Topological degeneracy
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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.[1] Topological degeneracy can be used as protected qubits and allow 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 on torus is equal to the number of quasiparticles types. The topological degeneracy on high genus Riemann surfaces depends on the topology of space and encode all information on the fusion of the quasiparticles.
The topological degeneracy also appear in the situation with trapped quasiparticles, where topological degeneracy depend on the number and the type of the trapped quasiparticles. Braiding those quasiparticles may lead 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 , where is the number of the defects (such as the number of vortices). Such topological degeneracy is referred as "Majorana zero-mode" on the defects. In contrast, there are many types of topological degeneracy for interacting systems. 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
- ^ 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"
- ^ 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
- ^ 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)
- ^ Xiao-Gang Wen, Phys. Rev. B, 40, 7387 (1989), "Vacuum Degeneracy of Chiral Spin State in Compactified Spaces"
- ^ 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"