Temperature dependence of the conductivity sum rule in the normal state due to inelastic scattering

We examine the temperature dependence of the optical sum rule in the normal state due to interactions. To be concrete, we adopt a weak-coupling approach that uses an electron-boson exchange model to describe inelastic scattering of the electrons with a boson, in the Migdal approximation. While a number of recent works attribute the temperature dependence in the normal state to that which arises in a Sommerfeld expansion, we show that in a wide parameter regime this contribution can be quite small. Instead, most of the temperature dependence arises from the zeroth-order term in the ``expansion,'' through the temperature dependence of the spectral function and the interaction parameters contained therein. For low boson frequencies, this circumstance causes a linear T dependence in the sum rule. We develop some analytical expressions and understanding of the temperature dependence.