  Computing superconducting properties starting from an exactly solvable model
for a doped Mott insulator stands as a grand challenge. We have recently shown
that this can be done starting from the Hatsugai-Kohmoto (HK) model which can
be understood generally as the minimal model that breaks the non-local $\mathbb
Z_2$ symmetry of a Fermi liquid, thereby constituting a new quartic fixed point
for Mott physics [Phillips et al., Nature Physics 16, 1175 (2020); Huang et
al., Nature Physics (2022)]. In the current work, we compute the
thermodynamics, condensation energy, and electronic properties such as the NMR
relaxation rate $1/T_1$ and ultrasonic attenuation rate. Key differences arise
with the standard BCS analysis from a Fermi liquid: 1) the free energy exhibits
a local minimum at $T_p$ where the pairing gap turns on discontinuously above a
critical value of the repulsive HK interaction, thereby indicating a
first-order transition, 2) a tri-critical point emerges, thereby demarcating
the boundary between the standard second-order superconducting transition and
the novel first-order regime, 3) Mottness changes the sign of the quartic
coefficient in the Landau-Ginzburg free-energy fuctional relative to that in
BCS, 4) as this obtains in the strongly interacting regime, it is Mott physics
that underlies the generic first-order transition, 5) the condensation energy
exceeds that in BCS theory suggesting that multiple Mott bands might be a way
of enhancing superconducting, 6) the heat-capacity jump is non-universal and
increases with the Mott scale, 7) Mottness destroys the Hebel-Slichter peak in
NMR, and 8) Mottness enhances the fall-off of the ultrasonic attenuation at the
pairing temperature $T_p$. As several of these properties are observed in the
cuprates, our analysis here points a way forward in computing superconducting
properties of strongly correlated electron matter.
