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The discovery of ultracold dilute liquids has significantly elevated ourinterest in various phenomena which go under the name of beyond-mean-field(BMF) physics. In these lecture notes we give an elementary introduction to thequantum stabilization and liquefaction of a collapsing weakly interactingBose-Bose mixture. A detailed derivation of the leading BMF correction, alsoknown as the Lee-Huang-Yang (LHY) term, in this system is presented in a mannersuitable for further generalizations and extensions. Although the LHY term is anonanalytic function of the density $n$, under certain conditions the leadingBMF correction becomes analytic and can be expanded in integer powers of $n$,effectively introducing three-body and higher-order interactions. We discusswhy and how well the Bogoliubov approach can predict these few-bodyobservables.

Beyond-mean-field effects in mixtures: few-body and many-body aspects