Charge Susceptibility and Kubo Response in Hatsugai-Kohmoto-related Models

We study the charge susceptibility for the band Hatsugai-Kohmoto (HK) and orbital (OHK) models. The charge susceptibility takes on the form of a modified Lindhard function with lower and upper Hubbard bands, thereby giving rise to a multi-pole structure. The particle-hole continuum consists of hot spots along the $\omega$ vs $q$ axis arising from inter-band transitions. Such transitions, which are strongly suppressed in non-interacting systems, obtain here because of the non-rigidity of the Hubbard bands. This modified Lindhard function gives rise to a plasmon dispersion that is inversely dependent on the momentum, resulting in an additional contribution to the conventional f-sum rule. This extra contribution originates from a long-range diamagnetic contribution to the current. This results in a non-commutativity of the long-wavelength ($q\rightarrow 0$) and thermodynamic ($L\rightarrow\infty$) limits. When the correct limits are taken, we find that the Kubo response computed with either open or periodic boundary conditions yields identical results that are consistent with the continuity equation. We also show that the long wavelength pathology of the current noted previously also plagues the Anderson impurity model interpretation of dynamical mean-field theory (DMFT).