Finite-temperature phase transition to a quantum spin liquid in a three-dimensional Kitaev model on a hyperhoneycomb lattice

Abstract

The quantum spin liquid is an enigmatic entity that is often hard to characterize within the conventional framework of condensed matter physics. We here present theoretical and numerical evidence for the characterization of a quantum spin liquid phase extending from the exact ground state to a finite critical temperature. We investigate a three-dimensional variant of the Kitaev model 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 on a diamond lattice. Analyzing this effective model by Monte Carlo simulation, we find a phase transition from the quantum spin liquid to a paramagnet at a finite critical temperature ${T}_{c}$, accompanied by a divergent singularity of the specific heat. 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 temperature and approaches a nonzero Van Vleck--type constant at low temperature. Although the susceptibility changes continuously at ${T}_{c}$, its temperature derivative shows a critical divergence at ${T}_{c}$. We also clarify that the dynamical spin correlation function is momentum independent but shows quantized peaks corresponding to discretized excitations. Although the phase transition accompanies no apparent symmetry breaking in terms of the Ising-type variables or of 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 can be interpreted as 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. The 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 $\ensuremath{\beta}$-${\text{Li}}_{2}$${\text{IrO}}_{3}$ is mentioned.