$\mathbb{Z}_2$ 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. From extensive numerics, 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 (temperature of the extremum 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, our computational approach reproduces the key features of the doping dependence of the compressibility demonstrated in recent cold-atom quantum simulators of the Hubbard model, thereby corroborating our conclusions on $\mathbb{Z}_2$ universality.