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Noncollinear Magnetic Modulation of Weyl Nodes in Ferrimagnetic Mn$_3$Ga

The tetragonal ferrimagnetic Mn$_3$Ga exhibits a wide range of intriguing magnetic properties. Here, we report the emergence of topologically nontrivial nodal lines in the absence of spin orbit coupling (SOC) which are protected by both mirror and $C_{4z}$ rotational symmetries. In the presence of SOC we demonstrate that the doubly degenerate nontrivial crossing points evolve into $C_{4z}$-protected Weyl nodes with chiral charge of $\pm$2. Furthermore, we have considered the experimentally reported noncollinear ferrimagnetic structure, where the magnetic moment of the Mn$_I$ atom (on the Mn-Ga plane) is tilted by an angle $θ$ with respect to the crystallographic $c$ axis. The evolution of the Weyl nodes with $θ$ reveals that the double Weyl nodes split into a pair of charge-1 Weyl nodes whose separation can be tuned by the magnetic orientation in the noncollinear ferrimagnetic structure.