Vanishing edge currents and orbital angular momentum in non-$p$-wave topological chiral superconductors

The edge currents of two dimensional topological chiral superconductors with nonzero Cooper pair angular momentum-e.g., chiral $p$-, $d$-, and $f$-wave superconductivity-are studied. Bogoliubov-de Gennes and Ginzburg-Landau calculations are used to show that in the continuum limit, only chiral $p$-wave states have a nonzero edge current and orbital angular momentum. Outside this limit, when lattice effects become important, edge currents in non-$p$-wave superconductors are comparatively smaller, but can be nonzero. Using Ginzburg-Landau theory, a simple criterion is derived for when edge currents vanish for non-$p$-wave chiral superconductivity on a lattice. The implications of our results for putative chiral superconductors such as Sr$_2$RuO$_4$ and UPt$_3$ are discussed.