Quantum theory of phonon-induced anomalous Hall effect in two-dimensional massive Dirac metals

Phonon-induced anomalous Hall or thermal Hall effects have been observed in various systems in recent experiments. However, the theoretical studies on this subject are still incomplete and not unified, and the current works mainly focus on the semiclassical Boltzmann equation approach. In this work, we present a systematic and fundamental quantum field theory study on the phonon-induced anomalous Hall effect, including both the side jump and skew scattering contributions, in a two-dimensional massive Dirac metal, which is considered as the minimum anomalous Hall system. We reveal a significant difference from the anomalous Hall effect induced by the widely studied Gaussian disorder which is known to be insensitive to temperature. While the anomalous Hall effect induced by phonons (deformation potential) approaches that by Gaussian disorder at high temperature, it behaves very differently at low temperature. Our work provides a microscopic and quantitative description of the crossover from the low to high temperature regime of the phonon-induced anomalous Hall conductivity, which may be observed in two-dimensional Dirac metals with broken time-reversal symmetry.