Flat band of topological states bound to a mobile impurity

I consider a particle in the topologically non-trivial Su-Schrieffer-Heeger (SSH) model interacting strongly with a mobile impurity, whose quantum dynamics is described by a topologically trivial Hamiltonian. A particle in the SSH model admits a topological zero-energy edge mode when a hard boundary is placed at a given site of the chain, which may be modelled by a static impurity. By solving the two-body problem analytically I show that, when the impurity is mobile, the topological edge states of the Su-Schrieffer-Heeger model remain fully robust and a flat band of bound states at zero energy is formed as long as the continuum spectrum of the two-body problem remains gapped, without the need for any boundaries in the system. This is guaranteed for a sufficiently heavy impurity. As a consequence of the infinite degeneracy of the zero energy modes, it is possible to spatially localise the particle-impurity bound states, effectively making the impurity immobile. These effects can be readily observed using two-dimensional photonic lattices.