Topological phase transitions driven by topological number change and next-nearest-neighbor hopping in skyrmion crystals

By considering the bulk-edge correspondence of skyrmion crystal this work focused on relation topological properties to polarity ($Q_{\rm{sk}}$), next-nearest-neighbor hopping ($t'$), and exchange coupling ($J$). We found that by continually increasing skyrmions from 1 2, monopole-lattice phase dipole-lattice both span a wide range in parameter space demonstrating robustness. While next-nearest-neighbor-hopping electrons coexists with nearest-neighbor flux pattern emergent magnetic field traversing two-dimensional SkX is reshaped giving rise different Hall effect. $t'$ 0 $t$ fixed be 1, we Chern numbers electronic bands change unity indefinite values transition. also considered effect finite $J$ results obtained adiabatic limit hold around $J=t$.