Sum rule for thermal conductivity and dynamical thermal transport coefficients in condensed matter

We display an interesting sum rule for the dynamical thermal conductivity for many standard models of condensed matter in terms of the expectation of a thermal operator. We present the thermal operator for several model systems of current interest, which enable an evaluation of the sum rule and the Lorentz number, the thermoelectric figure of merit as well as the thermopower at high frequencies. As a by-product, we present exact formulas for the $T=0$ chemical potential $\mu \left(0\right)$ for charged many-body systems, including the Hubbard model, in terms of expectation values of extensive operators. Simple estimates are provided for the thermopower of an infinitely correlated band model on the triangular lattice, modeling the physics of the sodium cobalt oxide system. The present result goes beyond the Heikes-Mott formula for the thermopower, and contains an additional transport correction that is sensitive to the lattice topology as well as the sign of hopping.