  We study a tight-binding model on the honeycomb lattice of chiral $d$-wave
superconductivity that breaks time-reversal symmetry. Due to its nontrivial
sublattice structure, we show that it is possible to construct a
gauge-invariant time-reversal-odd bilinear of the pairing potential. The
existence of this bilinear reflects the sublattice polarization of the pairing
state. We show that it generates persistent loop current correlations around
each lattice site and opens a topological mass gap at the Dirac points,
resembling Haldane's model of the anomalous quantum Hall effect. In addition to
the usual chiral $d$-wave edge states, there also exist electron-like edge
resonances due to the topological mass gap. We show that the presence of
loop-current correlations directly leads to a nonzero intrinsic ac Hall
conductivity, which produces the polar Kerr effect without an external magnetic
field. Similar results also hold for the nearest-neighbor chiral $p$-wave
pairing. We briefly discuss the relevance of our results to superconductivity
in twisted bilayer graphene.
