  A three-dimensional Kitaev model on a hyperhoneycomb lattice is investigated
numerically at finite temperature. The Kitaev model is one of the solvable
quantum spin models, where the ground state is given by gapped and gapless spin
liquids, depending on the anisotropy of the interactions. This model can be
rewritten as a free Majorana fermion system coupled with $Z_2$ variables. The
density of states of Majorana fermions shows an excitation gap in the gapped
region, while it is semimetallic in the gapless region reflecting the Dirac
node. Performing the Monte Carlo simulation, we calculate the temperature
dependence of the Majorana spectra. We find that the semimetallic dip is filled
up as temperature increases in the gapless region, but surprisingly, the
spectrum develops an excitation gap in the region near the gapless-gapped
boundary. Such changes of the low-energy spectrum appear sharply at the
transition temperature from the spin liquid to the paramagnetic state. The
results indicate that thermal fluctuations of the $Z_2$ fields significantly
influence the low-energy state of Majorana fermions, especially in the spin
liquid formation.
