  We propose a theoretical framework to investigate elementary excitations at
finite temperatures within a localized electron model that describes the
interactions between multiple degrees of freedom, such as quantum spin models
and Kugel-Khomskii models. Thus far, their excitation structures have been
mainly examined using the linear flavor-wave theory, an SU($N$) generalization
of the linear spin-wave theory. These techniques introduce noninteracting
bosonic quasiparticles as elementary excitations from the ground state, thereby
elucidating numerous physical phenomena, including excitation spectra and
transport properties characterized by topologically nontrivial band structures.
Nevertheless, the interactions between quasiparticles cannot be ignored in
systems exemplified by $S=1/2$ quantum spin models, where strong quantum
fluctuations are present. Recent studies have investigated the effects of
quasiparticle damping at zero temperature in such models. In our study,
extending this approach to the flavor-wave theory for general localized
electron models, we construct a comprehensive method to calculate excitation
spectra with the quasiparticle damping at finite temperatures. We apply our
method to the Kitaev model under magnetic fields, a typical example of models
with topologically nontrivial magnon bands. Our calculations reveal that chiral
edge modes undergo significant damping in weak magnetic fields, amplifying the
damping rate by the temperature increase. This effect is caused by collisions
with thermally excited quasiparticles. Since our approach starts from a general
Hamiltonian, it will be widely applicable to other localized systems, such as
spin-orbital coupled systems derived from multi-orbital Hubbard models in the
strong correlation limit.
