Vanishing of power-law corrections to Kubo's formula for the Hall current at incommensurate magnetic fields
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| Formato: | Preprint |
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2026
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| author | Mazzini, Gabriele Monaco, Domenico |
| author_facet | Mazzini, Gabriele Monaco, Domenico |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_21417 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Vanishing of power-law corrections to Kubo's formula for the Hall current at incommensurate magnetic fields Mazzini, Gabriele Monaco, Domenico Mathematical Physics Mesoscale and Nanoscale Physics 81Q15, 81Q20, 81V70 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. |
| title | Vanishing of power-law corrections to Kubo's formula for the Hall current at incommensurate magnetic fields |
| topic | Mathematical Physics Mesoscale and Nanoscale Physics 81Q15, 81Q20, 81V70 |
| url | https://arxiv.org/abs/2601.21417 |