Edge state behavior of interacting Bosons in a Su-Schrieffer-Heeger lattice

In the low momentum regime, the Su-Schrieffer-Heeger (SSH) model's key characteristics are encapsulated by a Dirac-type Hamiltonian in continuum space, i.e., the localized states emerge at the boundaries. Building on this, we have developed an effective Hamiltonian to model ultra cold interacting Bosons on an SSH like lattice through variational minimization under the mean field approximation. To pinpoint the boundary states, we have developed an algorithm by generalizing the imaginary time propagator, where a initial state evolves under the squared Hamiltonian to converge to the targeted state. This algorithm has broader applicability, enabling the identification of specific eigenstates in various contexts. Furthermore, we draw a parallel to an experimentally physical setup involving a gas of ultra cold Bosons confined to an array of potential wells with alternating depths. By establishing the system's analogy with the SSH system, we apply our algorithm to investigate boundary states in the presence of interaction, demonstrating how these findings align with those of the continuous system.