Phonon Stability and Sound Velocity of Bose Mixture Droplet

When the mean-field energy is as small as the quantum fluctuation, the self-energies under Bogoliubov approximation do not necessarily include all the contributions of the same order. For quantum droplets of a Bose mixture, the lower Bogoliubov mode is unstable at small momentum. Considering all the same order self-energies, the excitation spectrum is qualitatively different from the Bogoliubov mode. We develop Beliaev's method to calculate the elementary excitation and phonon velocity accurate to the order of gas parameter (na³)^½ in the homogeneous droplet. We discuss quasi-particles' Beliaev damping and find that density phonons are almost undamped . The sound velocity is found to be positive, which determines the nonzero superfluid critical velocity of droplet. In experiments, our results can be tested by measuring sound speed, superfluid critical velocity or collective mode.