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Topological degeneracy

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In quantum many-body physics, topological degeneracy is a phenomenon in which the ground state of a gapped many-body Hamiltonian becomes degenerate in the limit of large system size such that the degeneracy cannot be lifted by any local perturbations.[1]

Applications

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.

Background

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.

Topological degeneracy of topological order can be defined on a closed space or an open space with gapped boundaries or gapped domain walls,[6] including both Abelian topological orders [7] [8] and non-Abelian topological orders. [9] [10] The application of these types of systems for quantum computation has been proposed.[11] In certain generalized cases, one can also design the systems with topological interfaces enriched or extended by global symmetries.[12]


The topological degeneracy also appear in non-interacting fermion systems (such as p+ip superconductors[13]) 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.[14] [15] In contrast, there are many types of topological degeneracy for interacting systems.[16] [17] [18] A systematic description of topological degeneracy is given by tensor category (or monoidal category) theory.

See also

References

  1. ^ Wen, X. G.; Niu, Q. (1 April 1990). "Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces" (PDF). Physical Review B. 41 (13). American Physical Society (APS): 9377–9396. doi:10.1103/physrevb.41.9377. ISSN 0163-1829.
  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. ^ Wen, X. G. (1 September 1989). "Vacuum degeneracy of chiral spin states in compactified space". Physical Review B. 40 (10). American Physical Society (APS): 7387–7390. doi:10.1103/physrevb.40.7387. ISSN 0163-1829.
  6. ^ Kitaev, Alexei; Kong, Liang (July 2012). "Models for gapped boundaries and domain walls". Commun. Math. Phys. 313 (2). Springer-Verlag: 351. arXiv:1104.5047. doi:10.1007/s00220-012-1500-5. ISSN 1432-0916.
  7. ^ Wang, Juven; Wen, Xiao-Gang (13 March 2015). "Boundary Degeneracy of Topological Order". Physical Review B. 91 (12). American Physical Society (APS): 125124. arXiv:1212.4863. doi:10.1103/PhysRevB.91.125124. ISSN 2469-9969.
  8. ^ Kapustin, Anton (19 March 2014). "Ground-state degeneracy for abelian anyons in the presence of gapped boundaries". Physical Review B. 89 (12). American Physical Society (APS): 125307. arXiv:1306.4254. doi:10.1103/PhysRevB.89.125307. ISSN 2469-9969.
  9. ^ Wan, Hung; Wan, Yidun (18 February 2015). "Ground State Degeneracy of Topological Phases on Open Surfaces". Physical Review Letter. 114 (7). American Physical Society (APS): 076401. arXiv:1408.0014. doi:10.1103/PhysRevLett.114.076401. ISSN 1079-7114.
  10. ^ Lan, Tian; Wang, Juven; Wen, Xiao-Gang (18 February 2015). "Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy". Physical Review Letter. 114 (7). American Physical Society (APS): 076402. arXiv:1408.6514. doi:10.1103/PhysRevLett.114.076402. ISSN 1079-7114.
  11. ^ Bravyi, S. B.; Kitaev, A. Yu. "Quantum codes on a lattice with boundary". arXiv:quant-ph/9811052. {{cite journal}}: Cite journal requires |journal= (help)
  12. ^ Wang, Juven; Wen, Xiao-Gang; Witten, Edward (August 2018). "Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions". Physical Review X. 8 (3). American Physical Society (APS): 031048. arXiv:1705.06728. doi:10.1103/PhysRevX.8.031048. ISSN 2160-3308.
  13. ^ Read, N.; Green, Dmitry (15 April 2000). "Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect". Physical Review B. 61 (15). American Physical Society (APS): 10267–10297. arXiv:cond-mat/9906453. doi:10.1103/physrevb.61.10267. ISSN 0163-1829.
  14. ^ Kitaev, A Yu (1 September 2001). "Unpaired Majorana fermions in quantum wires". Physics-Uspekhi. 44 (10S). Uspekhi Fizicheskikh Nauk (UFN) Journal: 131–136. arXiv:cond-mat/0010440. doi:10.1070/1063-7869/44/10s/s29. ISSN 1468-4780.
  15. ^ Ivanov, D. A. (8 January 2001). "Non-Abelian Statistics of Half-Quantum Vortices inp-Wave Superconductors". Physical Review Letters. 86 (2). American Physical Society (APS): 268–271. arXiv:cond-mat/0005069. doi:10.1103/physrevlett.86.268. ISSN 0031-9007.
  16. ^ Bombin, H. (14 July 2010). "Topological Order with a Twist: Ising Anyons from an Abelian Model". Physical Review Letters. 105 (3). American Physical Society (APS): 030403. arXiv:1004.1838. doi:10.1103/physrevlett.105.030403. ISSN 0031-9007.
  17. ^ Barkeshli, Maissam; Qi, Xiao-Liang (24 August 2012). "Topological Nematic States and Non-Abelian Lattice Dislocations". Physical Review X. 2 (3). American Physical Society (APS): 031013. arXiv:1112.3311. doi:10.1103/physrevx.2.031013. ISSN 2160-3308.
  18. ^ You, Yi-Zhuang; Wen, Xiao-Gang (17 October 2012). "Projective non-Abelian statistics of dislocation defects in aZNrotor model". Physical Review B. 86 (16). American Physical Society (APS): 161107(R). arXiv:1204.0113. doi:10.1103/physrevb.86.161107. ISSN 1098-0121.
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