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In a binary quantum droplet, the interspecies attraction dominates over theintraspecies repulsions and the mean-field energy is unstable. The mechanicalstability is restored by the repulsive Lee-Huang-Yang (LHY) energy [1]. In theBogoliubov theory of the binary quantum droplet, there are two branches ofgapless excitations. The lower branch describes the phonon excitation and itsenergy is imaginary in the long-wavelength limit, implying dynamicalinstability. Recently it is found that the phonon energy is renormalized byhigher-order quantum fluctuations and the dynamical instability is removed [2].In this work, we study a binary quantum droplet in the path integral formalismto construct an effective model with the correct phonon energy. By integratingout the upper excitation branch, we obtain an effective single-mode modeldescribing density fluctuations, and derive the extended Gross-Pitaevskiiequation. In this approach, the LHY energy in the extended-GP equation ispurely positive without any assumption of neglecting the imaginary part. Thiseffective single-mode model can be also used outside and close to the quantumdroplet region such as in the LHY gas.

The Effective Single-mode model of a Binary Boson Mixture in the Quantum Droplet Region