The Fermionic spectrum, phase transition and domain walls of confined $^3$He-A film

The edge states of a $^3$He-A film are Weyl Fermions propagating on the edge in a direction determined by the chirality of the bulk phase. Under lateral confinement, the wave functions for counter-propagating Weyl Fermions on opposing edges overlap. We show that the edge states hyrbridize and form a band, and the continuum states exhibit band gaps. We report self-consistent calculations of the reduction in the spontaneous edge mass currents due to hybridization as a function of lateral confinement, $D$. Strong lateral confinement leads to a sequence of quantum phase transitions. The A phase undergoes a transition to a pair density wave (PDW) phase with broken translational symmetry at $D_{c_1}\approx 13\xi_0$, and a transition to a polar state at $D_{c_2}\approx 9\xi_0$. The PDW phase for $D < D_{c_1}$ is periodic array of chiral domains separated by domain walls\footnote{Y. Tsutsumi, \textbf{J. Low Temp. Phys.} 175, 2014} with currents that conflict with the direction of edge currents. We report self-consistent calculations of the PDW phase near $D_{c_1}$ that resolves the competition in energy between edge states and Fermions bound to the domain wall. The resulting pattern of circulating currents also resolves the apparent violation of current conser