  Quantum droplets have been realized in experiments on binary boson mixtures
and dipolar Bose gases. In these systems, the mean-field energy of the
Bose-Einstein condensation is attractive, and the repulsive Lee-Huang-Yang
energy is crucial for stability. The Bogoliubov theory incorrectly predicts
that the phonon mode is dynamically unstable in the long-wavelength limit. In
this work, we go beyond the Bogoliubov theory to study how the phonon mode is
stabilized in the quantum droplet of a binary boson mixture. Similar to
Beliaev's approach to a single-component Bose gas, we compute higher-order
contributions to the self-energy of the boson propagator. We find that the
interaction between spin and phonon excitations is the key for the phonon
stability. We obtain the sound velocity which can be tested by measuring the
superfluid critical velocity of the droplet in experiments. Beliaev damping of
this quantum droplet is also discussed.
