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Motivated by the recent synthesis of two insulating Li$_2$IrO$_3$ polymorphs,where Ir$^{4+}$ $S_{eff}$=1/2 moments form 3D ("harmonic") honeycomb structureswith threefold coordination, we study magnetic Hamiltonians on the resulting$\beta$-Li$_2$IrO$_3$ hyperhoneycomb lattice and $\gamma$-Li$_2$IrO$_3$stripyhoneycomb lattice. Experimentally measured magnetic susceptibilitiessuggest that Kitaev interactions, predicted for the ideal 90$^\circ$ Ir-O-Irbonds, are sizable in these materials. We first consider pure Kitaevinteractions, which lead to an exactly soluble 3D quantum spin liquid (QSL)with emergent Majorana fermions and Z$_2$ flux loops. Unlike 2D QSLs, the 3DQSL is stable to finite temperature, with $T_c \approx |K|/100$. On includingHeisenberg couplings, exact solubility is lost. However, by noting that theshortest closed loop $\ell$ is relatively large in these structures, weconstruct an $\ell\rightarrow \infty$ approximation by defining the model onthe Bethe lattice. The phase diagram of the Kitaev-Heisenberg model on thislattice is obtained directly in the thermodynamic limit, using tensor networkstates and the infinite-system time-evolving-block-decimation (iTEBD)algorithm. Both magnetically ordered and gapped QSL phases are found, thelatter being identified by an entanglement fingerprint.

Three dimensional quantum spin liquids in models of harmonic-honeycomb iridates and phase diagram in an infinite-D approximation