  Several useful thermodynamic relations are derived for metal-insulator
transitions, as generalizations of the Clausius-Clapeyron and Eherenfest
theorems. These relations hold in any spatial dimensions and at any
temperatures. First, they relate several thermodynamic quantities to the slope
of the metal-insulator phase boundary drawn in the plane of the chemical
potential and the Coulomb interaction in the phase diagram of the Hubbard
model. The relations impose constraints on the critical properties of the Mott
transition. These thermodynamic relations are indeed confirmed to be satisfied
in the cases of the one- and two-dimensional Hubbard models. One of these
relations yields that at the continuous Mott transition with a diverging charge
compressibility, the doublon susceptibility also diverges. The constraints on
the shapes of the phase boundary containing a first-order metal-insulator
transition at finite temperatures are clarified based on the thermodynamic
relations. For example, the first-order phase boundary is parallel to the
temperature axis asymptotically in the zero temperature limit. The
applicability of the thermodynamic relations are not restricted only to the
metal-insulator transition of the Hubbard model, but also hold in correlated
systems with any types of phases in general. We demonstrate such examples in an
extended Hubbard model with intersite Coulomb repulsion containing the charge
order phase.
