Relating the Hall conductivity to the many-body Chern number using Fermi's Golden rule and Kramers-Kronig relations

Abstract

This pedagogical piece provides a surprisingly simple demonstration that the quantized Hall conductivity of correlated insulators is given by the many-body Chern number, a topological invariant defined in the space of twisted boundary conditions. In contrast to conventional proofs, generally based on the Kubo formula, our approach entirely relies on combining Kramers-Kronig relations and Fermi's golden rule within a circular-dichroism framework. This pedagogical derivation illustrates how the Hall conductivity of correlated insulators can be determined by monitoring single-particle excitations upon a circular drive, a conceptually simple picture with direct implications for quantum-engineered systems, where excitation rates can be directly monitored.

8 pages, 2 figures + Appendix

References in corpus (9)

• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Quantum Spin Hall Insulator State in HgTe Quantum Wells
• Simplified topological invariants for interacting insulators
• Experimental Measurement of the Quantum Metric Tensor and Related Topological Phase Transition with a Superconducting Qubit
• Dichroic f-sum rule and the orbital magnetization of crystals
• Realization of a fractional quantum Hall state with ultracold atoms
• Connecting the Many-Body Chern Number to Luttinger's Theorem through Středa's Formula
• Tracing the nonequilibrium topological state of Chern insulators
• Global and Local Topological Quantized Responses from Geometry, Light and Time

Cited by in corpus (4)

• Theory of Generalized Landau Levels and Implication for non-Abelian States
• Circular Dichroism on the Edge of Quantum Hall Systems: From Many-Body Chern Number to Anisotropy Measurements
• Quantized and nonquantized Hall response in topological Hatsugai-Kohmoto systems
• Pressure-Tunable Generalized Wigner Crystal and Fractional Chern Insulator in twisted MoTe${}_2$