  A major challenge to the chiral $p$-wave hypothesis for the pairing symmetry
of the unconventional superconductor Sr$_2$RuO$_4$ is the null result of
sensitive scanning magnetometry experiments designed to detect the expected
spontaneous charge currents. Motivated by junction tunneling conductance
measurements which indicate the quenching of superconductivity at the surfaces
of even high-purity samples, we examine the spontaneous currents in a chiral
$p$-wave superconductor near a normal metal / superconductor interface using
the lattice Bogoliubov-de Gennes equations and Ginzburg-Landau theory, and find
that the edge current is suppressed by more than an order of magnitude compared
to previous estimates. These calculations demonstrate that interface details
can have a quantitatively meaningful effect on the expectations for
magnetometry experiments.
