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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.

This article deals with some elliptic complex equations of first order, i.e. the generalized Beltrimi equation with two degenerate lines in the discussed multiply connected domain. We first propose the new well-posed-ness of discontinuous Riemman-Hilbert problem, give estimates of solutions for the modified boundary value problem. Afterwards by using the method of parameter extension, the existence of continuous solutions for the generalized Beltrimi equation is verified. In the article, the proof of Hölder continuity of a singular double integer is very difficult and interesting. The above problem possesses the important applications to the Tricomi problem of mixed type equations of second order. AMS Subject Classification: 35J60, 35J70, 35J56Key Words: elliptic complex equations, two degenerate lines, the new posedness of discontinuous Riemann-Hilbert problem, unique solvability of continuous solutions, Hölder continuity of singular integer