Quantum metal-insulator transition at half-filling

For the Su-Schrieffer-Heeger-Hubbard model with spin-1/2 electrons at half-filling, we show that there is a metal-insulator transition when both the phonon and electron fluctuations are taken into account. At anti-adiabatic limit, the transition happens at U/t 0 = 2πλ, where λ is the dimension-less e-p coupling constant.