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