Phonon thermal Hall as a lattice Aharonov-Bohm effect

In a growing list of insulators, experiments find that magnetic field induces a misalignment between the heat flux and the thermal gradient vectors. This phenomenon, known as the phonon thermal Hall effect, implies energy flow without entropy production along the orientation perpendicular to the temperature gradient. The experimentally-measured thermal Hall angle in various insulators does not exceed a bound and becomes maximal at the temperature of peak longitudinal thermal conductivity. The present paper aims to propose a scenario providing and explanation for these two experimental facts. It begins by noticing that at this temperature, T_{max} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>m</mml:mi> <mml:mi>a</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> </mml:msub> </mml:math> , Normal phonon-phonon collisions become most frequent in comparison with Umklapp and boundary scattering events. Furthermore, the Born-Oppenheimer approximated molecular wave functions are known to acquire a phase in the presence of a magnetic field. In an anharmonic crystal, in which tensile and compressive strain do not cancel out, this field-induced atomic phase gives rise to a phonon Berry phase and generates phonon-phonon interference. The rough amplitude of the thermal Hall angle expected in this picture is set by the phonon wavelength, \lambda_{ph} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mi>h</mml:mi> </mml:mrow> </mml:msub> </mml:math> , and the crest atomic displacement, \delta u_m <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>δ</mml:mi> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:mrow> </mml:math> at T_{max} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>m</mml:mi> <mml:mi>a</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> </mml:msub> </mml:math> . The derived expression is surprisingly close to what has been experimentally found in black phosphorus, germanium and silicon.