We study in depth the charge susceptibility for the band Hatsugai-Kohmoto and orbital models. As either of these models describes a Mott insulator, the charge susceptibility takes on the form of a modified density response function with lower and upper Hubbard bands, thereby giving rise to a multipole structure. The particle-hole continuum consists of hot spots along the $\omega$ vs $q$ axis arising from interband transitions. Such transitions, which are strongly suppressed in noninteracting systems, obtain here because of the nonrigidity of the Hubbard bands. This modified density response 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 noncommutativity of the long-wavelength ($q\to 0$) and thermodynamic ($L\to \infty$) 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 contrary to recent claims. 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.

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• Charge dynamics
• 2-dimensional systems
• Mott insulators
• Dynamical mean field theory