Two classes of Mott insulator

There are two classes of Mott insulators in nature, distinguished by their responses to weak doping. With increasing chemical potential, type I Mott insulators undergo a first order phase transition from the undoped to the doped phase. In the presence of long-range Coulomb interactions, this leads to an inhomogeneous state exhibiting "micro-phase separation." In contrast, in type II Mott insulators charges go in continuously above a critical chemical potential. We show that if the insulating state has a broken symmetry, this increases the likelihood that it will be type I. There exists a close analogy between these two types of Mott insulators and the familiar type I and type II superconductors.