Renormalization Group Flows for Superfluid 3He under Confinement

Abstract

We theoretically investigate the effect of fluctuations in the order parameter on the phase transition of 3He under nanoscale confinement of one spatial dimension realized in recent experiments. We derive a new quasi two-dimensional free energy that relies on a 3×2 complex matrix instead of the 3×3 complex matrix order parameter found in the three-dimensional system. We minimize the quasi two-dimensional free energy and present two energetically degenerate phases that are predicted: the A-phase and the planar phase. Beyond the mean-field approximation, we calculate the RG flow in the three-dimensional, two-dimensional, and quasi two-dimensional limits. We derive the perturbative flow equations for all quartic coupling constants with non-trivial kinetic factor. We find that the B-phase is energetically favoured in the three-dimensional system in agreement with experiment. In contrast, in the quasi two-dimensional limit, weak-coupling perturbative renormalization group predicts the planar phase to be energetically favoured. However, strong-coupling corrections favour the A-phase observed in experiment. We derived FRG flow equations for the 3D, 2D and quasi-2D cases with non-trivial kinetic factor. In the quasi-2D limit, we found that the A-phase was favoured over the planar phase for certain levels of confinement. These confinement scales were also confirmed to have a prominent Fermi liquid to A-phase transition in experiment. Due to the unstable fixed point, we find that under confinement there is a fluctuation-induced first-order transition to the A-phase.