Persistent Homology-Based Indicator of Orientational Ordering in Two-Dimensional Quasi-Particle Systems Applied to Skyrmion Lattices

Two-dimensional (2D) particle systems, such as magnetic skyrmions, exhibit topological phase transitions between unique 2D phases. However, a simple and computationally efficient methodology to capture lattice configurational properties and construct an appropriate, easily calculable descriptor for phase identification remains elusive. Here, we 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 indicator, which selectively counts stable features in a persistence diagram, effectively traces the lattice's ordering changes, as confirmed by comparisons with the conventionally used measure of the ordering (the magnitude of the orientational order parameter $\langle|Ψ_6|\rangle$), typically used to identify lattice phases. We demonstrate the applicability of our indicator to experimental data, showing that it yields results consistent with those of simulations. This experimental validation highlights the robustness of the proposed method for real physical systems beyond idealized simulated systems. 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 interacting particles. As a key advantage, our indicator offers lower computational complexity than the conventionally used measures.