Universal Bose gases near resonance: A rigorous solution

We obtain a rigorous solution of universal Bose gases near resonance and offer an answer to one of the long-standing challenges of quantum gases at large scattering lengths, where the standard dilute theory breaks down. The solution was obtained by using an $\epsilon$ expansion near four spatial dimensions. In dimension $d=4-\epsilon$, the chemical potential of Bose gases near resonance is shown to approach the universal value ${\epsilon }^{\frac{2}{4-\epsilon }}{\epsilon }_F\sqrt{\frac{2}{3}}(1+0.474\epsilon -i1.217\epsilon +\cdots )$, where ${\epsilon }_F$ is the Fermi energy defined for a Fermi gas of density $n$, and the condensation fraction is equal to $\frac{2}{3}(1+0.0877\epsilon +\cdots )$. We also discuss the implications on ultracold gases in physical dimensions.

Figure 1

An $N$-body $L$-loop diagram with $N$ incoming and outgoing condensed lines (dashed); $L$ is the number of loops formed by propagators (solid lines). Each diagram has $N+(L-1)$ vertices. Near resonance, its contribution to the chemical potential is proportional to ${\epsilon }^{\frac{2}{4-\epsilon }}{\epsilon }^L$ independent of $N$.