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

TL;DR

This paper investigates how the correlation energy of N cobosons interacting via a BCS-like potential exhibits a surprising cancellation in 2D systems, a phenomenon linked to the unique density of states in 2D, but not present in other dimensions.

Contribution

It reveals a dimensionality-dependent cancellation of coboson correlation energy under BCS-like interactions, highlighting the special role of 2D systems and the form of the potential.

Findings

Cancellation occurs only in 2D systems with BCS-like potential.

No cancellation in 1D, 3D, 4D, or with arbitrary center-of-mass momentum.

The cancellation is linked to the constant density of states in 2D.

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.