  The combination of topology and quantum criticality can give rise to an
exotic mix of counterintuitive effects. Here, we show that unexpected
topological properties take place in a paradigmatic strongly-correlated
Hamiltonian: the 1D extended Bose-Hubbard model. In particular, we reveal the
presence of two distinct topological quantum critical points with localized
edge states and gapless bulk excitations. Our results show that the topological
critical points separate two phases, one topologically protected and the other
topologically trivial, both characterized by a long-range ordered string
correlation function. The long-range order persists also at the topological
critical points and it reflects the presence of localized edge states protected
by a finite charge gap. Finally, we introduce a super-resolution quantum gas
microscopy scheme for dipolar dysprosium atoms, which provides a reliable route
towards the experimental study of topological quantum critical points.
