Examining an Exactly Solvable First-Order Transition in a Modified Hubbard Model

A previous study (PRB {\bf 63}, 035014 (2000)) of the success of the different diagrammatic theories in reproducing the physics of the Hubbard model led to the introduction of a modified Hubbard interaction, the latter of which is given by $U~\sum_i \big(({\hat n}_{i,\uparrow} - \langle ({\hat n}_ {i,\uparrow} \rangle)~(({\hat n}_{i,\downarrow} - \langle ({\hat n}_{i,\downarrow} \rangle)\big)$. We show that the Hubbard model with this modified interaction leads to an exact phase diagram corresponding to a first-order phase transition, with a thermodynamic potential of the same form as that found for the familiar van der Waals equation of state. Then we show that the fully self-consistent T-Matrix Approximation for the repulsive Hubbard interaction accurately tracks the low electron density regime, and that {\it two} self-consistent solutions are found, corresponding to both the stable and metastable phases of this model. This is in contract to the minimally self-consistent T-Matrix theory that was shown to be successful for the attractive model (PRB {\bf 71}, 155111 (2005)).