Persistent Homology-Based Descriptor of Topological Ordering in Two-Dimensional Quasi-Particle Systems with Application to Skyrmion Lattices

Abstract: Two-dimensional (2D) quasi-particles systems, such as magnetic skyrmions, can exhibit a rich variety of topological phase transitions. However, the methodology for capturing the configurational properties of the lattice ordering and constructing an appropriate descriptor that can be easily calculated is not obvious. Here, we present a topological descriptor, "persistent diagram", and propose an indicator for topological phase transitions using persistent homology (PH). PH offers novel insights beyond conventional indicators by capturing topological features derived from the configurational properties of the lattice. The proposed persistent-homology-based topological indicator, which selectively counts stable features in the persistence diagram, effectively traces the lattice's topological changes, as confirmed by comparisons with the conventionally used measure of the ordering $\langle|\Psi_6|\rangle$, typically used to identify lattice phases. While our method is demonstrated in the context of skyrmion lattice systems, the approach is general and can be extended to other two-dimensional systems composed of repulsively interacting quasi-particles. Moreover, our indicator offers lower computational complexity than the conventionally used methods.