Topologically protected two-fluid edge states

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.