We study the localized magnetic states in the semi-Dirac-like system, and find that due to the existence of the saddle point, the magnetic region diminishes greatly with the energy of the saddle point approaching the impurity energy, and reaches a minimum at the energy of the saddle point equal to the impurity energy. A similar feature is observed in the magnetic moment of the impurity. This suppression behavior for the magnetization of the impurity can be understood from the saddle point induced mitigation of asymmetry on the density of state at impurity energy. In contrast, when the saddle point vanishes, due to the opening of the gap, the magnetic region exhibits a monotonous decay when the conduction band edge goes up through the impurity energy. The combined effect of the saddle point and the Coulomb interaction at the impurity on the local magnetization is also investigated.

We theoretically investigate the third-order nonlinear Hall effect by employing the quantum kinetic equation and present an analytic formula for the third-order harmonic conductivity, where the intraband and the mixed inter-band contributions caused by the external electric field as well as the multiphoton process dressed Berry curvature are unveiled. These results are applied in a two-dimensional (2D) Dirac system. We find that the third-order harmonic conductivity exhibits a crossover from the negative to positive value due to the intraband transition of the electrons in the low-frequency region, whereas a kink or dip structure due to the multiphoton process between the conduction and valence bands in the high-frequency region. In experiment, these features can be identified by detecting the transverse current. Our work deepens the understanding of the multiphoton process in the nonlinear transport of topological materials and enlightens a possible way to its characterization in experiments.

We investigate the localized magnetic state in the tilted Dirac cone system, wherein a lattice staggered potential (LSP) is introduced to create a gap between the conduction and valence bands. Our findings reveal that the breaking of symmetry between the sublattices results in depletion of the magnetic region of the impurity for positive LSP values, while a sharp strip is formed for negative LSP values with an increase in the tilt of the Dirac cone. Interestingly, within the magnetic region, the magnetic moment of the impurity remains constant at 0.8 Bohr magneton irrespective of the sign of LSP. However, the magnetic susceptibility at the edge of the magnetic region displays inconsistent behavior for positive and negative LSP values. We also analyze in detail the variations in the magnetic region, magnetic moment, and magnetic susceptibility with LSP strength at a fixed tilt.

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.