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

Figures (5)

• Figure 1: Schematic of the analysis workflow: (a) skyrmion coordinates, (b) Persistent Homology (PH) analysis, (c) calculation of a conventional lattice-ordering measure, and (d) comparison between the PH-based indicator and the conventional measure.
• Figure 2: Filtration and persistence diagrams (PDs) in the 0th and 1st homology dimensions. (a) Illustration of the topological features---connected components and loops---emerging during the filtration process, where the radius of disks centered at the skyrmion coordinates is varied. (b1) PD0 (0th-degree homology) and (b2) PD1 (1st-degree homology) with interpretations. For a detailed explanation of the PH filtration process, please refer to Ref. ref14.
• Figure 3: Average persistence diagrams (PDs) of the 0th- and 1st- degree homology, for three states under applied OOP magnetic fields of $B = 60$, $84$, and $108$$\upmu$T, corresponding to solid, hexatic, and liquid phases. The ”Birth” and ”Death” represent the specific times in a filtration process of persistent homology in which topological features emerge and disappear, respectively. The color map represents the multiplicity of generators (scatter plots) in the PD. Insets in the 0th-degree homology PDs display the corresponding real-space configurations, identified using a machine-learning-based, pixel-wise classification algorithm ref33.
• Figure 4: Inverse analysis for the two states at $B = 60$ and $108$$\upmu$T, tracing specific generators in the persistence diagram back to real-space configurations. In each state, the points labeled (a), (b), and (c) correspond to persistent homology generators with large lifetime (a), small lifetime and small birth value (b), and small lifetime and large birth value (c), respectively.
• Figure 5: Correlation between the Persistent Generator Count with Relative Stability $\langle \tilde{\chi} \rangle$ and the conventional orientational order parameter $\langle|\Psi_6|\rangle$ in the experimental data. First derivatives are compared using Gaussian Process Regression. A good correlation is indicated by the Pearson correlation coefficients $r = 0.993$ and $r = 0.861$.