Topology of three-dimensional Dirac semimetals and quantum spin Hall systems without gapless edge modes

The quantum spin Hall states are usually expected to possess gapless, helical edge modes. We show that the generic, n-fold-symmetric, momentum planes of three-dimensional, stable Dirac semimetals, which are orthogonal to the direction of nodal separation are examples of generalized quantum spin Hall systems, that possess quantized, spin or relative Chern numbers as bulk topological invariants, and gapped edge modes. We demonstrate these planes and the celebrated Bernevig-Zhang-Hughes model support identical quantized, non-Abelian Berry flux of magnitude 2π. Hence, they display identical quantized, topological response such as spin-charge separation and pumping of one Kramers-pair or SU(2) doublet, when probed with a magnetic flux tube. The Dirac points are identified as unit-strength, monopoles of SO(5) Berry connection, describing topological phase transitions between generalized quantum spin Hall and trivial insulators. Our work identifies precise bulk invariant and quantized response of Dirac semimetals, which are not diagnosed by nested Wilson loops and filling anomaly of corner-localized-states, and shows that many two-dimensional higher-order topological insulators can be understood as generalized quantum spin Hall systems.