Z_2 Universality of the Mott Transition

We demonstrate that the Mott transition exhibits universal scaling as a consequence of the breaking of a $\mathbb{Z}_2$ symmetry in momentum space. A direct consequence of this discrete symmetry breaking is the charge or Mott gap itself. We proffer that it is the charge compressibility that acts as the underlying order parameter as it is zero in the insulator and non-zero in the metallic state. Additionally, the Widom line (temperatures of the extrema of the compressibility) obeys a universal scaling of $T_m=0.39U$ deep into the insulating state directly from $Z_2$ universality. Furthermore, the temperature at which the second derivative of the compressibility has a minimum is independent of lattice geometry, exhibiting a universal scaling of $|U-U_c|^\alpha$ where $\alpha\approx 1$. Finally, a minimum number of momentum mixing resembles some crutial features in the doped regime achieved in recent cold-atom quantum simulators.