Exact Bond Ordered Ground State for the Transition Between the Band and the Mott Insulator

We derive an effective Hamiltonian $H_{eff}$ for an ionic Hubbard chain, valid for $t\ll U,\mathrm{\Delta }$, where $t$ is the hopping, $U$ the Coulomb repulsion, and $\mathrm{\Delta }$ the charge transfer energy. $H_{eff}$ is the minimal model for describing the transition from the band insulator (BI) ($\Delta -U\gg t$)andtheMottinsulator(MI)($U-\Delta \gg t$). Using spin-particle transformations (Phys. Rev. Lett. $\mathbf{\text{86}}$, 1082 (2001)), we map $H_{eff}(U=\mathrm{\Delta })$ into an SU(3) antiferromagnetic Heisenberg model whose exact ground state is known. In this way, we show rigorously that a spontaneously dimerized insulating ferroelectric phase appears in the transition region between the BI and MI.