A first-frequency-moment constraint on antisymmetric optical conductivity vanishes exactly in zero-field moiré continuum models and equals a magnetic-flux-determined constant in Hofstadter models, implying spectral compensation for anomalous Hall absorption.
Lange, Memory-function approach to the Hall con- stant in strongly correlated electron systems, Physical Review B55, 3907 (1997)
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Optical Hall absorption sum rule and spectral compensation in time-reversal-breaking moir\'e and Hofstadter systems
A first-frequency-moment constraint on antisymmetric optical conductivity vanishes exactly in zero-field moiré continuum models and equals a magnetic-flux-determined constant in Hofstadter models, implying spectral compensation for anomalous Hall absorption.