Are the surface Fermi arcs in Dirac semimetals topologically protected?

Motivated by recent experiments probing double Fermi arcs on the surface of Dirac semimetals (DSMs) Na3Bi and Cd3As2, we raise the question posed in the title. We find that, in marked contrast to Weyl semimetals, the Fermi arcs of DSMs are not topologically protected in general, except at certain time-reversal invariant momenta. For a simple 4-band model with a pair of Dirac nodes at k = (0, 0, ±Q) gapless surface states are protected only at kz = 0. We identify symmetry allowed bulk perturbations that destroy Fermi arcs, but show that they are necessarily “small”, i.e., higher order than terms kept in usual k · p theory. We validate our conclusions about the absence of a topological invariant protecting the surface states in DSMs using a K-theory analysis for the space groups of Na3Bi and Cd3As2