By considering the topological Hall conductivity and the nanoribbon edge states, i.e., the bulk-boundary correspondence of the skyrmion crystal (SkX), this work focused on the relation of the topological properties to the next-nearest-neighbor (nnn) hopping ($t^{\prime }$), the exchange coupling strength (J), and the topological number of the skyrmion ($Q_{\mathrm{sk}}$). When the nnn hopping of the electrons coexists with the nearest-neighbor hopping, the flux pattern of the emergent magnetic field traversing the SkX is reshaped giving rise to different topological Hall effect. By increasing $t^{\prime }$ from 0 to t, we found the Chern numbers of the electronic bands change from unity to indefinite values demonstrating a phase transition. By continually increasing $Q_{\mathrm{sk}}$ from 1 to 2, we found topological phase transitions occur at $Q_{\mathrm{sk}}=1.3$ and 1.7. We also considered the effect of finite J and found the results obtained in the adiabatic limit ($J\gg t$) hold to around $J=t$.

通过考虑拓扑霍尔电导率和纳米带边缘状态，即天体离子晶体（SkX）的体-边界对应关系，这项工作着眼于拓扑特性与下一近邻（nnn）跳跃之间的关系（${Ť}^{\prime }$），交换耦合强度（J）和天蝎子的拓扑数（${问}_{\mathrm{sk}}$）。当电子的nnn跳变与最近邻的跳变并存时，穿过SkX的出射磁场的通量模式将被重塑，从而产生不同的拓扑霍尔效应。通过增加${Ť}^{\prime }$从0到t，我们发现电子带的Chern数从1变为不确定的值，表明发生了相变。通过不断增加${问}_{\mathrm{sk}}$ 从1到2，我们发现拓扑相变发生在 ${问}_{\mathrm{sk}}=1.3$和1.7。我们还考虑了有限J的影响，并发现在绝热极限（$Ĵ\gg Ť$）坚持 $Ĵ=Ť$。