  The long-sought quantum spin liquid is a quantum-entangled magnetic state
leading to the fractionalization of spin degrees of freedom. Quasiparticles
emergent from the fractionalization affect not only the ground state properties
but also thermodynamic behavior in a peculiar manner. We here investigate how
the spin dynamics evolves from the high-temperature paramagnet to the quantum
spin liquid ground state, for the Kitaev spin model describing the
fractionalization into itinerant matter fermions and localized $Z_2$ gauge
fluxes. Beyond the previous study [J. Yoshitake, J. Nasu, and Y. Motome, Phys.
Rev. Lett. $\textbf{117}$, 157203 (2016)], in which the mean-field nature of
the cluster dynamical mean-field theory prevented us from studying
low-temperature properties, we develop a numerical technique by applying the
continuous-time quantum Monte Carlo (CTQMC) method to statistical samples
generated by the quantum Monte Carlo (QMC) method in a Majorana fermion
representation. This QMC+CTQMC method is fully unbiased and enables us to
investigate the low-temperature spin dynamics dominated by thermally excited
gauge fluxes, including the unconventional phase transition caused by gauge
flux loops in three dimensions, which was unreachable by the previous methods.
We apply this technique to the Kitaev model in both two and three dimensions.
Our results clearly distinguish two cases: while the dynamics changes smoothly
through the crossover in the two-dimensional honeycomb case, it exhibits
singular behaviors at the phase transition in the three-dimensional
hyperhoneycomb case. We show that the low-temperature spin dynamics is a
sensitive probe for thermally fluctuating gauge fluxes that behave very
differently between two and three dimensions.
