Binding of fermionic mixtures in low dimensions
In this thesis, we study the problem of binding in ultra-cold fermionic mixtures composed of N_h identical heavy fermions interacting with N_l light fermions through a zero-range attractive potential in low-dimensional geometries. The binding phenomenon depends on the space dimension and on the heavy-light mass ratio and results from a subtle competition between the repulsive Pauli pressure of identical fermions and the heavy-light attraction. In one dimension, by using an exact few-body analysis, we establish and characterize states of a single light atom bound with up to five heavy atoms. Then, using a mean-field approach valid for large N_h, we find that such N_h+1 clusters are bound for any N_h as long as the mass ratio exceeds a threshold which scales as N_h^3. Extending this mean-field method to N_l>1, we show that N+1-type clusters can attract each other and form self-bound chains resembling charge-density waves in the thermodynamic limit. We also extend this mean-field theory to the two-dimensional N_h+N_l problem and find that N_h+1 clusters can form for arbitrarily large N_h in two dimensions as well. However, the mass-ratio threshold for binding scales as N_h^2. In order to find the energy and size of the clusters we take into account beyond-mean-field effects as the two-dimensional mean-field theory is scale invariant. Finally, we present numerical evidence that two two-dimensional N+1-type clusters always repel each other and form no self-bound state, contrary to the one-dimensional case.
Jury : I. Bouchoule, H.-W. Hammer (rapporteur), P. Massignan (rapporteur), C. Mora, D. Petrov (directeur de thèse)