Time-reversal symmetry breaking Weyl semimetal and tunable quantum anomalous Hall effect in a two-dimensional metal-organic framework

Exploring the relationship between magnetism and band topology has become a popular research topic in condensed matter physics. In this work, we investigate this interplay in a two-dimensional (2D) metal-organic framework (MOF) ${\mathrm{Mn}}_{3}{({\mathrm{C}}_{6}{\mathrm{O}}_{6})}_{2}$. We discover that this MOF has a ``soft'' ferromagnetic property, making it possible to control the topological phases by adjusting the magnetization direction. In its ground state, the MOF has Weyl points precisely located at the Fermi level and only in one spin channel, which means it can be described as an ideal, fully spin-polarized 2D Weyl semimetal. We determine that the stability of these Weyl points is protected by the vertical mirror symmetries. Particularly, the Weyl point can be preserved even with considering the spin-orbital coupling, with an in-plane magnetization direction parallel to the mirror normal vector. By altering the in-plane magnetization away from the normal vector of the vertical mirror symmetry, we demonstrate that the Weyl point is gapped, transformed into the quantum anomalous Hall (QAH) phase, resulting in a chiral edge state at the boundary. We calculate a Chern number to capture the main feature of the QAH, which indicates quantized Hall conductance. Furthermore, by rotating the magnetization direction in the plane, we discover that the Weyl point is a critical point where the QAH changes the sign of the Chern number, along with inversing the propagating direction of the chiral edge state. Overall, our findings highlight ${\mathrm{Mn}}_{3}{({\mathrm{C}}_{6}{\mathrm{O}}_{6})}_{2}$ as a promising material candidate for studying the interplay between topology and magnetism, Weyl semimetal, and QAH phases.