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We consider two-dimensional (2d) quantum many-body systems with long-rangeorders, where the only gapless excitations in the spectrum are Goldstone modesof spontaneously broken continuous symmetries. To understand the interplaybetween classical long-range order of local order parameters and quantum orderof long-range entanglement in the ground states, we study the topological pointdefects and textures of order parameters in such systems. We show that theuniversal properties of point defects and textures are determined by theremnant symmetry enriched topological order in the symmetry-breaking groundstates with a non-fluctuating order parameter, and provide a classification fortheir properties based on the inflation-restriction exact sequence. Wehighlight a few phenomena revealed by our theory framework. First, in theabsence of intrinsic topological orders, we show a connection between thesymmetry properties of point defects and textures to deconfined quantumcriticality. Second, when the symmetry-breaking ground state have intrinsictopological orders, we show that the point defects can permute different anyonswhen braided around. They can also obey projective fusion rules in the sensethat multiple vortices can fuse into an Abelian anyon, a phenomena for which wecoin "defect fractionalization". Finally, we provide a formula to compute thefractional statistics and fractional quantum numbers carried by textures(skyrmions) in Abelian topological orders.

Theory of topological defects and textures in two-dimensional quantum orders with spontaneous symmetry breaking