Remote control of spin polarization of topological corner states

In two-dimensional higher-order topological insulators, the corner states are separated by a non-negligible distance. The crystalline symmetries protect the robustness of their corner states with long-range entanglement, which are robust against time-reversal breaking perturbations. Here, we demonstrate the possibility of direct control of the topological corner states by introducing the spin degree of freedom in a rhombus-shaped Kekul\'{e} nanostructure with local magnetization and local electric potential. By applying a local magnetization on one corner, the other corner can also be strongly spin polarized. By further applying a local electric potential at the same corner, the sign of the spin polarization can be reversed at both corners. We demonstrate the material realization in a $\gamma$-graphyne nanostructure with Mn adsorption and Si replacement at one corner by using the first-principles calculations. Our studies give a showcase of the remote correlation of quantum states in higher-order topological materials for spintronic and quantum applications.