Intrinsic Moiré Higher-Order Topology Beyond Effective Moiré Lattice Models

Abstract

Moiré superlattices provide a compelling platform for exploring exotic correlated physics. Electronic interference within these systems often results in flat bands with localized electrons, which are typically described by effective moiré lattice models. While conventional models treat moiré sites as indivisible, analogous to atoms in a crystal, this picture overlooks a crucial distinction: unlike a true atom, a moiré site is composed of tens to thousands of atoms and is therefore spatially divisible. Here, we introduce a universal mechanism rooted in this spatial divisibility to create topological boundary states in moiré materials. Through tight-binding and density functional theory calculations, we demonstrate that cutting a moiré site with a physical boundary induces bulk topological polarization, generating robust boundary states with fractional charges. We further show that when the net edge polarization is canceled, this mechanism drives the system into an intrinsic moiré higher-order topological insulator (mHOTI) phase. As a concrete realization, we predict that twisted bilayer tungsten disulfide ($WS_2$) is a robust mHOTI with experimentally detectable corner states when its boundaries cut through moiré hole sites. Our findings generalize the theoretical framework of moiré higher-order topology, highlight the critical role of edge terminations, and suggest new opportunities for realizing correlated HOTIs and higher-order superconductivity in moiré platforms.

Figures (4)

• Figure 1: (a) In atomic lattices, each site corresponds to a single atom, making it impossible to "cut" the site. (b) In moiré superlattices, each moiré site corresponds to a spatially extended electron orbital spanning many atoms, allowing moiré lattice sites to be, in principle spatially divisible.
• Figure 2: Emergent topological boundary states in a 1D moiré superlattice. (a) Schematic of the simplified 1D moiré superlattice model. Lower panel shows the moiré potential and the charge density of the bottom band. (b) Moiré band structure with a conventional unit cell (inset) for $v/t=0.3$ and $L_M=10$. The red dashed line represents a fit of effective lattice model. Parity eigenvalues at high-symmetry points are labeled by “$\pm$”. (c) Moiré band structure with an unconventional unit cell (inset) for $v/t=-0.3$. (d) Open-boundary energy spectrum of (c), with $N=40$ unit cells. The x-axis represents the total number of electrons, where N, 2N, 3N correspond to the full filling of the first, second, and third bands in (c), respectively. Inset shows the wavefunctions of the two topological boundary states at full filling of the first band (dashed line).
• Figure 3: Emergent topological boundary states in a 2D moiré superlattice. (a) Schematic of the square lattice model with a moiré potential. The solid box marks the moiré unit cell. (b) Moiré band structure for $v/t=-0.3$ and $L_M=10$. Inset shows the moiré Brillouin zone. Parity eigenvalues at high-symmetry points are labeled by “$\pm$”. (c) Ribbon spectrum of the band structure in (b), finite along the $y$-direction ($N_y=10$). Inset shows the ribbon geometry. (d) Energy spectrum of the band structure in (b), of a finite flake with $N=N_x\times N_y=10\times 10=100$ unit cells. Inset shows the charge density of edge states (bottom red) and corner states (top blue).
• Figure 4: 2D mHOTI in tbWS$_2$. (a) Atomic structure and moiré unit cell of tbWS$_2$. (b) Moiré potential felt by holes at VBM in tbWS$_2$. (c) Moiré band structure for a twist angle of $\theta = 2^\circ$. Inset shows the Brillouin zone. (d) First Moiré band gap and fractional corner charge $Q$ as a function of twist angles. (e) Ribbon spectrum corresponding to the band structure in (c). (f) Energy spectrum under full open-boundary conditions for a $10\times10$ supercell. Inset shows the charge density of edge states and HOTI corner states.