Gapped Dirac electrons generate a valley-Chern-number-dependent photon-phonon coupling and a frequency-induced chiral phonon splitting that survives at zero total Chern number.
(B27) Using the properties of gamma matrices, we can show Πµν Aa(p) = Πνµ aA(−p)
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Chern-Simons type cross-correlations and geometric Born effective charge of phonons
Gapped Dirac electrons generate a valley-Chern-number-dependent photon-phonon coupling and a frequency-induced chiral phonon splitting that survives at zero total Chern number.