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Two recent theoretical works studied the role of Kitaev interactions in thenewly observed incommensurate magnetic order in the hyper-honeycomb($\beta$-Li$_2$IrO$_3$) and stripy-honeycomb ($\gamma$-Li$_2$IrO$_3$) iridates.Each of these works analyzed a different model ($JK\Gamma$ versus coupledzigzag chain model) using a contrasting method (classical versus soft-spinanalysis). The lack of commonality between these works precludes meaningfulcomparisons and a proper understanding of these unusual orderings. In thisstudy, we complete the unfinished picture initiated by these two works bysolving both models with both approaches for both 3D honeycomb iridates.Through comparisons between all combinations of models, techniques, andmaterials, we find that the bond-isotropic $JK\Gamma$ model consistentlypredicts the experimental phase of $\beta$-Li$_2$IrO$_3$ regardless of themethod used, while the experimental phase of $\gamma$-Li$_2$IrO$_3$ can begenerated by the soft-spin approach with eigenmode mixing irrespective of themodel used. To gain further insights, we solve a 1D quantum spin-chain modelrelated to both 3D models using the density matrix renormalization group methodto form a benchmark. We discover that in the 1D model, incommensuratecorrelations in the classical and soft-spin analysis survive in the quantumlimit only in the presence of the symmetric-off-diagonal exchange $\Gamma$found in the $JK\Gamma$ model. The relevance of these results to the realmaterials are also discussed.

Two iridates, two models, two approaches: a comparative study on magnetism in 3D honeycomb materials