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Motivated by the recent realization of the three-dimensional hyperhoneycomband stripyhoneycomb lattices in lithium iridate (Li$_2$IrO$_3$), we study thepossible spin-singlet superconducting states on the whole series of harmonichoneycomb lattices. Beginning with an isolated out-of-plane twist making thehoneycomb lattice three-dimensional, we find that the chiral d$\pm$id state,well-known from the honeycomb lattice, is realized in the largest members ofthe series at low to intermediate doping. Along the twist, four chiral edgestates form a two-dimensional dispersive band. Reducing the distance betweenthe twists to form the smaller members of the harmonic honeycomb lattices, thedegeneracy between the d-wave states is lifted, which finally destroys thechiral state. By analyzing the hyper- and stripyhoneycomb lattices andgeneralizing using the $D_{2h}$ point group of all the harmonic honeycomblattices, we show that the superconducting state often belongs to the trivialirreducible representation. This state has nodal lines at low to intermediatedoping, which is possible because the full lattice symmetry allows sign changesbetween different sets of bonds. We also find time-reversal symmetry brokenstates, which are either fully gapped or feature nodal points, in certain partsof the phase diagram. Finally, we draw a comparison between the statesclassified in terms of the $D_{2h}$ symmetries and those observed on the$D_{6h}$ honeycomb lattice.

From chiral d-wave to nodal line superconductivity in the harmonic honeycomb lattices