Persistent homology as efficient phase descriptor for 2D skyrmion lattices

Abstract

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 indicator for phase identification remains elusive. Here, we propose an indicator for topological phase transitions using persistent homology (PH). PH provides a complementary topological description by capturing persistent 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 ⟨|Ψ6|⟩), 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. The proposed topological indicator remains computationally tractable for the system sizes considered here, while preserving the topological invariant-based structural information.