  Edge states reveal the nontrivial topology of energy band in the bulk. As
localized states at boundaries, many-body edge states may obey a special
symmetry that is broken in the bulk. When local particle-particle interaction
is induced, they may support a particular property. We consider an extended
two-dimensional Su-Schrieffer-Heeger Hubbard model and examine the appearance
of $\eta$-pairing states, which are excited eigenstates related to
superconductivity. In the absence of Hubbard interaction, the energy band is
characterized by topologically invariant polarization in association with edge
states. In the presence of on-site Hubbard interaction, $\eta$-pairing edge
states appear in the topologically nontrivial phase, resulting in the
condensation of pairs at the boundary. In addition, as Hamiltonian eigenstates,
the edge states contain paired fermions and unpaired fermions. Neither affects
the other; they act as two-fluid states. From numerical simulations of
many-body scattering processes, a clear manifestation and experimental
detection scheme of topologically protected two-fluid edge states are provided.
