Pair size versus coherence length and quantum geometry in the spin-orbit coupled BCS-BEC crossover
We investigate the interplay between pairing correlations and quantum geometry in the spin-orbit-coupled BCS-BEC crossover. Working in the helicity basis, we derive the exact pair-size tensor for the two-body bound states and show that it separates into intraband and quantum-geometric interband contributions, with the latter determined by the quantum metric. We extend this decomposition to two distinct many-body length scales at zero temperature: the Cooper-pair size, obtained from mean-field BCS theory, and the coherence length, obtained from the Gaussian fluctuation theory. Although these quantities almost coincide in the BCS regime, they describe distinct physical properties away from it, with the pair size characterizing the internal extent of a pair and the coherence length characterizing the long-wavelength response of the order parameter. We evaluate both quantities for three-dimensional Rashba, three-dimensional Weyl, and two-dimensional Rashba spin-orbit coupling models. We find that the pair size decreases monotonically with increasing spin-orbit coupling as the pair localizes, whereas the coherence length develops a pronounced minimum in the crossover regime before growing again in the BEC regime as the composite bosons become weakly interacting. This minimum is tied to the collapse of the noninteracting Fermi surface onto the ring or sphere of helicity-band minima. The quantum-geometric contributions reach up to nearly one third of the pair size and about 40\% of the coherence length, tracking these same changes in the underlying helicity Fermi-surface topology.