We present numerical evidence for the presence of a finite-temperature ($T$) phase transition separating paramagnet and quantum spin liquid in a three-dimensional variant of the Kitaev model defined on a hyperhoneycomb lattice in the limit of strong
anisotropy; the model is mapped onto an effective Ising-type model, where elementary excitations consist of closed loops of flipped Ising-type variables. Analyzing this effective model by Monte Carlo simulation, we find a phase transition from quantum spin liquid to paramagnet at a finite critical temperature $T_c$. We also compute the magnetic properties in terms of the original quantum spins. We find that the magnetic susceptibility exhibits a broad hump above $T_c$, while it obeys the Curie law at high $T$ and approaches a nonzero Van Vleck-type constant at low $T$. Although the susceptibility changes continuously at $T_c$, its $T$ derivative shows critical divergence at $T_c$. We also clarify that the dynamical spin correlation function is momentum independent but shows quantized peaks corresponding to the discretized excitations. Although the phase transition accompanies no apparent symmetry breaking in terms of the Ising-type variables as well as the original quantum spins, we characterize it from a topological viewpoint. We find that, by defining the flux density for loops of the Ising-type variables, the transition is interpreted as the one occurring from the zero-flux quantum spin liquid to the nonzero-flux paramagnet; the latter has a Coulombic nature due to the local constraints. The role of global constraints on the Ising-type variables is examined in comparison with the results in the two-dimensional loop model. A correspondence of our model to the Ising model on a diamond lattice is also discussed. A possible relevance of our results to the recently-discovered hyperhoneycomb compound, $beta$-Li$_2$IrO$_3$, is mentioned.
Two-dimensional orbital compass model is studied as an interacting itinerant electron model. A Hubbard-type tight-binding model, from which the orbital compass model is derived in the strong coupling limit, is identified. This model is analyzed by th
e random-phase approximation (RPA) and the self-consistent RPA methods from the weak coupling. Anisotropy for the orbital fluctuation in the momentum space is qualitatively changed by the on-site Coulomb interaction. This result is explained by the fact that the dominant fluctuation is changed from the intra-band nesting to the inter-band one by increasing the interaction.
