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Interplay of symmetry and topology in correlated bosons

Discovery of topological insulators (TIs) taught us an important lesson about how symmetry can reshape and enrich our understanding of gapped quantum phases. Unlike in weakly-correlated semiconductors, strong interactions are necessary to reach a topological phase in boson/spin systems. What are the bosonic analogs of TIs, which support symmetry-protected gapless boundary excitations in spite of a gapped "trivial" bulk? In this talk I'll provide a simple physical picture of these bosonic TIs, by considering how to quantum disorder a symmetry-breaking phase. Effective field theories are derived, which naturally encodes symmetry properties and allows us to classify bosonic TIs. This framework is generalized to topologically ordered phases with symmetries, such as gapped Z2 spin liquids. In the end we discuss its possible application to understand the spin liquid ground state of kagome Heisenberg model.