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{{Use American English|date=January 2019}}{{Short description|Phenomenon in many-body quantum systems
'''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 perturbation]]s 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>
}}
In [[quantum mechanics|quantum]] [[many-body physics]], '''topological degeneracy''' is a phenomenon in which the [[ground state]] of a [[gapped Hamiltonian|gapped many-body Hamiltonian]] becomes degenerate in the [[thermodynamic limit|limit of large system size]] such that the degeneracy cannot be lifted by any [[local perturbation]]s.<ref>{{cite journal | last1=Wen | first1=X. G. |author-link=Xiao-Gang Wen| last2=Niu | first2=Q. | title=Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces | journal=Physical Review B | publisher=American Physical Society (APS) | volume=41 | issue=13 | date=1 April 1990 | issn=0163-1829 | doi=10.1103/physrevb.41.9377 | pmid=9993283 | pages=9377–9396|url=http://dao.mit.edu/~wen/pub/topWN.pdf | bibcode=1990PhRvB..41.9377W}}</ref>


==Applications==
==Applications==


Topological degeneracy can be used to protect qubits which allows [[Topological quantum computer|topological quantum computation]].<ref>{{cite journal | last1=Nayak | first1=Chetan | last2=Simon | first2=Steven H. |author-link2=Steven H. Simon| last3=Stern | first3=Ady |author-link3=Ady Stern| last4=Freedman | first4=Michael |author-link4=Michael Freedman| last5=Das Sarma | first5=Sankar |author-link5=Sankar Das Sarma| title=Non-Abelian anyons and topological quantum computation | journal=Reviews of Modern Physics | publisher=American Physical Society (APS) | volume=80 | issue=3 | date=2008-09-12 | issn=0034-6861 | doi=10.1103/revmodphys.80.1083 | pages=1083–1159|arxiv=0707.1889 | bibcode=2008RvMP...80.1083N| s2cid=119628297 }}</ref> It is believed that topological degeneracy implies [[topological order]] (or long-range entanglement <ref>{{cite journal | last1=Chen | first1=Xie | last2=Gu | first2=Zheng-Cheng | last3=Wen | first3=Xiao-Gang |author-link3=Xiao-Gang Wen| title=Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order | journal=Physical Review B | volume=82 | issue=15 | date=2010-10-26 | issn=1098-0121 | doi=10.1103/physrevb.82.155138 | page=155138|arxiv=1004.3835 | bibcode=2010PhRvB..82o5138C| s2cid=14593420 }}</ref>) in the ground state.<ref name=wen>{{cite journal | last=Wen | first=X. G. |author-link=Xiao-Gang Wen| title= Topological Orders in Rigid States | journal=International Journal of Modern Physics B | publisher=World Scientific Pub Co Pte Lt | volume=04 | issue=2 | year=1990 | issn=0217-9792 | doi=10.1142/s0217979290000139 | pages=239–271|archive-url=https://web.archive.org/web/20070806075129/http://dao.mit.edu/~wen/pub/topo.pdf|archive-date=2007-08-06|url=http://dao.mit.edu/~wen/pub/topo.pdf | bibcode=1990IJMPB...4..239W}}</ref> Many-body states with topological degeneracy are described by [[topological quantum field theory]] at low energies.
Topological degeneracy can be used to protect qubits which allows [[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{{whom}} that topological degeneracy implies [[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{{fact}}. Many-body states with topological degeneracy are described by [[topological quantum field theory]] at low energies.


==Background==
==Background==


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>
Topological degeneracy was first introduced to physically define topological order.<ref>{{cite journal | last=Wen | first=X. G. |author-link=Xiao-Gang Wen| title=Vacuum degeneracy of chiral spin states in compactified space | journal=Physical Review B | publisher=American Physical Society (APS) | volume=40 | issue=10 | date=1 September 1989 | issn=0163-1829 | doi=10.1103/physrevb.40.7387 | pmid=9991152 | pages=7387–7390| bibcode=1989PhRvB..40.7387W }}</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.
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 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
⚫
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
walls,<ref name="1104.5047">{{cite journal | last1=Kitaev | first1=Alexei | last2=Kong | first2=Liang | title=Models for gapped boundaries and domain walls | journal=Commun. Math. Phys. | volume=313 | issue=2 | pages=351–373 | date=July 2012 | issn= 1432-0916 | doi=10.1007/s00220-012-1500-5 |arxiv=1104.5047| bibcode=2012CMaPh.313..351K | s2cid=3070055 }}</ref> including both Abelian topological orders <ref name="1212.4863">{{cite journal | last1=Wang | first1=Juven | last2=Wen | first2=Xiao-Gang | title=Boundary Degeneracy of Topological Order | journal=Physical Review B | volume=91 | issue=12 | date=13 March 2015 | issn= 2469-9969 | doi=10.1103/PhysRevB.91.125124 | page=125124 |arxiv=1212.4863| bibcode=2015PhRvB..91l5124W | s2cid=17803056 }}</ref><ref name="1306.4254">{{cite journal | last=Kapustin | first=Anton | title=Ground-state degeneracy for abelian anyons in the presence of gapped boundaries | journal=Physical Review B | publisher=American Physical Society (APS) | volume=89 | issue=12 | date=19 March 2014 | issn= 2469-9969 | doi=10.1103/PhysRevB.89.125307 | page=125307 |arxiv=1306.4254 | bibcode=2014PhRvB..89l5307K| s2cid=33537923 }}</ref>
and non-Abelian topological orders.
<ref name="1408.0014">{{cite journal | last1=Wan | first1=Hung | last2=Wan | first2=Yidun | title=Ground State Degeneracy of Topological Phases on Open Surfaces | journal=Physical Review Letters | volume=114 | issue=7 | date=18 February 2015 | issn= 1079-7114 | doi=10.1103/PhysRevLett.114.076401 | pmid=25763964 | page=076401 |arxiv=1408.0014 | bibcode=2015PhRvL.114g6401H| s2cid=10125789 }}</ref>
<ref name="1408.6514">{{cite journal | last1=Lan | first1=Tian | last2=Wang | first2=Juven | last3=Wen | first3=Xiao-Gang | title=Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy | journal=Physical Review Letters | volume=114 | issue=7 | date=18 February 2015 | issn= 1079-7114 | doi=10.1103/PhysRevLett.114.076402 | page=076402 |arxiv=1408.6514 | pmid=25763965 | bibcode=2015PhRvL.114g6402L| s2cid=14662084 }}</ref> The application of these types of systems for [[quantum computation]] has been proposed.<ref name="quant-ph/9811052">{{cite journal | last1=Bravyi | first1=S. B. | last2=Kitaev | first2=A. Yu. | title=Quantum codes on a lattice with boundary |arxiv=quant-ph/9811052| year=1998 | bibcode=1998quant.ph.11052B }}</ref> In certain generalized cases, one can also design the systems with topological interfaces enriched or extended by global or gauge symmetries.<ref name="1705.06728">{{cite journal | last1=Wang | first1=Juven | last2=Wen | first2=Xiao-Gang | last3=Witten | first3=Edward | title=Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions | journal=Physical Review X | volume=8 | issue=3 | date=August 2018 | issn= 2160-3308 | doi=10.1103/PhysRevX.8.031048 | page= 031048 |arxiv=1705.06728| bibcode=2018PhRvX...8c1048W | s2cid=119117766 }}</ref>

⚫
The topological degeneracy also appear in non-interacting fermion systems (such as p+ip superconductors<ref>{{cite journal | last1=Read | first1=N. | last2=Green | first2=Dmitry | title=Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect | journal=Physical Review B | volume=61 | issue=15 | date=15 April 2000 | issn=0163-1829 | doi=10.1103/physrevb.61.10267 | pages=10267–10297|arxiv=cond-mat/9906453 | bibcode=2000PhRvB..6110267R| s2cid=119427877 }}</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
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).
<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>{{cite journal | last=Kitaev | first=A Yu | title=Unpaired Majorana fermions in quantum wires | journal=Physics-Uspekhi | publisher=Uspekhi Fizicheskikh Nauk (UFN) Journal | volume=44 | issue=10S | date=1 September 2001 | issn=1468-4780 | doi=10.1070/1063-7869/44/10s/s29 | pages=131–136|arxiv=cond-mat/0010440 | bibcode=2001PhyU...44..131K| s2cid=9458459 }}</ref>
Such topological degeneracy is referred as "Majorana zero-mode" on the defects.<ref>
<ref>{{cite journal | last=Ivanov | first=D. A. | title=Non-Abelian Statistics of Half-Quantum Vortices inp-Wave Superconductors | journal=Physical Review Letters | volume=86 | issue=2 | date=8 January 2001 | issn=0031-9007 | doi=10.1103/physrevlett.86.268 | pmid=11177808 | pages=268–271|arxiv=cond-mat/0005069| bibcode=2001PhRvL..86..268I | s2cid=23070827 }}</ref>
Alexei Kitaev, arXiv:cond-mat/0010440;
In contrast, there are many types of topological degeneracy for interacting systems.<ref>{{cite journal | last=Bombin | first=H. | title=Topological Order with a Twist: Ising Anyons from an Abelian Model | journal=Physical Review Letters | volume=105 | issue=3 | date=14 July 2010 | issn=0031-9007 | doi=10.1103/physrevlett.105.030403 | pmid=20867748 | page=030403|arxiv=1004.1838 | bibcode=2010PhRvL.105c0403B| s2cid=5285193 }}</ref>
Unpaired Majorana fermions in quantum wires</ref>
<ref>{{cite journal | last1=Barkeshli | first1=Maissam | last2=Qi | first2=Xiao-Liang | title=Topological Nematic States and Non-Abelian Lattice Dislocations | journal=Physical Review X | volume=2 | issue=3 | date=24 August 2012 | issn=2160-3308 | doi=10.1103/physrevx.2.031013 | page=031013|arxiv=1112.3311| bibcode=2012PhRvX...2c1013B |doi-access=free}}</ref>
<ref>
<ref>{{cite journal | last1=You | first1=Yi-Zhuang | last2=Wen | first2=Xiao-Gang | title=Projective non-Abelian statistics of dislocation defects in aZNrotor model | journal=Physical Review B | publisher=American Physical Society (APS) | volume=86 | issue=16 | date=17 October 2012 | issn=1098-0121 | doi=10.1103/physrevb.86.161107 | page=161107(R)|arxiv=1204.0113| bibcode=2012PhRvB..86p1107Y | s2cid=119266900 }}</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.
A systematic description of topological degeneracy is given by tensor category (or [[monoidal category]]) theory.


Latest revision as of 03:19, 11 July 2022

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

[edit]

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

[edit]

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 or gauge 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

[edit]

References

[edit]
  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. Bibcode:1990PhRvB..41.9377W. doi:10.1103/physrevb.41.9377. ISSN 0163-1829. PMID 9993283.
  2. ^ Nayak, Chetan; Simon, Steven H.; Stern, Ady; Freedman, Michael; Das Sarma, Sankar (2008-09-12). "Non-Abelian anyons and topological quantum computation". Reviews of Modern Physics. 80 (3). American Physical Society (APS): 1083–1159. arXiv:0707.1889. Bibcode:2008RvMP...80.1083N. doi:10.1103/revmodphys.80.1083. ISSN 0034-6861. S2CID 119628297.
  3. ^ Chen, Xie; Gu, Zheng-Cheng; Wen, Xiao-Gang (2010-10-26). "Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order". Physical Review B. 82 (15): 155138. arXiv:1004.3835. Bibcode:2010PhRvB..82o5138C. doi:10.1103/physrevb.82.155138. ISSN 1098-0121. S2CID 14593420.
  4. ^ Wen, X. G. (1990). "Topological Orders in Rigid States" (PDF). International Journal of Modern Physics B. 04 (2). World Scientific Pub Co Pte Lt: 239–271. Bibcode:1990IJMPB...4..239W. doi:10.1142/s0217979290000139. ISSN 0217-9792. Archived from the original (PDF) on 2007-08-06.
  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. Bibcode:1989PhRvB..40.7387W. doi:10.1103/physrevb.40.7387. ISSN 0163-1829. PMID 9991152.
  6. ^ Kitaev, Alexei; Kong, Liang (July 2012). "Models for gapped boundaries and domain walls". Commun. Math. Phys. 313 (2): 351–373. arXiv:1104.5047. Bibcode:2012CMaPh.313..351K. doi:10.1007/s00220-012-1500-5. ISSN 1432-0916. S2CID 3070055.
  7. ^ Wang, Juven; Wen, Xiao-Gang (13 March 2015). "Boundary Degeneracy of Topological Order". Physical Review B. 91 (12): 125124. arXiv:1212.4863. Bibcode:2015PhRvB..91l5124W. doi:10.1103/PhysRevB.91.125124. ISSN 2469-9969. S2CID 17803056.
  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. Bibcode:2014PhRvB..89l5307K. doi:10.1103/PhysRevB.89.125307. ISSN 2469-9969. S2CID 33537923.
  9. ^ Wan, Hung; Wan, Yidun (18 February 2015). "Ground State Degeneracy of Topological Phases on Open Surfaces". Physical Review Letters. 114 (7): 076401. arXiv:1408.0014. Bibcode:2015PhRvL.114g6401H. doi:10.1103/PhysRevLett.114.076401. ISSN 1079-7114. PMID 25763964. S2CID 10125789.
  10. ^ Lan, Tian; Wang, Juven; Wen, Xiao-Gang (18 February 2015). "Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy". Physical Review Letters. 114 (7): 076402. arXiv:1408.6514. Bibcode:2015PhRvL.114g6402L. doi:10.1103/PhysRevLett.114.076402. ISSN 1079-7114. PMID 25763965. S2CID 14662084.
  11. ^ Bravyi, S. B.; Kitaev, A. Yu. (1998). "Quantum codes on a lattice with boundary". arXiv:quant-ph/9811052. Bibcode:1998quant.ph.11052B. {{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): 031048. arXiv:1705.06728. Bibcode:2018PhRvX...8c1048W. doi:10.1103/PhysRevX.8.031048. ISSN 2160-3308. S2CID 119117766.
  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): 10267–10297. arXiv:cond-mat/9906453. Bibcode:2000PhRvB..6110267R. doi:10.1103/physrevb.61.10267. ISSN 0163-1829. S2CID 119427877.
  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. Bibcode:2001PhyU...44..131K. doi:10.1070/1063-7869/44/10s/s29. ISSN 1468-4780. S2CID 9458459.
  15. ^ Ivanov, D. A. (8 January 2001). "Non-Abelian Statistics of Half-Quantum Vortices inp-Wave Superconductors". Physical Review Letters. 86 (2): 268–271. arXiv:cond-mat/0005069. Bibcode:2001PhRvL..86..268I. doi:10.1103/physrevlett.86.268. ISSN 0031-9007. PMID 11177808. S2CID 23070827.
  16. ^ Bombin, H. (14 July 2010). "Topological Order with a Twist: Ising Anyons from an Abelian Model". Physical Review Letters. 105 (3): 030403. arXiv:1004.1838. Bibcode:2010PhRvL.105c0403B. doi:10.1103/physrevlett.105.030403. ISSN 0031-9007. PMID 20867748. S2CID 5285193.
  17. ^ Barkeshli, Maissam; Qi, Xiao-Liang (24 August 2012). "Topological Nematic States and Non-Abelian Lattice Dislocations". Physical Review X. 2 (3): 031013. arXiv:1112.3311. Bibcode:2012PhRvX...2c1013B. 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. Bibcode:2012PhRvB..86p1107Y. doi:10.1103/physrevb.86.161107. ISSN 1098-0121. S2CID 119266900.
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