Topology of SO(5) monopoles and three dimensional Dirac semimetals*

The topological properties of Kramers degenerate band structures generally emerge from non-trivial textures of underlying SO(5) Berry’s vector potential. The linear touching between a pair of Kramers degenerate energy bands at isolated points of momentum space along an n-fold axis of rotation gives rise to three dimensional Dirac semimetals, where the Dirac points act as singularities of SO(5) gauge fields. Even though considerable progress has been made in recent years toward understanding the stability of Dirac points in the presence of parity, time reversal and discrete rotational symmetries, as of now there is no clear definition of topological invariants for Dirac points and any arbitrary momentum plane that is perpendicular to the direction of nodal separation. We will show how the implementation of global rotational symmetry causes an U(1) x U(1) abelian gauge fixing of SO(5) vector potential, allowing us to define monopole invariants and quantized ``spin Chern" numbers respectively for the Dirac points and the two dimensional planes perpendicular to the direction of nodal separation. We also demonstrate that generic form of Kramers degenerate Dirac semimetals do not support protected helical Fermi arcs.