The notion of fractional charge is not well defined in 1D Luttinger liquid (despite many papers say that the charge is fractionalized in 1D Luttinger liquid). In fact, it is hard to define fractional charge in any gapless state if the low energy excitations are not described by free quasiparticles.

For gapped states, fractional charge in 1D is due to translation symmetry breaking, or due to projective representation of the symmetry group (symmetry protected topological phase), while fractional charge in 2D and higher is due to topological order
(ie long-range entanglement). See A physical understanding of fractionalization and Why is fractional statistics and non-Abelian common for fractional charges?

For example, topological insulators have no topological order (ie have only short-range entanglement). As a result, topological insulators have no quasiparticles with fractional charges. On the other hand, FQH states have non-trivial topological orders (ie long-range entanglement). As a result, FQH states have both fractional charges and fractional statistics.

This post imported from StackExchange Physics at 2014-04-05 04:41 (UCT), posted by SE-user Xiao-Gang Wen