Resolving the Puzzles in Computing the Intrinsic Thermal Hall Conductivity by a Semi-Classical Interpretation

The intrinsic thermal Hall conductivity is usually calculated using linear response theory or by analyzing boundary modes. However, these calculations appear puzzling and require more thorough understanding: (1) Even though the total bulk-plus-boundary thermal Hall conductivity converges in the zero temperature limit, the bulk thermal Hall conductivity diverges, which is confusing; (2) The bulk thermal Hall conductivity depends not only on the Berry curvature, but also on an additional term, while the total thermal Hall conductivity is completely independent of this term; (3) In the phonon case, the boundary is sometimes modeled by a high potential outside the sample, which is unnatural since phonons do not have chemical potential; (4) In the phonon case, some former way of modeling requires the knowledge of the Berry curvature of the high-energy phonons, on which the thermal Hall conductivity seems unlikely to depend. In this work, we present a suitable physical modeling of the boundary, and derive a corresponding semi-classical single-particle action. This solves the puzzles above and in particular, gives a clear interpretation of the linear response calculations. Our work provides a simple and intuitive picture for analyzing the thermal Hall effect.