  We investigate the effect of a magnetic field on the extended Kitaev spin
liquid state through phonon dynamics. Using a constrained fermionic
self-consistent mean-field method, we analyze the quantum spin liquid (QSL)
ground state for the extended Kitaev model with both the Zeeman term and the
perturbative three-spin interaction term $\kappa$. Our results demonstrate the
dependence of the stability of the Kitaev QSL state on the field direction,
consistent with findings in the literature. Additionally, we calculate the
phonon dynamics for acoustic phonons coupled to the Majorana fermion
excitations of the Kitaev spin liquid state, discussing the temperature and
field evolution of these quantities.
