  Exploring topological phases in interacting systems is a challenging task. We
investigate many-body topological physics of interacting fermions in an
extended Su-Schrieffer-Heeger (SSH) model, which extends the two sublattices of
SSH model into four sublattices and thus is dubbed SSH4 model, based on the
density-matrix renormalization-group numerical method. The interaction-driven
phase transition from topological insulator to charge density wave (CDW) phase
can be identified by analyzing the variations of entanglement spectrum,
entanglement entropies, energy gaps, CDW order parameter, and fidelity. We map
the global phase diagram of the many-body ground state, which contains
nontrivial topological insulator, trivial insulator and CDW phases,
respectively. In contrast to interacting SSH model, in which the phase
transitions to the CDW phase are argued to be first-order phase transitions,
the phase transitions between the CDW phase and topologically
trivial/nontrivial phases are shown to be continuous phase transitions.
Finally, we {also} show the phase diagram of interacting spinful SSH4 model,
where the attractive (repulsive) on-site spin interaction amplifies
(suppresses) the CDW phase. The models analyzed here can be implemented with
ultracold atoms on optical superlattices.
