diagram of the half-filled Peierls-Hubbard model on the triangular lattice. The phase boundary is estimated from the Gutzwiller MD simulations. The nearest-neighbor hopping $t^{(0)}_{1} = 1$ serves as the units for energy and temperature. The critical line $U_c(T)$ ends at $T = 0$ and $T_c \approx 0.9$. (b) The spatially averaged double-occupancy $\langle D \rangle = \sum_i^N D_i/N$ versus the temperature with the Hubbard repulsion set at $U = 13.1$. The shaded area marked the coexistence region of the phase transition. ](phase-diagram.pdf){width="0.99\columnwidth"}

A schematic phase diagram of the half-filled Peierls-Hubbard model is shown in Fig. \[fig:phase-diagram\](a) based on the Gutzwiller MD simulations. Similar phase diagrams are obtained from the finite-temperature Gutzwiller calculation [@wang10; @mezio17] as well as DMFT [@georges96; @bulla01; @liebsch09]. The first-order MIT line $U_c(T)$ ends at two critical points at $T = 0$ and a finite $T_c > 0$. The first analytical theory of the zero-temperature MIT was proposed by Brinkman and Rice using the Gutzwiller method [@brinkman70]. In the Brinkman-Rice scenario, the double-occupancy decreases linearly with increasing $U$ according to: $D(U) = \frac{1}{4}(1 - U/U_c)$, where the critical $U_c = 8 |E_{\rm band}|$ and $E_{\rm band}$ is the electron band energy. It is worth noting that the 