Mysterious dimensionality effect: the cancellation of the $N$-coboson correlation energy under a BCS-like potential

Abstract

We use Richardson-Gaudin exact equations to derive the ground-state energy of $N$ composite bosons (cobosons) interacting via a potential which acts between fermion pairs having zero center-of-mass momentum, that is, a potential similar to the reduced BCS potential used in conventional superconductivity. Through a density expansion, we show that while, for 2D systems, the $N$-coboson correlation energy undergoes a surprising cancellation which leaves the interaction part with a $N(N-1)$ dependence only, such a cancellation does not exist in 1D, 3D, and 4D systems --- which corresponds to 2D parabolic traps --- nor when the cobosons interact via a similar short-range potential but between pairs having an arbitrary center-of-mass momentum. This shows that the previously-found cancellation which exists for the Cooper-pair correlation energy results not only from the very peculiar form of the reduced BCS potential, but also from a quite mysterious dimensionality effect, the density of states for Cooper pairs feeling the BCS potential being essentially constant, as for 2D systems.