The band-touching points of stable, three-dimensional, Kramers-degenerate, Dirac semimetals are singularities of a five-component, unit vector field and non-Abelian, SO(5)-Berry's connections, whose topological classification is an important, open problem. We solve this problem by performing second homotopy classification of Berry's connections. Using Abelian projected connections, the generic planes, orthogonal to the direction of nodal separation, and lying between two Dirac points are shown to be higher-order topological insulators, which support quantized, chromo-magnetic flux or 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 jumps by ±1. Using these bulk invariants, we determine the topological universality class of different types of Dirac semimetals. We also describe a universal recipe for computing quantized, non-Abelian flux for Dirac materials from the windings of spectra of planar Wilson loops, displaying SO(5)-gauge invariance. With non-perturbative, analytical solutions of surface-states, we show the absence of helical Fermi arcs, and predict the fermiology and the spin-orbital textures. We also discuss the similarities and important topological distinction between the surface-states Hamiltonian and the generator of Polyakov loop of Berry's connections.

Recent theoretical works on effective, four-band models of three-dimensional, Dirac semimetals suggest the generic planes in momentum space, orthogonal to the direction of nodal separation, and lying between two Dirac points are higher-order topological insulators, supporting gapped, edge-states. Furthermore, the second homotopy classification of four-band models shows the higherorder topological insulators support quantized, non-Abelian Berry's flux and the Dirac points are monopoles of SO(5) Berry's connections. Due to the lack of suitable computational scheme, such bulk topological properties are yet to be determined from the ab initio band structures of Dirac materials. In this work, we report first, comprehensive topological classification of ab initio band structures of Na3Bi, by computing Wilson loops of non-Abelian, Berry's connections for several, Kramers-degenerate bands. Our work shows the quantized, non-Abelian, Berry's flux can be used as a stable, bulk invariant for describing higher-order topology and topological phase transitions.

Multiple works suggest the possibility of classification of quantum spin Hall effect with magnetic flux tubes, which cause separation of spin and charge degrees of freedom and pumping of spin or Kramers-pair. However, the proof of principle demonstration of spin-charge separation is yet to be accomplished for realistic, ab initio band structures of spin-orbit-coupled materials, lacking spin-conservation law. In this work, we perform thought experiments with magnetic flux tubes on $$\beta$$ β -bismuthene, and demonstrate spin-charge separation, and quantized pumping of spin for three insulating states, that can be accessed by tuning filling fractions. With a combined analysis of momentum-space topology and real-space response, we identify important role of bands supporting even integer invariants, which cannot be addressed with symmetry-based indicators. Our work sets a new standard for the computational diagnosis of two-dimensional, quantum spin-Hall materials by going beyond the $$\mathbb {Z}_{2}$$ Z 2 paradigm and providing an avenue for precise determination of the bulk invariant through computation of quantized, real-space response.