Triviality of the BCS-BEC crossover in extended dimensions: Implications for the ground state energy

Cold atom traps and certain neutron star layers may contain fermions with separation much larger than the range of pairwise potentials yet much shorter than the scattering length. Such systems can display universal characteristics independent of the details of the short range interactions. Our main result is that for space dimensions $D$ smaller than two and larger than four a specific extension of this problem is amenable to exact results. In particular, the energy per particle at the BCS-BEC crossover point is equal to the energy of the free fermion system in all $D⩽2$ whereas this energy is rigorously nonpositive (and potentially vanishing) in all $D⩾4$. We discuss the $D=3$ case. A particular unjustified recipe suggests $\xi =1∕2$ in $D=3$.

Figure 1

A schematic (thick lines) of the binding spherically symmetric potential $V({\mathbf{r}}_i-{{\mathbf{r}}^{\prime }}_j)\equiv V\left({\mathbf{r}}_{ij}\right)$ (in three dimensions) which is nonzero only inside a sphere of radius $r_0$ about the origin, $V\left(\mathbf{r}\right)=-V_0\Theta (r_0-\mid \mathbf{r}\mid )$. A cartoon of a scaled very weakly bound $s$-wave state $[\chi \left({\mathbf{r}}_{ij}\right)=\mid {\mathbf{r}}_{ij}\mid \varphi \left({\mathbf{r}}_{ij}\right)]$ is shown. To attain exactly one zero-energy bound state we need to fit exactly one-quarter period for $\chi (\mid \mathbf{r}\mid )\sim \sin \kappa r$ and set $\kappa r_0=\frac{\pi }{2}$ with $\kappa =\sqrt{2\mu V_0}∕\hslash$. Here, $\mu =m∕2$ is the reduced mass of a fermionic pair. The zero-energy bound state wave function in general dimension $D$ is derived in the Appendix.