BCS-BEC crossover induced by a synthetic non-Abelian gauge field

We investigate the ground state of interacting spin-$\half$ fermions (3D) at a finite density ($\rho \sim \kf^3$) in the presence of a uniform non-Abelian gauge field. The gauge field configuration (GFC) described by a vector $\blam \equiv (\lambda_x, \lambda_y, \lambda_z)$, whose magnitude $\lambda$ determines the gauge coupling strength, generates a generalized Rashba spin-orbit interaction. For a weak attractive interaction in the singlet channel described by a small negative scattering length $(\kf |\as| \lesssim 1)$, the ground state in the absence of the gauge field ($\lambda=0$) is a BCS (Bardeen-Cooper-Schrieffer) superfluid with large overlapping pairs. With increasing gauge coupling strength, a non-Abelian gauge field engenders a crossover of this BCS ground state to a BEC (Bose-Einstein condensate) ground state of bosons even with a weak attractive interaction that fails to produce a two-body bound state in free vacuum. For large gauge couplings $(\lambda/\kf \gg 1)$, the BEC attained is a condensate of bosons whose properties are solely determined by the gauge field (and not by the scattering length so long as it is non-zero) -- we call these bosons "rashbons". In the absence of interactions ($\as = 0^-$), the shape of the Fermi surface of the system undergoes a topological transition at a critical gauge coupling $\lambda_T$. For high symmetry gauge field configurations we show that the crossover from the BCS superfluid to the rashbon BEC occurs in the regime of $\lambda$ near $\lambda_T$. In the context of cold atomic systems, this work makes an interesting suggestion of obtaining BCS-BEC crossover through a route other than tuning the interaction between the fermions.