Temperature evolution of spin dynamics in extended Kitaev models within an Abrikosov fermion theory beyond mean field

Abstract: The $S=1/2$ Kitaev model provides an exactly solvable setting in which spin fractionalization produces itinerant Majorana fermions coupled to an emergent $\mathbb{Z}_2$ gauge field. In realistic Kitaev magnets, however, non-Kitaev interactions break integrability and make their finite-temperature dynamics difficult to calculate. We develop a finite-temperature parton mean-field theory based on the Abrikosov-fermion representation and incorporate the effects of exchange interactions through the random phase approximation to calculate the dynamical spin structure factor. For the pure Kitaev model, the method reproduces a strong, nearly dispersionless peak at a finite low energy together with a broad continuum at higher energies. It also reproduces the momentum dependence of the dynamical spin structure factor and its temperature evolution, including the downward shift of the low-energy spectral weight, in qualitative agreement with continuous-time quantum Monte Carlo calculations. Within the present theory, the contraction of the continuum is traced to the temperature-induced reduction of the bond amplitudes, which narrows the parton bands and compresses the two-parton phase space. For ferromagnetic Kitaev coupling, the energy of the low-energy peak is minimized at the $Γ$ point for a ferromagnetic Heisenberg interaction and maximized there for an antiferromagnetic interaction. In the ferromagnetic Kitaev-$Γ$ model, the static response develops a finite-momentum softening, indicating an instability toward the period-three magnetic order reported in numerical studies. For uniform-ferromagnetic states selected by a $[111]$ magnetic field, we find that a broad parton continuum coexists with magnonlike dispersive structures, producing spectral features consistent with a spinon-to-magnon crossover.