  At the Mott transition, electron-electron interaction changes a metal, in
which electrons are itinerant, to an insulator, in which electrons are
localized. This phenomenon is central to quantum materials. Here we contribute
to its understanding by studying the two-dimensional Hubbard model at finite
temperature with plaquette cellular dynamical mean-field theory. We provide an
exhaustive thermodynamic description of the correlation-driven Mott transition
of the half-filled model by calculating pressure, charge compressibility,
entropy, kinetic energy, potential energy and free energy across the
first-order Mott transition and its high-temperature crossover (Widom line).
The entropy is extracted from the Gibbs-Duhem relation and shows complex
behavior near the transition, marked by discontinuous jumps at the first-order
boundary, singular behavior at the Mott endpoint and inflections marking sharp
variations in the supercritical region. The free energy allows us to identify
the thermodynamic phase boundary, to discuss phases stability and
metastability, and to touch upon nucleation and spinodal decomposition
mechanisms for the transition. We complement this thermodynamic description of
the Mott transition by an information-theoretic description. We achieve this by
calculating the local entropy, which is a measure of entanglement, and the
single-site total mutual information, which quantifies quantum and classical
correlations. These information-theoretic measures exhibit characteristic
behaviors that allow us to identify the first-order coexistence regions, the
Mott critical endpoint and the crossovers along the Widom line in the
supercritical region.
