Size shrinking of composite bosons for increasing density in the BCS to Bose-Einstein crossover

TL;DR

This paper investigates how the size of composite bosons decreases as density increases in the BCS-BEC crossover, confirming this trend analytically in both 2D and 3D, but highlighting limitations of mean-field theory in 2D.

Contribution

It provides a general physical argument and analytical confirmation that composite boson size shrinks with increasing density, and discusses the limitations of mean-field approximation in 2D.

Findings

Composite boson size decreases with density in both 2D and 3D.

Mean-field solutions confirm size shrinking in 3D, but not fully in 2D.

Born approximation fails in 2D at low energy, affecting mean-field accuracy.

Abstract

We consider a system of fermions in the continuum case at zero temperature, in the strong-coupling limit of a short-range attraction when composite bosons form as bound-fermion pairs. We examine the density dependence of the size of the composite bosons at leading order in the density ("dilute limit"), and show on general physical grounds that this size should decrease with increasing density, both in three and two dimensions. We then compare with the analytic zero-temperature mean-field solution, which indeed exhibits the size shrinking of the composite bosons both in three and two dimensions. We argue, nonetheless, that the two-dimensional mean-field solution is not consistent with our general result in the "dilute limit", to the extent that mean field treats the scattering between composite bosons in the Born approximation which is known to break down at low energy in two dimensions.