Band-insulator–metal–Mott-insulator transition in the half-filled $t\text{−}{t}^{\prime }$ ionic Hubbard chain

We investigate the ground-state phase diagram of the half-filled $t\text{−}{t}^{\prime }$ repulsive Hubbard model in the presence of a staggered ionic potential $\Delta$ using the continuum-limit bosonization approach. We find that with increasing on-site repulsion $U$, depending on the value of the next-nearest-neighbor hopping amplitude $t^{\prime }$, the model shows three different versions of the ground-state phase diagram. For $t^{\prime }<t_c^{{*}^{\prime }}$, the ground-state phase diagram consists of the following three insulating phases: band insulator at $U<U_c$, ferroelectric insulator at $U_c<U<U_s$, and correlated (Mott) insulator at $U>U_c$. For $t^{\prime }>t_c^{\prime }$ there is only one transition from a spin-gapped metallic phase at $U<U_c$ to a ferroelectric insulator at $U>U_c$. Finally, for intermediate values of the next-nearest-neighbor hopping amplitude $t_c^{{*}^{\prime }}<t^{\prime }<t_c^{\prime }$ we find that with increasing on-site repulsion, at $U_{c1}$ the model undergoes a second-order commensurate-incommensurate-type transition from a band insulator into a metallic state and at larger $U_{c2}$ there is a Kosterlitz-Thouless-type transition from a metal to a ferroelectric insulator.

Figure 1

(Color online) Qualitative sketch of the ground-state phase diagram of the half-filled $t\text{−}{t}^{\prime }$ ionic Hubbard chain for $t=1$ and fixed $\Delta >0$ as a function of the parameters $t^{\prime }\ge 0$ and $U\ge 0$. At $U=0$ the critical value of the next-nearest-neighbor hopping amplitude $t_c^{\prime }>0.5t$ corresponds to the insulator-to-metal transition point in the case of a free ionic chain. At $t^{\prime }=0$, two critical values of the on-site Hubbard repulsion $U=U_c^{ch}$ and $U=U_c^{sp}$ correspond, respectively, to the (charge) transition from the BI to a FI and to the (spin) transition from the FI phase to the MI in the ground state of the standard ionic Hubbard model. Coordinates of the tricritical point $A(U_c^{*},{t_c^{*}}^{\prime })$ determine, at $t=1$ and for the given $\Delta >0$, the minimal critical value of the parameter ${t_c^{*}}^{\prime }>0.5t$ below which the metallic phase is absent and the minimal critical value of the on-site repulsion $U_c^{*}$, below which the FI phase is absent. Solid lines mark the phase transition lines where the charge gap closes. The dashed line corresponds to the phase transition line where the spin closes. The dash-dotted line $t^{\prime }=0.5t$ is given as a guide to the eye.

Figure 2

(Color online) Qualitative sketch of behavior of the single-particle (solid line), spin (dashed line), and optical (dash-dotted line) gap as a function of the on-site repulsion $U$ based on the exact numerical results obtained in Refs. 9, 10.

Figure 3

(Color online) Sector of the ground-state phase diagram of the half-filled $t\text{−}{t}^{\prime }$ ionic Hubbard model where, for $t_c^{{*}^{\prime }}<t^{\prime }<t_c^{\prime }$ and with increasing on-site repulsion, at $U_{c1}$ the model undergoes a second-order commensurate-incommensurate-type transition from a band insulator into a metallic state $\left(A_1\right)$ and at larger $U_{c2}$ a Kosterlitz-Thouless-type transition from a metal into a ferroelectric insulator $\left(A_2\right)$. Coordinates of the tricritical point $A(U_c^{*},t_c^{{*}^{\prime }})$ determine the minimal critical value of the parameter $t_c^{{*}^{\prime }}>0.5t$ (for given $t$ and $\Delta$) below which the metallic phase is absent and the minimal critical value of the on-site repulsion $U_c^{*}$ below which the FI phase is absent. Solid lines mark the phase transition lines. The dash-dotted line $t^{\prime }=0.5t$ is given as a guide to the eye.