Quantum kinetic equation and thermal conductivity tensor for bosons

TL;DR

This paper derives a comprehensive quantum kinetic equation for bosons that accounts for boundaries and inhomogeneities, enabling precise analysis of thermal transport and Hall effects without auxiliary fields.

Contribution

It introduces a phase space formalism for bosonic quantum kinetics, including Berry curvature and energy magnetization, applicable to complex inhomogeneous systems.

Findings

Derivation of exact quantum kinetic equations for bosons in full phase space

Recovery of the semiclassical Boltzmann equation at first order

Self-contained derivation of the intrinsic thermal Hall effect for bosons

Abstract

We systematically derive the quantum kinetic equation in full phase space for any quadratic hamiltonian of bosonic fields, including in the absence of translational invariance. This enables the treatment of boundaries, inhomogeneous systems and states with non-trivial textures, such as skyrmions in the context of magnetic bosons. We relate the evolution of the distribution of bosons in phase space to single-electron, band-diagonal, physical quantities such as Berry curvature and energy magnetization, by providing a procedure to "diagonalize" the Hamiltonian in phase space, using the formalism of the Moyal product. We obtain exact equations, which can be expanded order by order, for example in the "smallness" of the spatial gradients, providing a "semiclassical" approximation. In turn, at first order, we recover the usual full Boltzmann equation and give a self-contained and exact derivation of the intrinsic thermal Hall effect of bosons. The formulation clarifies the contribution from "energy magnetization" in natural manner, and does not require the inclusion of Luttinger's pseudo-gravitational field to obtain thermal transport quantities.