# Topological degeneracy

Topological degeneracy is a phenomenon in quantum many-body physics, that the ground state of a gapped many-body system becomes 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]

## ApplicationsEdit

Topological degeneracy can be used to protect qubits which allows topological quantum computation.[2] It is believed that topological degeneracy implies topological order (or long-range entanglement [3]) in the ground state.[4] Many-body states with topological degeneracy are described by topological quantum field theory at low energies.

## BackgroundEdit

Topological degeneracy was first introduced to physically define topological order.[5] 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 topological defects (such as vortices, dislocations, holes in 2D sample, ends of a 1D sample, etc.), where the topological degeneracy depends on the number of defects. Braiding those topological defect 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[6]) 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 ${\displaystyle 2^{N_{d}/2}/2}$ , where ${\displaystyle N_{d}}$  is the number of the defects (such as the number of vortices). Such topological degeneracy is referred as "Majorana zero-mode" on the defects.[7] [8] In contrast, there are many types of topological degeneracy for interacting systems.[9] [10] [11] A systematic description of topological degeneracy is given by tensor category (or monoidal category) theory.

## ReferencesEdit

1. ^
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, Topological Orders in Rigid States. Int. J. Mod. Phys. B4, 239 (1990)
5. ^ Xiao-Gang Wen, Phys. Rev. B, 40, 7387 (1989), "Vacuum Degeneracy of Chiral Spin State in Compactified Spaces"
6. ^ 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"
7. ^ Alexei Kitaev, arXiv:cond-mat/0010440; Unpaired Majorana fermions in quantum wires
8. ^ D. A. Ivanov, Phys. Rev. Lett. 86, 268 (2001); arXiv:cond-mat/0005069; Non-abelian statistics of half-quantum vortices in p-wave superconductors
9. ^ H. Bombin, Phys. Rev. Lett. 105, 030403 (2010), arXiv:1004.1838. Topological Order with a Twist: Ising Anyons from an Abelian Model
10. ^ M. Barkeshli, X.-L. Qi, arXiv:1112.3311; Topological Nematic States and Non-Abelian Lattice Dislocations
11. ^ Yi-Zhuang You, Xiao-Gang Wen, arXiv:1204.0113; Projective non-Abelian Statistics of Dislocation Defects in a Z_N Rotor Model