Ground state angular momentum, spectral asymmetry, and topology in chiral superfluids and superconductors

Recently it was discovered that the ground state orbital angular momentum in two-dimensional chiral superfluids with pairing symmetry $(p_x+i{p}_y{)}^{\nu }$ depends on the winding number $\nu$ in a striking manner. The ground state value for the $\nu =1$ case is $L_z=\mathrm{\hslash }N\mathrm{/}2$ as expected by counting the Cooper pairs, while a dramatic cancellation takes place for $\nu >1$. The origin of the cancellation is associated with the topological edge states that appear in a finite geometry and give rise to a spectral asymmetry. Here we study the reduction of orbital angular momentum for different potential profiles and pairing strengths, showing that the result $L_z=\mathrm{\hslash }N\mathrm{/}2$ is robust for $\nu =1$ under all studied circumstances. We study how angular momentum depends on the gap size $\mathrm{\Delta }\mathrm{/}{E}_F$ and obtain the result $L_z=\frac{\hbar\nu}{2} N(1-\frac{\mu}{E_F})$for$\nu=2,3$\mathrm{.}Thus,thegap-dependenceof$L_z$for$\nu<4$entersatmostthroughthechemicalpotentialwhile$\nu\geq4$ is qualitatively different. In addition, we generalize the spectral asymmetry arguments to \emph{total} angular momentum in the ground state of triplet superfluids where due to a spin-orbit coupling $L_z$ is not a good quantum number. We find that the ground state total angular momentum also behaves very differently depending on total angular momentum of the Cooper pairs.