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Near Linearity of the Macroscopic Hall Current Response in Infinitely Extended Gapped Fermion Systems
Marius Wesle1, Giovann Marcelli2, Tadahiro Miyao3
1Fachbereich Mathematik, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany.
Communications in Mathematical Physics
|July 7, 2025
Summary
This study shows that topological insulators are excellent insulators, with longitudinal currents vanishing exponentially with electric field strength. The Hall conductivity, however, remains robust and measurable, offering a precise way to characterize these materials.
Area of Science:
- Condensed Matter Physics
- Topological Matter
- Quantum Transport Phenomena
Background:
- Topological insulators are materials with unique electronic properties, acting as insulators in the bulk but conducting on the surface.
- Understanding their response to external electric fields is crucial for applications and fundamental physics.
- Previous studies often focused on voltage drops, not constant electric fields on macroscopic systems.
Purpose of the Study:
- To investigate the current response of gapped topological insulators to a constant external electric field.
- To determine the longitudinal and Hall current densities in such systems.
- To derive a formula for Hall conductivity in interacting fermion systems.
Main Methods:
- Utilized the non-equilibrium almost-stationary states approach.
- Analyzed a macroscopic, infinitely extended system of fermions on a lattice with short-range interactions.
- Considered systems with a gapped ground state and magnetic translation invariance.
Main Results:
- Longitudinal current density induced by a constant electric field is vanishingly small (O(ε^∞)).
- Hall current density is linear in the electric field strength (ε) up to exponentially small corrections.
- Derived a generalized double commutator formula for Hall conductivity in interacting systems.
Conclusions:
- The system behaves as an ideal insulator regarding longitudinal current.
- Hall conductivity (σ_H) is constant within gapped phases and robust against interactions.
- For 2D systems, Hall conductance precisely measures σ_H with vanishing experimental variance.
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