Emergence of a diverse array of phases in an exactly solvable model

Numerical solvers for strongly correlated electronic systems poses a significant challenge in condensed matter theory. In recent years, extensive exploration of strongly correlated effects has been carried out based on exactly solvable Hatsugai-Kohmoto-like models. In this work, we investigate a simple two-site cluster extension of the Hatsugai-Kohmoto model. The model exhibits a rich variety of phases, including the charge-$4e$ phase, charge-$2e$ phase, metallic phase, pseudogap phase, and Luttinger liquid phase. These findings contribute to the understanding of strongly correlated physics and superconducting physics.