$Z_2$ gauge theory description of the Mott transition in infinite dimensions

The infinite-dimensional half-filled Hubbard model can be mapped exactly with no additional constraint onto a model of free fermions coupled in a $Z_2$ gauge-invariant manner to auxiliary Ising spins in a transverse field. In this slave-spin representation, the zero-temperature insulator-to-metal transition translates into spontaneous breaking of the local $Z_2$ gauge symmetry, which is not forbidden in infinite dimensions, thus endowing the Mott transition of an order parameter that is otherwise elusive in the original fermion representation. We demonstrate this interesting scenario by exactly solving the effective spin-fermion model by dynamical mean-field theory both at zero and at finite temperature.

Figure 1

Sketch of the phase diagram of the paramagnetic half-filled Hubbard model on a Bethe lattice [23]. $D=2t$ is half the bandwidth. At $U/D=2.7$, the first-order metal-insulator transition occurs at the temperature $T\approx 0.0037D$.

Figure 2

Left panels: Zero-temperature spectral functions of the physical and auxiliary fermions, $A\left(\omega \right)$ and $A_d\left(\omega \right)$, shown as black and red curves, respectively, for a range of $U$. Right panels: Corresponding Ising spin spectral functions $A_{\sigma }\left(\omega \right)$, which, being the imaginary part of a causal response function, is odd in frequency. For $U/D=3$ the system is in the insulating phase.

Figure 3

(a) Ising magnetization $\langle {\sigma }^x\rangle$ as a function of an external test field $h_x$ in the metallic phase at $U/D=2.6$ and at zero temperature. We note the finite value as $h_x\to 0$. (b) Zero-field Ising magnetization $\langle {\sigma }_x\rangle$ (full black line) and the quasiparticle renormalization factor (full red line) as a function of $U$. For comparison, we also plot ${\langle {\sigma }_x\rangle }^2$ (blue dashed line) and $\sqrt{1-U^2/U_c^2}$ (orange dashed line).

Figure 4

Ising magnetization $\langle {\sigma }^x\rangle$ as a function of an external test field $h_x$ in the metallic phase at $U/D=2$ (top panel) and the insulating one at $U/D=4$ (bottom panel), for different temperatures.

Figure 5

Correlation function $C_{\text{spin}}^{>}\left(\omega \right)$ of the Ising operator ${\sigma }^x$ at $U/D=2.7$, i.e., in the metal phase not far from the transition, for different temperatures. We note the gradual emergence of a $\delta$ peak at $\omega =0$ on decreasing the temperature toward $T=0$.

Figure 6

Spectral functions of the physical electron (top panel), of the auxiliary fermion (middle panel), and of the Ising spin (bottom panel) at $U=2.7D$ upon increasing the temperature across the first-order metal-insulator transition. The insets show the close-ups to the low-frequency behavior. The right-hand inset in the top panel shows the density of states at the chemical potential of the physical electrons at $U=2.7D$ upon increasing the temperature.

Figure 7

(a) Physical-electron spectral function intensity at $\omega =0$ and the slave-spin susceptibility $d\langle {\sigma }_x\rangle /d{h}_x$ at constant temperature $T>T_c$ as functions of $U$. The circles indicate the inflection points on both curves that may serve as possible definitions of the crossover between the bad metal and the Mott insulator. (b) Auxiliary-fermion and (c) slave-spin spectral functions across the metal-insulator crossover.

Figure 8

Impurity free energy for the model with the source term of Eq. (42). We plot the results vs $h$ for a range of temperatures, both in the metallic and in the insulating phase.