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We present numerical evidence for the presence of a finite-temperature ($T$)phase transition separating paramagnet and quantum spin liquid in athree-dimensional variant of the Kitaev model defined on a hyperhoneycomblattice in the limit of strong anisotropy; the model is mapped onto aneffective Ising-type model, where elementary excitations consist of closedloops of flipped Ising-type variables. Analyzing this effective model by MonteCarlo simulation, we find a phase transition from quantum spin liquid toparamagnet at a finite critical temperature $T_c$. We also compute the magneticproperties in terms of the original quantum spins. We find that the magneticsusceptibility exhibits a broad hump above $T_c$, while it obeys the Curie lawat high $T$ and approaches a nonzero Van Vleck-type constant at low $T$.Although the susceptibility changes continuously at $T_c$, its $T$ derivativeshows critical divergence at $T_c$. We also clarify that the dynamical spincorrelation function is momentum independent but shows quantized peakscorresponding to the discretized excitations. Although the phase transitionaccompanies no apparent symmetry breaking in terms of the Ising-type variablesas well as the original quantum spins, we characterize it from a topologicalviewpoint. We find that, by defining the flux density for loops of theIsing-type variables, the transition is interpreted as the one occurring fromthe zero-flux quantum spin liquid to the nonzero-flux paramagnet; the latterhas a Coulombic nature due to the local constraints. The role of globalconstraints on the Ising-type variables is examined in comparison with theresults in the two-dimensional loop model. A correspondence of our model to theIsing model on a diamond lattice is also discussed. A possible relevance of ourresults to the recently-discovered hyperhoneycomb compound,$\beta$-Li$_2$IrO$_3$, is mentioned.

Finite-Temperature Phase Transition to a Quantum Spin Liquid in a Three-Dimensional Kitaev Model on a Hyperhoneycomb Lattice