  Phonon Hall effect (PHE) has attracted a lot of attention in recent years
with many theoretical and experimental explorations published. While
experiments work on complicated materials, theoretical studies are still
hovering around the phenomenon-based models. Moreover, previous microscopic
theory was found unable to explain large thermal Hall conductivity obtained by
experiments in strontium titanate (STO). Therefore, as a first attempt to
bridge this gap, we implement first-principles calculations to explore the PHE
in real materials. Our work provides a new benchmark of the PHE in sodium
chloride (NaCl) under a large external magnetic field. Moreover, we demonstrate
our results in barium titanate (BTO), and discuss the results in STO in detail
about their deviation from experiments. As a possible future direction, we
further propose that the inner electronic Berry curvature plays an important
role in the PHE in STO.
