Bulk topology of stable three-dimensional Dirac semimetals

The band-touching points of stable, three-dimensional, Kramers-degenerate, Dirac semimetals act as the singularities of non-Abelian, SO(5)-Berry’s connections, and their homotopy classification is still an important, open problem. In this work, we solve this problem by performing second homotopy classification of Berry’s connections, and determine the topological universality classes of stable Dirac semimetals. The generic two-dimensional planes, orthogonal to the direction of nodal separation, and lying between two Dirac points are shown to be exotic topological insulators, supporting quantized, chromo-magnetic flux or a relative Chern number, and gapped edge states. The Dirac points are identified as a pair of unit-strength, SO(5)- monopole and anti-monopole, where the relative Chern number can jump by ±1. With analytical solutions of surface states, we show the non-existence of helical Fermi arcs, which are often considered to be the smoking gun signatures of bulk topology. Finally, we outline a general recipe for computing bulk invariants of all Dirac materials from the winding of gauge-invariant eigenvalues of planar Wilson loops, without relying on any non-universal properties of surface states.