$\mathbb{Z}_2$ Universality of the Mott Transition

Abstract

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 $|U-U_c|^α$ where $α\approx 1$. 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.

Figures (5)

• Figure 1: (a) Low-temperature compressibility, $\chi$, as a function of $U/W$ in band HK model, where $W$ is the bandwidth, in $d=1$, $2$, $3$ and $4$. (b) The compressibility as a function of temperature for a metal (yellow), an insulator (green), and a metal around the Mott transition (purple). (c) Low-temperature compressibility $\chi$ as a function of $U$in $4$-MMHK model on 2-d square lattice with $t'/t=-0.25$. (d) Low-temperature compressibility $\chi$ as a function of $U$in $4$-MMHK model on a 2-d triangular lattice.
• Figure 2: (a) The widom lines: $T_m$, the temperature at which the compressibility $\chi$ has an extremum, as a function of $U$ for band HK model for $d=1, 2, 3, 4$ and $d=\infty$ dimensions. (b) Scaled compressibility $\chi/\chi_c$ versus scaled temperature $T/T_c$ for $U>U_c$ for band HK model on $2$D square lattice. (c) The widom line for $4$-MMHK model on a 2-d square lattice with $t'/t=-0.25$. (d) The widom line for $4$-MMHK model on a 2-d triangular lattice.
• Figure 3: The gap size $\Delta$ as a function of $U$ for (a) band HK model on $2$-d square lattice with $t'=0$, (b) $4$-MMHK model on square lattice with $t'/t=-0.25$, (c) $4$-MMHK model on triangular lattice. The Mott transition is marked by the vertical dashed line.
• Figure 4: Dots: The quantum simulator compressibility as a function of doping level in $2$D square lattice Hubbard model with $U/t=7.00(4)$ (permission from coldatom), connected by solid lines. Dashed lines: computed compressibility in the doped 4-MMHK model on $2$D square lattice at $U/t = 7$. The yellow lines are high-temperature $(T /t = 0.35)$ data, the blue lines are data at low temperature $(T /t = 0.08)$.
• Figure 5: (a) double occupancy $D$ as a function of $U$ for band HK model in $d=1$, $2$, $3$, $4$, and $\infty$. (b) Double occupancy $D_4$ as a function of $U$ in $4$-MMHK model on 2-d square lattice with $t'/t=-0.25$. (c) Double occupancy $D_4$ as a function of $U$ in $4$-MMHK model on 2-d triangular lattice.