Non-Abelian topological defects and strain mapping in 2D moir\'e materials

We present a general method to analyze the topological nature of the domain boundary connectivity that appeared in relaxed moir\'e superlattice patterns at the interface of 2-dimensional (2D) van der Waals (vdW) materials. At large enough moir\'e lengths, all moir\'e systems relax into commensurated 2D domains separated by networks of dislocation lines. The nodes of the 2D dislocation line network can be considered as vortex-like topological defects. We find that a simple analogy to common topological systems with an $S^1$ order parameter, such as a superconductor or planar ferromagnet, cannot correctly capture the topological nature of these defects. For example, in twisted bilayer graphene, the order parameter space for the relaxed moir\'e system is homotopy equivalent to a punctured torus. Here, the nodes of the 2D dislocation network can be characterized as elements of the fundamental group of the punctured torus, the free group on two generators, endowing these network nodes with non-Abelian properties. Extending this analysis to consider moir\'e patterns generated from any relative strain, we find that antivortices occur in the presence of anisotropic heterostrain, such as shear or anisotropic expansion, while arrays of vortices appear under twist or isotropic expansion between vdW materials. Experimentally, utilizing the dark field imaging capability of transmission electron microscopy (TEM), we demonstrate the existence of vortex and antivortex pair formation in a moir\'e system, caused by competition between different types of heterostrains in the vdW interfaces. We also present a methodology for mapping the underlying heterostrain of a moir\'e structure from experimental TEM data, which provides a quantitative relation between the various components of heterostrain and vortex-antivortex density in moir\'e systems.