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'''Topological degeneracy''' is a phenomenon in quantum many-body physics,
that the ground state of a gapped many-body system become degenerate
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>
 
Topological degeneracy
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>
It is believed that the appearance of  topological degeneracy implies the [[topological order]] (or long-range entanglements<ref>
Xie Chen, Zheng-Cheng Gu, [[Xiao-Gang Wen]],
[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)
</ref>) in the ground state. Many-body states with topological degeneracy
are described by [[topological quantum field theory]] at low energies.
 
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>
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.
 
The topological degeneracy also appears in the situation with trapped quasiparticles, where the topological degeneracy depends on the number and the
type of the trapped quasiparticles. Braiding those  quasiparticles
leads to topologically protected non-Abelian [[geometric phase]], which can be used to perform topologically protected [[quantum computation]].
The topological degeneracy also appear in non-interacting fermion systems
(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
where number of the degenerate states is given by <math>2^{N_d/2}/2</math>, where
<math>N_d</math> is the number of the defects (such as the number of vortices).
Such topological degeneracy is referred as "Majorana zero-mode" on the defects.<ref>
Alexei Kitaev, arXiv:cond-mat/0010440;
Unpaired Majorana fermions in quantum wires</ref>
<ref>
D. A. Ivanov, Phys. Rev. Lett. 86, 268 (2001); arXiv:cond-mat/0005069;
Non-abelian statistics of half-quantum vortices in p-wave superconductors
</ref>
In contrast, there are many types of topological degeneracy for
interacting systems.<ref>
H. Bombin, Phys. Rev. Lett. 105, 030403 (2010),
arXiv:1004.1838.
Topological Order with a Twist: Ising Anyons from an Abelian Model
</ref>
<ref>M. Barkeshli, X.-L. Qi, arXiv:1112.3311;
Topological Nematic States and Non-Abelian Lattice Dislocations
</ref>
<ref>
Yi-Zhuang You, Xiao-Gang Wen, arXiv:1204.0113;
Projective non-Abelian Statistics of Dislocation Defects in a Z_N Rotor Model
</ref>
A systematic description of topological degeneracy is given by
tensor category (or [[monoidal category]]) theory.
 
==See also==
 
* [[Topological order]]
* [[Quantum topology]]
* [[Topological defect]]
* [[Topological quantum field theory]]
* [[Topological quantum number]]
* [[Majorana fermion]]
 
== References ==
{{Reflist}}
 
[[Category:Quantum phases]]
[[Category:Condensed matter physics]]

Revision as of 23:00, 29 March 2013

Topological degeneracy is a phenomenon in quantum many-body physics, that the ground state of a gapped many-body system become degenerate 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.[1]

Topological degeneracy can be used as protected qubits which allows us to perform topological quantum computation.[2] It is believed that the appearance of topological degeneracy implies the topological order (or long-range entanglements[3]) in the ground state. Many-body states with topological degeneracy are described by topological quantum field theory at low energies.

Topological degeneracy was first introduced to physically define topological order.[4] 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.

The topological degeneracy also appears in the situation with trapped quasiparticles, where the topological degeneracy depends on the number and the type of the trapped quasiparticles. Braiding those quasiparticles leads to topologically protected non-Abelian geometric phase, which can be used to perform topologically protected quantum computation.

The topological degeneracy also appear in non-interacting fermion systems (such as p+ip superconductors[5]) with trapped defects (such as vortices). In non-interacting fermion systems, there is only one type of topological degeneracy where number of the degenerate states is given by 2Nd/2/2, where Nd is the number of the defects (such as the number of vortices). Such topological degeneracy is referred as "Majorana zero-mode" on the defects.[6] [7] In contrast, there are many types of topological degeneracy for interacting systems.[8] [9] [10] A systematic description of topological degeneracy is given by tensor category (or monoidal category) theory.

See also

References

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  1. ↑ Xiao-Gang Wen and Qian Niu, Phys. Rev. B41, 9377 (1990), "Ground state degeneracy of the FQH states in presence of random potential and on high genus Riemann surfaces"
  2. ↑ 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); arXiv:0707.1889
  3. ↑ Xie Chen, Zheng-Cheng Gu, Xiao-Gang Wen, Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order Phys. Rev. B 82, 155138 (2010)
  4. ↑ Xiao-Gang Wen, Phys. Rev. B, 40, 7387 (1989), "Vacuum Degeneracy of Chiral Spin State in Compactified Spaces"
  5. ↑ 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"
  6. ↑ Alexei Kitaev, arXiv:cond-mat/0010440; Unpaired Majorana fermions in quantum wires
  7. ↑ D. A. Ivanov, Phys. Rev. Lett. 86, 268 (2001); arXiv:cond-mat/0005069; Non-abelian statistics of half-quantum vortices in p-wave superconductors
  8. ↑ H. Bombin, Phys. Rev. Lett. 105, 030403 (2010), arXiv:1004.1838. Topological Order with a Twist: Ising Anyons from an Abelian Model
  9. ↑ M. Barkeshli, X.-L. Qi, arXiv:1112.3311; Topological Nematic States and Non-Abelian Lattice Dislocations
  10. ↑ Yi-Zhuang You, Xiao-Gang Wen, arXiv:1204.0113; Projective non-Abelian Statistics of Dislocation Defects in a Z_N Rotor Model
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