No bulk thermal currents in massive Dirac fermions

We calculate the energy current flowing in the bulk of a (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mn>2</a:mn><a:mo>+</a:mo><a:mn>1</a:mn></a:math>)-dimensional system of massive Dirac fermions and along a (<b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mn>1</b:mn><b:mo>+</b:mo><b:mn>1</b:mn></b:math>)-dimensional domain wall generated by flipping the sign of the particle mass. We show that, at low temperatures and in the long-wavelength limit, the system does not support a bulk thermal Hall current proportional to the temperature gradient. The only such contribution is due to states localized at the domain wall. This puts an end to a controversy existing in the literature and amends previous results obtained via first-order perturbation calculations.