Weyl nodal lines, Weyl points and the tunable quantum anomalous Hall effect in two-dimensional multiferroic metal oxynitride: TlNO.

The investigation of two-dimensional (2D) multiferroic and topological quantum phases is a significant topic in current condensed matter physics. In this study, we discover quantum topological phases in the multiferroic material TlNO. We observe that its ferroelectric (FE) phase displays a ferromagnetic ground state with magnetization favoring in-plane orientation. In the absence of spin-orbit coupling (SOC), a Weyl nodal loop around the Fermi level is evident, representing a 1D band crossing between spin-up and spin-down states. When spin-orbit coupling is taken into account, setting the magnetization in-plane, the Weyl nodal loop becomes gapped. Additionally, a pair of 2D Weyl nodes appear on the high-symmetry path, protected by a vertical mirror symmetry allowed by the magnetization. Remarkably, we prove that the Weyl nodes are situated at the topological phase transition between two quantum anomalous Hall (QAH) phases with opposite Chern numbers. Therefore, by adjusting the magnetization, it is possible to switch the propagation direction of chiral edge states. Furthermore, from its ferroelectric state to a paraelectric state, the time-reversal symmetry breaking nodal line is transformed into a Weyl point, achieving 100% spin polarization. Particularly, the Weyl points remain robust against SOC when the vertical mirror symmetry is preserved. Importantly, we also demonstrate that the Weyl point also represents the transition point where the QAH phase changes the sign of its Chern number. Overall, our study provides new insights into the study of multiferroic and topological phenomena in 2D materials and offers a potential avenue for controlling QAH phases.