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| '''Topological degeneracy''' is a phenomenon in quantum many-body physics, | | The author's name is Chung and she feels comfy when individuals use the complete name. One of the extremely best issues in the world for me is karaoke but I don't have the time lately. Accounting is exactly where his main earnings [http://photo.net/gallery/tag-search/search?query_string=arrives arrives] from. Her spouse and her chose to reside in New Jersey. Check out my web site right here: http://www.abc12.com/story/26765494/weight-destroyer-review-does-weight-destroyer-really-work-read-my-story-and-find-out<br><br>Feel free to surf to my web blog :: [http://www.abc12.com/story/26765494/weight-destroyer-review-does-weight-destroyer-really-work-read-my-story-and-find-out Abc13 Weight Destroyer] |
| that the ground state of a gapped many-body system become degenerate
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| in the large system size limit, and that such a degeneracy cannot be lifted by any local perturbations as long as the system size is large.<ref>[[Xiao-Gang Wen]] and [[Qian Niu]], [http://dao.mit.edu/~wen/pub/topWN.pdf Phys. Rev. '''B41''', 9377 (1990), "Ground state degeneracy of the FQH states in presence of random potential and on high genus Riemann surfaces"] </ref> | |
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| Topological degeneracy
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| can be used as protected qubits which allows us to perform [[Topological quantum computer|topological quantum computation]].<ref>Chetan Nayak, [[Steven H. Simon]], [[Ady Stern]], [[Michael Freedman]], [[Sankar Das Sarma]], "Non-Abelian Anyons and Topological Quantum Computation", Rev. Mod. Phys. 80, 1083 (2008); [http://www.arxiv.org/abs/0707.1889 arXiv:0707.1889] </ref>
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| It is believed that the appearance of topological degeneracy implies the [[topological order]] (or long-range entanglements<ref>
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| Xie Chen, Zheng-Cheng Gu, [[Xiao-Gang Wen]],
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| [http://arxiv.org/abs/1004.3835 Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order] Phys. Rev. B 82, 155138 (2010)
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| </ref>) in the ground state. Many-body states with topological degeneracy
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| are described by [[topological quantum field theory]] at low energies.
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| Topological degeneracy was first introduced to physically define topological order.<ref> [[Xiao-Gang Wen]], Phys. Rev. B, '''40''', 7387 (1989), "Vacuum Degeneracy of Chiral Spin State in Compactified Spaces" </ref>
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| In two-dimensional space, the topological degeneracy depends on the topology of space, and the topological degeneracy on high genus Riemann surfaces encode all information on the [[quantum dimensions]] and the fusion algebra of the quasiparticles. In particular, the topological degeneracy on torus is equal to the number of quasiparticles types.
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| The topological degeneracy also appears in the situation with trapped quasiparticles, where the topological degeneracy depends on the number and the
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| type of the trapped quasiparticles. Braiding those quasiparticles
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| leads to topologically protected non-Abelian [[geometric phase]], which can be used to perform topologically protected [[quantum computation]].
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|
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| The topological degeneracy also appear in non-interacting fermion systems
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| (such as p+ip superconductors<ref>N. Read and D. Green, Phys. Rev. B61, 10267 (2000); arXiv:cond-mat/9906453; "Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect"</ref>) with trapped defects (such as vortices). In non-interacting fermion systems, there is only one type of topological degeneracy
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| where number of the degenerate states is given by <math>2^{N_d/2}/2</math>, where
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| <math>N_d</math> is the number of the defects (such as the number of vortices).
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| Such topological degeneracy is referred as "Majorana zero-mode" on the defects.<ref>
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| Alexei Kitaev, arXiv:cond-mat/0010440;
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| Unpaired Majorana fermions in quantum wires</ref>
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| <ref>
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| D. A. Ivanov, Phys. Rev. Lett. 86, 268 (2001); arXiv:cond-mat/0005069;
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| Non-abelian statistics of half-quantum vortices in p-wave superconductors
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| </ref>
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| In contrast, there are many types of topological degeneracy for
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| interacting systems.<ref>
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| H. Bombin, Phys. Rev. Lett. 105, 030403 (2010),
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| arXiv:1004.1838.
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| Topological Order with a Twist: Ising Anyons from an Abelian Model
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| </ref>
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| <ref>M. Barkeshli, X.-L. Qi, arXiv:1112.3311;
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| Topological Nematic States and Non-Abelian Lattice Dislocations
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| </ref>
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| <ref>
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| Yi-Zhuang You, Xiao-Gang Wen, arXiv:1204.0113;
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| Projective non-Abelian Statistics of Dislocation Defects in a Z_N Rotor Model
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| </ref>
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| A systematic description of topological degeneracy is given by
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| tensor category (or [[monoidal category]]) theory.
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| ==See also==
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| * [[Topological order]]
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| * [[Quantum topology]]
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| * [[Topological defect]]
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| * [[Topological quantum field theory]]
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| * [[Topological quantum number]]
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| * [[Majorana fermion]]
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| == References ==
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| {{Reflist}}
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| [[Category:Quantum phases]]
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| [[Category:Condensed matter physics]]
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