Quantum Monte Carlo study of bilayer ionic Hubbard model

The interaction-driven insulator-to-metal transition has been reported in the ionic Hubbard model (IHM) for moderate interaction $U$, while its metallic phase only occupies a narrow region in the phase diagram. To explore the enlargement of the metallic regime, we extend the ionic Hubbard model to two coupled layers and study the interplay of interlayer hybridization $V$ and two types of intralayer staggered potentials $\mathrm{\Delta }$: one with the same (in-phase) and the other with a $\pi$-phase shift (antiphase) potential between layers. Our determinant quantum Monte Carlo (DQMC) simulations at lowest accessible temperatures demonstrate that the interaction-driven metallic phase between Mott and band insulators expands in the $\mathrm{\Delta }\text{−}V$ phase diagram of bilayer IHM only for in-phase ionic potentials; while antiphase potential always induces an insulator with charge density order. This implies possible further extension of the ionic Hubbard model from the bilayer case here to a realistic three-dimensional model.

Figure 1

Phase diagrams of bilayer ionic Hubbard model at lowest accessible temperature $\beta t=20$ for (a) $U/t=0$ and (b) $U/t=2$ determined via spectral functions and temperature dependence of local Green's functions. Blue (red) color is to differentiate two types of staggered potentials: ${\theta }_2=0$ in-phase and ${\theta }_2=\pi$ antiphase. Only systems with ${\theta }_2=0$ can host an interaction-driven insulator-metal transition (yellow patch). The gray patched regime is unresolved ultimately due to unaffordable larger lattice sizes and lower temperatures. BI, S-BI, and AF-I denote band insulator, singlet insulator, and antiferromagnetic insulator, respectively. The system size is $16\times 16\times 2$.

Figure 2

Comparison of local density of states for potential valley site with varying interlayer hybridization $V$ starting from metallic $\left(\mathrm{\Delta }/t=0.5\right)$ and insulating $\left(\mathrm{\Delta }/t=1.0\right)$ phases in both cases of ${\theta }_2=0$ in-phase and ${\theta }_2=\pi$ antiphase staggered potentials. The hybridization pushes the in-phase (antiphase) system towards a metal (insulator). ${\mathrm{\Delta }}_0$ and ${\mathrm{\Delta }}_{\pi }$ denote ${\theta }_2=0$ and ${\theta }_2=\pi$ separately for simplicity.

Figure 3

Comparison of local density of states for potential valley site with large interlayer hybridization $V\gg \mathrm{\Delta }$. The splitting between the bonding and antibonding bands induced by $V$ results in the formation of tightly bound singlets. The symmetry/asymmetry of the two band peaks indicates different potential environment imposed on the singlets between in-phase (a) and antiphase (b) cases.

Figure 4

Temperature evolution of local $\beta G(\beta /2)$ at potential valley sites for in-phase and antiphase ionic potentials for bilayer $N=14\times 14\times 2$. At weak $\mathrm{\Delta },\beta G(\beta /2)$ decreases with lowering temperature implying the insulating nature of both systems. At larger $\mathrm{\Delta }/t\ge 0.3$, only in-phase ionic potential can host the interaction-driven metallic phase, although the available data are limited by the sign problem for unaffordable larger lattice sizes and lower temperatures.

Figure 5

Antiferromagnetic structure factor of (a) spin density wave $S_{\text{sdw}}$ and (b) charge density wave $S_{\text{cdw}}$ with increasing interlayer hybridization $V$ for both in-phase and antiphase staggered potential systems.

Figure 6

The local $\beta G(\beta /2)$'s dependence on lattice sizes for metals and insulators determined in $N=16\times 16\times 2$ systems. For in-phase potential, the insulators do not show finite-size effects while metals show noticeable effects that do not change the phase qualitatively. The two curves for antiphase potential are only for comparison since Fig. 4 shows that this type of potential cannot host metallic phases.

Figure 7

Finite-size effects of the antiferromagnetic structure factors. Strong finite-size effect of $S_{\text{sdw}}/N$ indicates the absence of antiferromagnetic spin order in the thermodynamic systems; while the lack of $S_{\text{cdw}}/N$'s size dependence confirms the charge density order due to the staggered potential.

Figure 8

Comparison of local $G\left(\tau \right)$ at potential valley site between bilayer with in-phase and antiphase ionic potentials. The curve slope at $\tau =\beta /2$ provides hints on the spectral gap. The flatness of the curves in (a) signal the metallic behavior; while (c) implies a transition from an insulator (blue) to a metal (red). In addition, (b) and (d) indicate the insulating phase.

Figure 9

Similar to Fig. 8 except for the amplitude of ionic potentials. (a) A transition from a metal to an insulator while (b) characterizes an insulator.