Trapped fermions in a synthetic non-Abelian gauge field

On increasing the coupling strength ($\lambda$) of a non-Abelian gauge field that induces a generalized Rashba spin-orbit coupling, the topology of the Fermi surface of a homogeneous gas of non-interacting fermions of density $\rho \sim k_F^3$ undergoes a change at a critical value, $\lambda_T \approx k_F$ [PRB {\bf 84}, 014512 (2011)]. We analyze how this affects the size/shape of a cloud of fermions trapped in a harmonic potential. We develop an adiabatic formulation, with Pancharatnam-Berry phase terms, for the one particle states in a trap with the gauge field. Local density approximation reveals that the cloud shrinks in a {\em characteristic fashion with increasing $\lambda$} and predicts a spherical cloud for all gauge field configurations. We show, via a calculation of the cloud shape using exact eigenstates, that for certain gauge fields there is systematic anisotropy in the cloud shape that increases with increasing gauge coupling $\lambda$. An important spin-off of our adiabatic formulation is that it reveals exciting possibilities for the cold-atom realization of interesting Hamiltonians (eg. quantum hall spherical geometry) by using a non-Abelian gauge field in conjunction with another potential.