Pairing and condensation in a resonant Bose-Fermi mixture Elisa Fratini and Pierbiagio Pieri Dipartimento di Fisica, Scuola di Scienze e Tecnologie, Universit` a di Camerino, Via Madonna delle Carceri 9, I-62032 Camerino, Italy (Dated: October 30, 2018) We study by diagrammatic means a Bose-Fermi mixture, with boson-fermion coupling tuned by a Fano-Feshbach resonance. For increasing coupling, the growing boson-fermion pairing correlations progressively reduce the boson condensation temperature and make it eventually vanish at a critical coupling. Such quantum critical point depends very weakly on the population imbalance and for vanishing boson densities coincides with that found for the polaron-molecule transition in a strongly imbalanced Fermi gas, thus bridging two quite distinct physical systems. PACS numbers: 03.75.Ss,03.75.Hh,32.30.Bv,74.20.-z body calculations (beyond mean-field). We thus consider the following (grand-canonical) Hamiltonian: One of the main reasons behind the recent great in- terest in ultracold gases, is the possibility to reproduce physical systems relevant to other areas in physics, with a flexibility and a degree of tunability of physical parame- ters which is unimaginable in the original system of inter- est. The very same flexibility can also be used, however, to construct novel physical systems. A noticeable exam- ple of such novel systems, namely, resonant Bose-Fermi mixtures, will be here at issue. We will be interested in particular in the competition between boson-fermion pairing correlations and bosonic condensation, occuring in a mixture of single-component bosons and fermions when the boson-fermion pairing is made progressively stronger by means of a Fano-Feshbach resonance. Initial theoretical studies of ultracold Bose-Fermi mix- tures considered mainly non-resonant systems [1–5], where boson-fermion pairing is irrelevant, and studied, within mean-field like treatments, the tendency towards collapse or phase separation (as motivated by the first experimental results on non-resonant Bose-Fermi mix- tures [6]). Bose-Fermi mixtures in the presence of a Fano- Feshbach resonance have been subsequently considered in [7–17], mostly for narrow resonances [7, 10–13, 16] and/or in specific contexts such as optical lattices, re- duced dimensionality, zero temperature or vanishing den- sitiy of one component [8, 9, 14, 15, 17]. Most of current experiments on Bose-Fermi mix- tures [19–23] appear however to be closer to the case of a broad resonance, which is characterized by the small- ness of the effective range parameter r 0 of the interac- tion potential with respect to both the average inter- particle distance and the scattering length [18]. Un- der these conditions, the resonant Bose-Fermi mixture is accurately described by a minimal Hamiltonian, made just by bosons and fermions mutually interacting via an attractive point-contact potential. This Hamiltonian is simpler and more “fundamental” in character than its counterpart for a narrow resonance because of the ab- sence of any other parameter besides the strength of the boson-fermion interaction. In addition, its simplicity allows the implementation of more sophisticated many- The expected behavior of the system is clear in the two opposite limits of the boson-fermion coupling. In the weak-coupling limit, where the scattering length a is small and negative (such that ( k$_{F}$ a ) − 1 ≪ − 1), the In particular, we will use the same dimensionless cou- pling strength ( k$_{F}$ a ) − 1 normally used for two-component Fermi gases, where the wave-vector k$_{F}$ ≡ (3 π $^{2}$n ) 1 / 3 (note that k$_{F}$ coincides with the noninteracting Fermi wave- vector k 0 F = (6 π $^{2}$n$_{F}$ ) 1 / 3 only for n$_{B}$ = n$_{F}$ ). The effective coupling strength in the many-body sys- tem is determined by comparing the boson-fermion scat- tering length a with the average interparticle distance n − 1 / 3 (with n = n$_{B}$ + n$_{F}$ , where n$_{B}$ and n$_{F}$ are the bo- son and fermion particle number density, respectively). The Hamiltonian (1) does not contain explicitly the boson-boson interaction. Provided it is repulsive and non-resonant, this interaction yields in fact a mean-field shift, which consists in a simple redefinition of the bo- son chemical potential (attractive boson-boson interac- tions would lead to mechanical instability and are ex- cluded from our consideration). S-wave interaction be- tween fermions is finally excluded by Pauli principle. Here ψ † $_{s}$( r ), creates a particle of mass m$_{s}$ and chemical potential µ$_{s}$ at spatial position r , where s = B, F indi- cates the boson and fermion atomic species, respectively, while v 0 is the bare strength of the contact interaction (we set ¯ h = k$_{B}$ = 1 throughout this paper). Ultraviolet divergences associated with the use of a contact interac- tion in (1) are eliminated, as for two-component Fermi gases [24], by expressing the bare interaction v 0 in terms of the boson-fermion scattering length a via the (diver- gent) expression 1 /v 0 = m$_{r}$/ (2 πa ) − ∫ 2 m$_{r}$/ k $^{2}$d k / (2 π ) $^{3}$, where m$_{r}$ = m$_{B}$m$_{F}$ / ( m$_{B}$ + m$_{F}$ ) is the reduced mass. H = ∑ s ∫ d r ψ † $_{s}$( r )( − ∇ 2 2 m$_{s}$ − µ$_{s}$ ) ψ$_{s}$ ( r ) + v 0 ∫ d r ψ † $_{B}$( r ) ψ † $_{F}$( r ) ψ$_{F}$ ( r ) ψ$_{B}$ ( r ) (1) arXiv:1001.1971v1 [cond-mat.quant-gas] 12 Jan 2010