Optical Hall absorption sum rule and spectral compensation in time-reversal-breaking moir\'e and Hofstadter systems

TL;DR

This paper derives and demonstrates a sum rule for antisymmetric optical Hall absorption in topological systems, linking low- and high-frequency spectral weights and providing a framework for experimental analysis.

Contribution

It formulates a first-frequency-moment sum rule for optical Hall absorption and applies it to moiré and Hofstadter models, revealing universal and compensatory spectral behaviors.

Findings

In zero-field moiré models, the sum rule vanishes, indicating spectral compensation.

In Hofstadter models, the sum rule yields a universal value based on magnetic flux.

The sum rule links low- and high-frequency spectral contributions, aiding in topological material analysis.

Abstract

Optical spectroscopy provides a powerful, contact-free probe of topological quantum states, yet exact constraints on antisymmetric Hall absorption remain much less well developed than their longitudinal counterparts. Motivated by earlier Hall-conductivity sum rules, we formulate the corresponding first-frequency-moment constraint for the antisymmetric optical conductivity, whose imaginary part governs chirality-dependent absorption. We then demonstrate this sum rule in two classes of time-reversal-breaking topological systems. For a zero-field moir\'e continuum model hosting topological bands, the moment vanishes exactly, implying that any low-frequency anomalous Hall absorption must be compensated by higher-frequency spectral weight of the opposite sign. For a Hofstadter model under a uniform magnetic field, the same moment takes a universal value fixed by the magnetic flux density, independent of microscopic model details. By linking low- and high-frequency spectral contributions, this optical Hall absorption sum rule provides a rigorous framework for quantifying circular dichroism constraints and diagnosing Landau-level mixing. Our results show how a known Hall spectral constraint acquires new and experimentally relevant content in modern interacting topological materials.