A $Γ$-valley Moiré Platform for Tunable Square Lattice Hubbard Model

Abstract

Moiré superlattices have emerged as a premier platform for simulating the Hubbard model, yet achieving high tunability in square-lattice systems remains a key challenge. We demonstrate that $Γ$-valley twisted square homobilayers provide a faithful and highly tunable realization of $t-t'-U$ Hubbard model, extending the recent proposal in M-valley systems. We show that at small twist angles, an emergent layer-exchange symmetry decouples electronic states into flat bands residing on two nested square sublattices. An interlayer displacement field breaks this symmetry to induce controllable inter-sublattice hybridization, enabling wide-range experimental tuning of the effective hopping ratio $t'/t$. By establishing a direct correspondence between $Γ$- and M-valley systems, we provide a unified framework for understanding displacement-field tunability in square moiré physics. These findings establish $Γ$-valley twisted bilayers as a versatile platform for simulating the square-lattice Hubbard model and exploring its rich landscape of correlated phenomena.

Figures (6)

• Figure 1: (a) Band structure of the continuum model at $\vartheta=5^\circ$ with parameters $m=0.6m_{\text{e}}$, $a_0=5$ Å, $w_0=15$ meV, $w_1=55$ meV, $D=0$. The inset highlights the top flat bands. The two topmost bands share identical dispersion and are split by $2w_0$. The middle panels show the symmetry-adapted maximally localized Wannier functions for the topmost six bands. The black square indicates the moiré unit cell. Brown denotes layer-bonding states at the Wyckoff position $(0,0)$, while jasper denotes antibonding states at $(1/2,1/2)$, consistent with the color scheme in (b). (b) Schematic of the moiré square superlattice. Layer-bonding and antibonding states reside at B and A sublattice, respectively. The parameters $t$, $t'$, and $t"$ denote the first-, second-, and third-nearest-neighbor hoppings of the effective single-band model. Red and blue arrows indicate second-order virtual processes contributing to $t$ and $t'$, respectively.
• Figure 2: Dependence of the hopping parameters $t$, $t'$, $t"$ and their ratios $t'/t$, $t"/t$ on the displacement field $D$. (a),(b) Results for the topmost band (layer-bonding state localized on the B sublattice). (c),(d) Results for the second topmost band (layer-antibonding state localized on the A sublattice). The gray shaded region in (d) indicates the regime where $t\approx 0$, leading to a divergence of $t'/t$. The continuum model parameters are the same as in Fig. \ref{['fig1']}.
• Figure 3: Perturbative understanding of the displacement-field dependence of the hopping parameters $t$ and $t'$. (a) Layer-bonding state. (b) Layer-antibonding state. Here $\Delta t \equiv t(D)-t(0)$ and $\Delta t' \equiv t'(D)-t'(0)$. Symbols (squares and circles) denote $\Delta t$ and $\Delta t'$ obtained from single-band (SB) projections of the continuum model, as shown in Fig. \ref{['fig2']}. Solid and dashed lines represent the corresponding results for $\Delta t$ and $\Delta t'$, respectively, calculated perturbatively from the six-band tight-binding (sTB) model. In the shown range of $D$, the topmost six bands remain well isolated from higher-energy bands.
• Figure 4: $t$-$t'$-$U$ model for the second topmost band. (a),(b) Displacement-field dependence of the hopping parameters $t$, $t'$, $t"$ and their ratios $t'/t$, $t"/t$. (c) Bandwidth of the second topmost band ($W_2$) and its energy separations from the first ($W_{12}$) and third ($W_{23}$) bands. (d) Ratio $U/t$, obtained by projecting the screened Coulomb interaction onto the second topmost band with $\epsilon=75$ and $\xi=3$ nm. All results are calculated for $\theta=8^\circ$, $m=0.6m_{\mathrm{e}}$, $w_0=35$ meV, and $w_1=75$ meV.
• Figure 5: Lattice and band structures of monolayer. (a) Schematic of the 2D tight-binding model. The gray and black squares mark the primitive and enlarged unit cells, respectively. (b),(c) Band structures corresponding to the primitive and enlarged unit cells, respectively. The inset in (b) shows the original and reduced Brillouin zone.