# Whis is the difference between charge fractionalization in 1D and 2D?

Both 1D Polyacetelene and 2D fractional quantum Hall state can support fractional excitations.

But as I can see, there are some differences: the ground state of Polyacetelene breaks translational invariance but FQHE does not. In general a symmetry breaking state in 2D are considered as conventional state and does not support fractional excitations.

I wonder why is there such a difference between 1D and 2D fractionalizations?
Is there any 1D model can support fractional excitations from a symmetry preserved ground state ?

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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.

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Does long-range entanglement mean long-range interaction such as Coulomb interaction? The short-range entanglement in topological insulators means the nearest-neighboring hopping? – Timothy May 3 '13 at 0:42
No. Here we only consider systems with short range interactions and short range hopping. Even with short range interactions and short range hopping, the ground state can be long-range entangled. Long-range entanglement is a new concept, which was defined in arxiv.org/abs/1004.3835 . See also arxiv.org/abs/1210.1281 , – Xiao-Gang Wen May 3 '13 at 5:01

Is there any 1D model can support fractional excitations from a symmetry preserved ground state ?

Yes, in 1D there exist symmetry preserved groud states supporting fractional charge(at the boundary or domain wall). The S=1 SO(3) symmetric Haldane phase is an example, since the edge state carry spin-1/2 and is fractionalized. More examples can be found below: http://arxiv.org/abs/1204.0520 http://arxiv.org/abs/1209.2990

Generally, 1D gapped phases supporting fractional charge of the symmetry group are call symmetry protected topological (SPT) phases, which are classified by projective representatoin of the symmetry group (See Chen, Gu, Wen, Phys. Rev. B 83, 035107 (2011)).

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Fractional excitations are understood to be generic in 1D. An example with a "symmetry presreved" state (whatever that is supposed to mean in 1D) is the simple Luttinger liquid. The Luttinger liquid exhibits charge-fractionalization in to spin charge separation. This was first shown here, I believe.

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Thanks for answering. Since Luttinger liquid is a gapless let me rephrase my question: Is there any gapped 1D model can support fractional excitations from a symmetry preserved ground state ? – Lei Wang Apr 30 '13 at 6:11
(Anti)holons in the Hubbard model are fractional excitations that are gapped, with the gap being of the order of the on-site repulsion $U$. – delete000 Apr 30 '13 at 6:46