Double- and quad-helicoid surface states in magnetic Dirac semimetals protected by glide-time-reversal symmetry

The Dirac point in k-space is in general topologically trivial, because it is a superposition of two Weyl points with opposite charges. Meanwhile, a Z2 monopole charge was defined for Dirac semimetals with G and T symmetry (G: glide, T: time-reversal) in previous works, and they lead to double helicoid surface states. Moreover, when there are two glide symmetries, they lead to quad helicoid surface states. In this work, we show that even in Dirac semimetals with GT-symmetry but without time-reversal symmetry, one can define a new Z2 monopole charge and show its bulk-surface correspondence. We also show that the new Z2 charge is equal to the Z2 invariant for G-protected topological crystalline insulators, and to the second Stiefel-Whitney number for PT-protected nodal line semimetals, under some symmetries. We propose some candidate magnetic Dirac semimetal materials showing double- and quad-helicoid surface states.