Symmetry, Maximally Localized Wannier States, and a Low-Energy Model for Twisted Bilayer Graphene Narrow Bands

We build symmetry-adapted maximally localized Wannier states and construct the low-energy tight-binding model for the four narrow bands of twisted bilayer graphene. We do so when the twist angle is commensurate near the “magic” value and the narrow bands are separated from the rest of the bands by energy gaps. On each layer and sublattice, every Wannier state has three peaks near the triangular moiré lattice sites. However, each Wannier state is localized and centered around a site of the honeycomb lattice that is dual to the triangular moiré lattice. The space group and the time-reversal symmetries are realized locally. The corresponding tight-binding model provides a starting point for studying the correlated many-body phases.

Popular Summary

When two stacked layers of graphene are rotated relative to one another by a “magic” angle of about 1.10°, the stack behaves like a superconductor and as a type of insulator, depending on the concentration of charge carriers. This recent experimental discovery, which also shows a phase diagram curiously similar to unconventional superconductors, suggests that there is a fundamental connection between the insulator phase and superconductivity. To provide a starting point for exploring these connections, we develop a realistic but simple theoretical model that describes electron motion in twisted bilayer graphene.

Interference between the honeycomb structures of the graphene sheets creates a moiré interference pattern, which forms a triangular lattice. We develop mathematical descriptions for how electrons localize at the triangular lattice sites and for the associated tight-binding model, which describes electron motion due to tunneling between the sites. The localized electron states have an intricate pattern: Each is centered on the dual honeycomb lattice site with three peaks at its neighboring triangular lattice sites. The associated tight-binding model reproduces the four narrow conduction bands of twisted bilayer graphene almost exactly.

The structure of the localized electron states has deep implications for the nature of the electron-electron interactions within the four narrow bands. Our work should allow for quantitative descriptions of such interactions and systematic studies of the correlated phases in twisted bilayer graphene.

Figure 1

(a) The superlattice of twisted bilayer graphene. Blue (red) sites are the carbon atoms on the bottom (top) layers. The triangular lattice forms when twisted angle is commensurate. The plot shows the lattice when $m=2$ and $n=1$. (b) The center of the local Wannier states. Black dots are the sites of the triangular superlattice. Red and blue dots are two nonequivalent Wyckoff sites, where the local Wannier states are centered. In our construction, $w_1$ and $w_2$ are placed at one Wyckoff position, and $w_3$ and $w_4$ are placed at another position. Note that the Wychoff sites form an emergent honeycomb lattice.

Figure 2

Red dots: The four narrow bands produced from the tight-binding model with hopping parameters given in Ref. [20] with the twist angle of 1.30°. Blue dots: The interpolated band structure by the projection method.

Figure 3

(a),(b) The square of the magnitude of the Bloch states $|{\psi }_{\mathbf{\Gamma },E_{+},\epsilon }{|}^2$ and $|{\psi }_{\mathbf{\Gamma },E_{-},\epsilon }{|}^2$ and (c) the localization of the WSs obtained from the projected method. The four panels show $|w_1{|}^2$ at (upper left) the top layer sublattice $A$, (upper right) the top layer sublattice $B$, (lower left) the bottom layer sublattice $A$, and (lower right) the bottom layer sublattice $B$.

Figure 4

Comparison of the narrow-band structure produced by the model given by Ref. [20] (red solid line) and the tight-binding model based on the WSs (blue dots) with the range of hopping (a) $L_c=2L$, (b) $L_c=4L$, (c) $L_c=6L$, and (d) $L_c=8L$.