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The study of symmetry-protected topological states in presence of electroncorrelations has recently aroused great interest as rich and exotic phenomenacan emerge. Here, we report a concrete example by employing large-scaleunbiased quantum Monte Carlo study of the Kane-Mele model with cluster chargeinteractions. The ground-state phase diagram for the model at half filling isestablished. Our simulation identifies the coexistence of a symmetry-protectedtopological order with a symmetry-breaking Kekul$\acute{e}$ valence bond orderand shows that the spontaneous symmetry-breaking is accompanied by aninteraction-driven topological phase transition (TPT). This TPT featuresappearance of zeros of single-particle Green's function and gap closing in spinchannel rather than single-particle excitation spectrum, and thus has nomean-field correspondence.

Combined spontaneous symmetry-breaking and symmetry-protected topological order from cluster charge interaction