Thermal Vector Potential Theory of Transport Induced by a Temperature Gradient

A microscopic formalism to calculate thermal transport coefficients is presented based on a thermal vector potential, whose time derivative is related to a thermal force. The formalism is free from the unphysical divergences reported to arise when Luttinger’s formalism is applied naively, because the equilibrium (“diamagnetic”) currents are treated consistently. The mathematical structure for the thermal transport coefficients is shown to be identical with that for the electric ones if the electric charge is replaced by the energy. The results indicate that the thermal vector potential couples to the energy current via the minimal coupling.

Figure 1

Diagramatic representation of the contributions to the thermal Hall effect. Solid, wavy, and dotted lines denote the electron, the thermal vector potential ${\mathit{A}}_T$, and the electromagnetic vector potential $\mathit{A}$, respectively. Diagrams (a) and (b) correspond to contributions of Eqs. (17) and (18), respectively, while contributions of diagram (c) vanish.