LAUSR

Weyl semimetals (WSMs) have Weyl nodes where conduction and valence bands meet in the absence of inversion or time-reversal symmetry (TRS) or both. The TRS-broken WSM phase can be driven from a topological Dirac semimetal by magnetic field $\mathbf{B}$ or magnetic dopants, considering that Dirac semimetals have degenerate Weyl nodes stabilized by rotational symmetry, i.e., Dirac nodes. Here we develop a Wannier-function-based tight-binding (WF-TB) model to investigate the formation of Weyl nodes and nodal rings induced by $\mathbf{B}$ field in the topological Dirac semimetal ${\mathrm{Na}}_{3}\mathrm{Bi}$. The field is applied along the rotational axis. So far, studies of $\mathbf{B}$ field induced WSMs have been limited to cases with effective models. Remarkably, our study based on the WF-TB model shows that upon $\mathbf{B}$ field each Dirac node is split into four separate Weyl nodes along the rotational axis near the Fermi level; two nodes with Chern number $\ifmmode\pm\else\textpm\fi{}1$ (single Weyl nodes) and two with Chern number $\ifmmode\pm\else\textpm\fi{}2$ (double Weyl nodes). This result is in contrast to the common belief that each Dirac node consists of only two single Weyl nodes with opposite chirality. In the context of the $4\ifmmode\times\else\texttimes\fi{}4$ effective models, the existence of double Weyl nodes ensures nonzero cubic terms in momentum. We examine the evolution of Fermi arcs at a side surface as a function of chemical potential. The number of Fermi arcs at a given chemical potential is consistent with the corresponding Fermi surface Chern numbers. Our study also reveals the existence of nodal rings in the mirror plane near the Fermi level upon $\mathbf{B}$ field. These nodal rings persist with spin-orbit coupling. Our WF-TB model can be used to compute interesting features such as anomalous Hall and thermal conductivities, and our findings can be applied to other topological Dirac semimetals like ${\mathrm{Cd}}_{3}{\mathrm{As}}_{2}$.