Bose spectrum of superfluid systems of the $\text{He}^3$–$\text{He}^4$ type

The collective modes of Bose excitations of model Fermi–Bose systems are investigated. Three-dimensional models of the $\text{He}^3A$–$\text{He}^4$ and $\text{He}^3B$–$\text{He}^4$systems are investigated, and also models of $\text{He}^3$–$\text{He}^4$ films in which the $\text{He}^3$-component can be in superfluid phases $a$ and $b$. For all models, a hydrodynamic (effective) action functional is constructed which makes it possible to find the Bose spectrum of the considered systems. The interaction of the Fermi and Bose subsystems leads to interaction of the acoustic modes of both subsystems and hardly influences the other modes of the Bose spectrum. A biquadratic equation is obtained for the velocities of the acoustic modes; this equation does not change on the transition from the $A$ to the $B$ phase in the three-dimensional models or from the $a$ to the $b$ phase in the models of $\text{He}^3$–$\text{He}^4$ films.