From few-body atomic physics to many-body statistical physics:
The unitary Bose gas and the three-body hard-core model Ultracold atomic gases offer unprecedented possibilities to realize and manipulate quantum systems. The control on interparticle interactions allows one to reach the strongly-interacting regime, with both fermionic and bosonic atomic species. In the unitary limit, where the interaction strength is at its maximum, universal properties emerge. For bosonic atoms, these include the Efimov effect, the surprising existence of an infinite sequence of three-body bound states. In this thesis, we have studied a system of unitary bosons. Starting from the two- and three-body cases, we have shown that the chosen model correctly captures the universal features of the Efimov effect. For the corresponding many-body problem, we have developed a quantum Monte Carlo algorithm capable of realizing the different thermodynamic phases in which the system may exist: The high-temperature normal gas, Bose-Einstein condensate, and Efimov liquid. A single ingredient of our model would remain relevant in the infinite-temperature limit, namely the three-body hard-core repulsion, which constitutes a generalization of the classical hard-sphere potential. For this model, we have proposed a solution to the two- and three-dimensional packing problem, based on an analytical ansatz and on the simulated-annealing technique. Extending these results to finite pressure showed that the system has a discontinuous melting transition, which we identified through the Monte Carlo method. Keywords: Ultracold atoms, unitary Bose gas, Bose-Einstein condensation, Efimov effect, three-body hard-core model, Monte Carlo method.