查看“︁拓扑简并”︁的源代码

在量子多体物理中，'''拓扑简并'''是指有能隙的[[多体哈密顿量]]在大系统尺寸极限下的基态简并现象，这种基态简并不会被局域微扰破坏<ref name="#1">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] {{Wayback|url=http://www.arxiv.org/abs/0707.1889 |date=20210112193821 }}</ref>。

== 应用 ==
拓扑简并可以用于保护允许进行拓扑量子计算<ref name="#1"/>的[[量子比特]]。人们认为拓扑简并意味着基态中存在拓扑序(或长程纠缠<ref>
Xie Chen, Zheng-Cheng Gu, [[Xiao-Gang Wen]],
[[arxiv:1004.3835|Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order]] Phys. Rev. B 82, 155138 (2010)
</ref>)。<ref name="wen">[[Xiao-Gang Wen]], [http://dao.mit.edu/~wen/pub/topo.pdf Topological Orders in Rigid States.] {{Wayback|url=http://dao.mit.edu/~wen/pub/topo.pdf |date=20110720000932 }} ''Int. J. Mod. Phys''. '''B4''', 239 (1990)</ref> 具有拓扑简并的多体态可以用低能[[拓撲量子場論|拓扑量子场论]]描述。

== 背景 ==
拓扑简并最早是在定义拓扑序的时候引入的。<ref>{{Cite journal|title=Vacuum degeneracy of chiral spin states in compactified space|last=Wen|first=X. G.|authorlink=Xiao-Gang Wen|date=1 September 1989|journal=Physical Review B|publisher=American Physical Society (APS)|issue=10|doi=10.1103/physrevb.40.7387|volume=40|pages=7387–7390|issn=0163-1829|pmid=9991152}}</ref>在两维空间中，拓扑简并依赖于空间的拓扑性质，拓扑简并在高属黎曼面(high genus Riemann surfaces)包含了的所有量子维度上的信息也包含了准粒子的融合代数(fusion algebra)。 环面上的拓扑简并度与准粒子类型的数目相同。

在有拓扑缺陷（例如旋涡，位错，2D样品的孔洞，1D样品的末端，等等）的情况下拓扑简并也会出现，此时拓扑简并度与缺陷的数目相关。拓扑缺陷之间的编织（braiding）会出现拓扑保护非阿贝尔几何相，它可用于进行拓扑保护的[[量子计算机|量子计算]]。

拓扑序的拓扑简并可以定义在在一个封闭空间或有边界的开放空间或者有能隙的畴壁（domain wall）上<ref name="1104.5047">{{Cite journal|title=Models for gapped boundaries and domain walls|last=Kitaev|first=Alexei|last2=Kong|first2=Liang|date=July 2012|journal=Commun. Math. Phys.|issue=2|doi=10.1007/s00220-012-1500-5|volume=313|pages=351–373|arxiv=1104.5047|issn=1432-0916}}</ref>，这里拓扑序既包括阿贝尔拓扑序 <ref name="1212.4863">{{Cite journal|title=Boundary Degeneracy of Topological Order|last=Wang|first=Juven|last2=Wen|first2=Xiao-Gang|date=13 March 2015|journal=Physical Review B|issue=12|doi=10.1103/PhysRevB.91.125124|volume=91|page=125124|arxiv=1212.4863|issn=2469-9969}}</ref><ref name="1306.4254">{{Cite journal|title=Ground-state degeneracy for abelian anyons in the presence of gapped boundaries|last=Kapustin|first=Anton|date=19 March 2014|journal=Physical Review B|publisher=American Physical Society (APS)|issue=12|doi=10.1103/PhysRevB.89.125307|volume=89|page=125307|arxiv=1306.4254|issn=2469-9969}}</ref>也包括阿贝尔拓扑序。<ref name="1408.0014">{{Cite journal|title=Ground State Degeneracy of Topological Phases on Open Surfaces|last=Wan|first=Hung|last2=Wan|first2=Yidun|date=18 February 2015|journal=Physical Review Letters|issue=7|doi=10.1103/PhysRevLett.114.076401|volume=114|page=076401|arxiv=1408.0014|issn=1079-7114|pmid=25763964}}</ref><ref name="1408.6514">{{Cite journal|title=Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy|last=Lan|first=Tian|last2=Wang|first2=Juven|date=18 February 2015|journal=Physical Review Letters|issue=7|doi=10.1103/PhysRevLett.114.076402|volume=114|page=076402|arxiv=1408.6514|issn=1079-7114|last3=Wen|first3=Xiao-Gang}}</ref> 基于这些类型系统的[[量子计算机|量子计算]]已经被提出。<ref name="quant-ph/9811052">{{Cite journal|title=Quantum codes on a lattice with boundary|url=https://archive.org/details/arxiv-quant-ph9811052|last=Bravyi|first=S. B.|last2=Kitaev|first2=A. Yu.|year=1998|arxiv=quant-ph/9811052}}</ref>在某些情况下，还可以设计一些具有全局对称性，能够丰富或扩展拓扑接口的系统。<ref name="1705.06728">{{Cite journal|title=Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions|last=Wang|first=Juven|last2=Wen|first2=Xiao-Gang|date=August 2018|journal=Physical Review X|issue=3|doi=10.1103/PhysRevX.8.031048|volume=8|page=031048|arxiv=1705.06728|issn=2160-3308|last3=Witten|first3=Edward}}</ref>

拓扑简并也存在于有受限缺陷（trapped defects，例如涡旋）的无相互作用费米子系统（例如p+ip超导体<ref>{{Cite journal|title=Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect|last=Read|first=N.|last2=Green|first2=Dmitry|date=15 April 2000|journal=Physical Review B|issue=15|doi=10.1103/physrevb.61.10267|volume=61|pages=10267–10297|arxiv=cond-mat/9906453|issn=0163-1829}}</ref>）中。在无相互作用费米子系统中，只有一种类型的拓扑简并，简并态的数目是<math>2^{N_d/2}/2</math>，其中<math>N_d</math> 是缺陷的数目（例如涡旋的数目）。这种拓扑简并也被称之为缺陷上的"马约拉纳零模"。<ref>{{Cite journal|title=Unpaired Majorana fermions in quantum wires|last=Kitaev|first=A Yu|date=1 September 2001|journal=Physics-Uspekhi|publisher=Uspekhi Fizicheskikh Nauk (UFN) Journal|issue=10S|doi=10.1070/1063-7869/44/10s/s29|volume=44|pages=131–136|arxiv=cond-mat/0010440|issn=1468-4780}}</ref><ref>{{Cite journal|title=Non-Abelian Statistics of Half-Quantum Vortices inp-Wave Superconductors|last=Ivanov|first=D. A.|date=8 January 2001|journal=Physical Review Letters|issue=2|doi=10.1103/physrevlett.86.268|volume=86|pages=268–271|arxiv=cond-mat/0005069|issn=0031-9007|pmid=11177808}}</ref>相反的，有相互作用的系统中存在多种类型的拓扑简并。<ref>{{Cite journal|title=Topological Order with a Twist: Ising Anyons from an Abelian Model|last=Bombin|first=H.|date=14 July 2010|journal=Physical Review Letters|issue=3|doi=10.1103/physrevlett.105.030403|volume=105|page=030403|arxiv=1004.1838|issn=0031-9007|pmid=20867748}}</ref><ref>{{Cite journal|title=Topological Nematic States and Non-Abelian Lattice Dislocations|last=Barkeshli|first=Maissam|last2=Qi|first2=Xiao-Liang|date=24 August 2012|journal=Physical Review X|issue=3|doi=10.1103/physrevx.2.031013|volume=2|page=031013|arxiv=1112.3311|issn=2160-3308}}</ref><ref>{{Cite journal|title=Projective non-Abelian statistics of dislocation defects in aZNrotor model|last=You|first=Yi-Zhuang|last2=Wen|first2=Xiao-Gang|date=17 October 2012|journal=Physical Review B|publisher=American Physical Society (APS)|issue=16|doi=10.1103/physrevb.86.161107|volume=86|page=161107(R)|arxiv=1204.0113|issn=1098-0121}}</ref>张量范畴（或[[么半範疇]]）理论对拓扑简并进行了系统的描述。

== 参看 ==

* 拓扑序
* 量子拓扑
* [[拓扑缺陷]]
* [[拓撲量子場論|拓扑量子场论]]
* 拓扑量子数
* [[马约拉纳费米子]]

== 参考文献 ==
{{Reflist}}
[[Category:凝聚体物理学]]