Symmetry-enforced Moiré Topology

TL;DR

The paper addresses the challenge of predicting moiré band topology in 2D moiré materials without expensive first-principles calculations. It develops a symmetry-based framework that connects atomic symmetry data at the Γ point with the moiré symmetry group to enforce nontrivial topology in the low-energy moiré bands under a weak moiré potential, yielding 197 qualifying combinations and 92 candidate monolayers. A concrete moiré cubic Rashba model verifies the principle, showing insulator and semimetal topologies depending on irrep content and demonstrating robustness to twist angle and moiré details. The approach generalizes to other valleys and provides design guidelines for discovering new topologically nontrivial moiré materials, with implications for correlated phases and moiré-based devices.

Abstract

Topological flat bands in two-dimensional (2D) moiré materials have emerged as promising platforms for exploring the interplay between topology and correlation effects. However, realistic calculations of moiré band topology using density functional theory (DFT) are computationally inefficient due to the large number of atoms in a single moiré unit cell. In this work, we propose a systematic scheme to predict the topology of moiré bands from atomic symmetry data and moiré symmetry group, both of which can be efficiently extracted from DFT. Specifically, for $Γ$-valley electron gases, we find that certain combinations of atomic symmetry data and moiré symmetry groups can enforce nontrivial band topology in the low-energy moiré bands, as long as the moiré band gap is smaller than the atomic band splitting at the moiré Brillouin zone boundary. This symmetry-enforced nontrivial moiré topology, including both topological insulators and topological semimetals, is robust against various material-specific details such as the precise form and strength of the moiré potential or the exact twist angle. By exhaustively scanning all 2D atomic symmetry data and moiré symmetry groups, we identify 197 combinations that can yield symmetry-enforced nontrivial moiré topology, and we verify one such combination using a moiré model with cubic Rashba spin-orbit coupling. By screening the existing 2D material database, we currently identify 92 monolayer materials with (i) the low-energy bands near $Γ$ and (ii) the atomic symmetry data that belong to those combinations. Our approach is generalizable to other valleys and provides a useful guideline for experimental efforts to discover and design new topologically nontrivial moiré materials.

Figures (4)

• Figure 1: Theoretical Approaches to Identify Moiré Topology. All green blocks are easily computationally accessible, while the red block is computationally inefficient. Our work provides an efficient way to indicate moiré topology from the moiré symmetry group and atomic symemtry data.
• Figure 2: Schematics of Symmetry-Enforced Moiré Topology. (a) The band structure around $\Gamma$ point of a pristine layered material with point group $C_{6v}$. The low-energy band at $\Gamma$ is described by the 2D $\bar{\Gamma}_7$ irrep. (b) Band folding at zero moiré potential. At ${\text{K}}_M$, the folded bands form a three-dimensional representation $\bar{\text{K}}_4 \oplus \bar{\text{K}}_6$, while at ${\text{M}}_M$ they form a two-dimensional irrep $\bar{\text{M}}_5$. $\Delta_A$ marks the energy scale of the atomic band splitting at the mBZ boundary. (c) One possible moiré band structure in the presence of weak moiré potential, i.e., the energy splitting generated by $\Delta_M$ is smaller than $\Delta_A$. Here the top two bands (red) carry the $\bar{\text{K}}_6$ irrep, while the third top band has the $\bar{\text{K}}_4$ irrep, resulting in nontrivial topology in the isolated top two bands. (d) The other possible moiré band structure in the presence of weak moiré potential. The $\bar{\text{K}}_4$ band has higher energy than the two $\bar{\text{K}}_6$ bands, resulting in a topological semi-metallic phase for the red bands due to the unavoidable touching between the second and third top bands at $\text{K}_M$.
• Figure 3: Cubic Rashba SOC model under moiré potential. (a) Phase diagram of the topmost two bands of the moiré cubic Rashaba model in \ref{['eq_main:HammC']}. $E_\Delta$ denotes the band gap between the second and third topmost bands. The red dashed lines separate the insulator region and semimetal region. (b) The band dispersion and band irreps computed with the parameters labeled by the blue cross point in (a). The top two bands have $\mathbb{Z}_2 = 1$. (c) The band dispersion and band irreps computed with the parameters labeled by the red cross point in (a). The top two bands are connected to the top third band.
• Figure 4: (a) Band dispersion of BiBrTe. (b) Band dispersion of BiFTe. The irreps of VBM for both materials are $\bar{\Gamma}_4\bar{\Gamma}_5$. The inset figures show a zoom-in of the regions inside the dashed boxes.