  Real-space topological invariants were widely used to characterize
chiral-symmetric higher-order topological phases (HOTPs). However, a
momentum-space characterization to these HOTPs, which essentially reveals their
intrinsic bulk-boundary correspondence and facilitates their detection in
quantum simulation systems, is still lacking. Here, we propose an
experimentally observable momentum-space characterization to the
chiral-symmetric HOTPs by the concept of polarized topological charges. It
provides a unified description to topological phase transitions caused by the
closing and reopening of band gap not only of the bulk states but also the edge
states. Remarkably, these polarized topological charges can be identified by
measuring the pseudospin structures. A feasible scheme to detect the HOTPs in
the $^{87}$Rb cold atomic system is given. Our work opens an avenue for
characterization and experimental detection of the chiral-symmetric HOTPs in
momentum space.
