The field-angle anisotropy of proximate Kitaev systems under an in-plane magnetic field

We have investigated the field-angle behaviors of magnetic excitations under an in-plane magnetic field for proximate Kitaev systems. By employing the exact diagonalization method in conjunction with the linear spin wave theory, we have demonstrated that the magnetic excitation gap in the polarized phase is determined by the magnon excitation at $M$ points and has a strong anisotropy with respect to the field direction in the vicinity of the critical field limit. The specific heat from this magnon excitation bears qualitatively the same anisotropic behaviors as expected one for the non-Abelian spin liquid phase in the Kitaev model and experimentally observed one of the intermediate phases in $alpha$-RuCl$_3$.