Fingerprint of parity anomaly for localized magnetic states in quantum anomalous Hall systems

We investigate the local magnetic states of impurities in quantum anomalous Hall (QAH) systems and observe that with an increasing band gap, the magnetic region of impurities expands in the QAH phase, while it contracts in the ordinary insulator (OI) phase. During the transition between the QAH and the OI phase, the magnetization area undergoes a significant transformation from a broad region to a narrow strip, which serves as a distinctive characteristic of the parity anomaly in the localized magnetic states. Furthermore, the presence of the parity anomaly leads to notable alterations in the dependence of the magnetic moment and magnetic susceptibility on the Fermi energy. Additionally, we analyze the spectral function of the magnetic impurity as a function of Fermi energy for both the QAH and OI phases.