From Wikipedia, the free encyclopedia
Phenomenon in many-body quantum systems
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 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
2
N
d
/
2
/
2
{\displaystyle 2^{N_{d}/2}/2}
, where
N
d
{\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.
[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
^
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 .
^ 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 .
^ 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 .
^
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.
^
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 .
^ 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 .
^ 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 .
^ 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 .
^ 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 .
^ 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 .
^ Bravyi, S. B.; Kitaev, A. Yu. (1998). "Quantum codes on a lattice with boundary".
arXiv :
quant-ph/9811052 .
Bibcode :
1998quant.ph.11052B .
^ 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 .
^ 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 .
^ 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 .
^ 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 .
^ 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 .
^ 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 .
^ 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 .