  The chiral hinge modes are the key feature of a second order topological
insulator in three dimensions. Here we propose a quadrupole index in
combination of a slab Chern number in the bulk to characterize the flowing
pattern of chiral hinge modes along the hinges at the intersection of the
surfaces of a sample. We further utilize the topological field theory to
demonstrate the correspondent connection of the chiral hinge modes to the
quadrupole index and the slab Chern number, and present a picture of
three-dimensional quantum anomalous Hall effect as a consequence of chiral
hinge modes. The two bulk topological invariants can be measured in electric
transport and magneto-optical experiments. In this way we establish the
bulk-hinge correspondence in a three-dimensional second order topological
insulator.
