Real-Space Imaging and Topological Indices of Finite Skyrmion Systems

We investigated the topological properties of finite skyrmion systems under open boundary conditions using a tight-binding model. To characterize topological order in finite systems without translational symmetry, we evaluated the local Chern marker (LCM) [Formula: see text], which provides a real-space representation of the Chern number. The LCM is evaluated using the lowest occupied states, revealing a clear correlation between the skyrmion spin texture and the spatially resolved contribution to the Chern number. In addition, we computed the Bott index, a real-space topological invariant, by constructing the ground-state projector from occupied electronic states, while varying the electron filling for a fixed band structure. By systematically increasing the supercell size L and varying the number of occupied electronic states, we examined the convergence behavior of the Bott index. We demonstrated that the Bott index provides an accurate description of the electronic topology in finite skyrmion systems, and that, in the limit of large supercell size L, it converges to the Chern number obtained under periodic boundary conditions.