies $j_{\mathrm{Hall}}(\varepsilon) = \varepsilon \sigma_{\mathrm{Hall}} + \mathcal{O}(\varepsilon^{n+1})$, with all higher-order power-law corrections vanishing, thereby extending Kubo’s linear-response framework beyond the linear regime. The results generalize prior analyses to incommensurate magnetic flux and lattice-periodic perturbations, providing rigorous tools for topological transport in magnetic crystals and strengthening the connection between NEASS constructions and Hall transport in IQHE 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.