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We establish the topology of the spin-singlet superconducting states in thebare hyperhoneycomb lattice and derive analytically the full phase diagramusing only symmetry and topology in combination with simple energy arguments.The phase diagram is dominated by two states preserving time-reversal symmetry.We find that the line-nodal state dominating at low doping levels istopologically nontrivial and exhibits surface Majorana flat bands, which weshow perfectly match the bulk-boundary correspondence using Berry phaseapproach. At higher doping levels we find a fully gapped state with trivialtopology. By analytically calculating the topological invariant of the linenodes, we derive the critical point between the line-nodal and fully gappedstates as a function of both pairing parameters and doping. We find that theline-nodal state is favored not only at lower doping levels but also ifsymmetry-allowed deformations of the lattice is present. Adding simple energyarguments we establish that a fully gapped state with broken time-reversalsymmetry likely appears covering the actual phase transition. We find thistime-reversal symmetry broken state to be topologically trivial, while we findan additional point nodal state at very low doping levels to have nontrivialtopology with associated Fermi surface arcs. We eventually address therobustness of the phase diagram to generalized models also including adiabaticspin-orbit coupling, and show how all but the point nodal state are reasonablystable.

Topological nodal superconducting phases and topological phase transition in the hyperhoneycomb lattice