Electrical transport in the Hatsugai-Kohmoto model

We show that in models with the Hatsugai-Kohmoto type of interaction that is local in momentum space thus infinite range in real space, Kubo formulas neither reproduce the correct thermodynamic susceptibilities, nor yield sensible transport coefficients. Using Kohn's trick to differentiate between metals and insulators by threading a flux in a torus geometry, we uncover the striking property that Hatsugai-Kohmoto models with an interaction-induced gap in the spectrum sustain a current that grows as the linear size at any nonzero flux and which can be either diamagnetic or paramagnetic.