在量子多體物理中,拓撲簡併是指有能隙的多體哈密頓量在大系統尺寸極限下的基態簡併現象,這種基態簡併不會被局域微擾破壞[1]

應用

編輯

拓撲簡併可以用於保護允許進行拓撲量子計算[1]量子比特。人們認為拓撲簡併意味著基態中存在拓撲序(或長程糾纏[2])。[3] 具有拓撲簡併的多體態可以用低能拓撲量子場論描述。

背景

編輯

拓撲簡併最早是在定義拓撲序的時候引入的。[4]在兩維空間中,拓撲簡併依賴於空間的拓撲性質,拓撲簡併在高屬黎曼面(high genus Riemann surfaces)包含了的所有量子維度上的信息也包含了准粒子的融合代數(fusion algebra)。 環面上的拓撲簡併度與准粒子類型的數目相同。

在有拓撲缺陷(例如旋渦,位錯,2D樣品的孔洞,1D樣品的末端,等等)的情況下拓撲簡併也會出現,此時拓撲簡併度與缺陷的數目相關。拓撲缺陷之間的編織(braiding)會出現拓撲保護非阿貝爾幾何相,它可用於進行拓撲保護的量子計算

拓撲序的拓撲簡併可以定義在在一個封閉空間或有邊界的開放空間或者有能隙的疇壁(domain wall)上[5],這裡拓撲序既包括阿貝爾拓撲序 [6][7]也包括阿貝爾拓撲序。[8][9] 基於這些類型系統的量子計算已經被提出。[10]在某些情況下,還可以設計一些具有全局對稱性,能夠豐富或擴展拓撲接口的系統。[11]

拓撲簡併也存在於有受限缺陷(trapped defects,例如渦旋)的無相互作用費米子系統(例如p+ip超導體[12])中。在無相互作用費米子系統中,只有一種類型的拓撲簡併,簡併態的數目是 ,其中  是缺陷的數目(例如渦旋的數目)。這種拓撲簡併也被稱之為缺陷上的"馬約拉納零模"。[13][14]相反的,有相互作用的系統中存在多種類型的拓撲簡併。[15][16][17]張量範疇(或么半範疇)理論對拓撲簡併進行了系統的描述。

參看

編輯

參考文獻

編輯
  1. ^ 1.0 1.1 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); arXiv:0707.1889頁面存檔備份,存於網際網路檔案館
  2. ^ 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)
  3. ^ Xiao-Gang Wen, Topological Orders in Rigid States.頁面存檔備份,存於網際網路檔案館Int. J. Mod. Phys. B4, 239 (1990)
  4. ^ Wen, X. G. Vacuum degeneracy of chiral spin states in compactified space. Physical Review B (American Physical Society (APS)). 1 September 1989, 40 (10): 7387–7390. ISSN 0163-1829. PMID 9991152. doi:10.1103/physrevb.40.7387. 
  5. ^ Kitaev, Alexei; Kong, Liang. Models for gapped boundaries and domain walls. Commun. Math. Phys. July 2012, 313 (2): 351–373. ISSN 1432-0916. arXiv:1104.5047 . doi:10.1007/s00220-012-1500-5. 
  6. ^ Wang, Juven; Wen, Xiao-Gang. Boundary Degeneracy of Topological Order. Physical Review B. 13 March 2015, 91 (12): 125124. ISSN 2469-9969. arXiv:1212.4863 . doi:10.1103/PhysRevB.91.125124. 
  7. ^ Kapustin, Anton. Ground-state degeneracy for abelian anyons in the presence of gapped boundaries. Physical Review B (American Physical Society (APS)). 19 March 2014, 89 (12): 125307. ISSN 2469-9969. arXiv:1306.4254 . doi:10.1103/PhysRevB.89.125307. 
  8. ^ Wan, Hung; Wan, Yidun. Ground State Degeneracy of Topological Phases on Open Surfaces. Physical Review Letters. 18 February 2015, 114 (7): 076401. ISSN 1079-7114. PMID 25763964. arXiv:1408.0014 . doi:10.1103/PhysRevLett.114.076401. 
  9. ^ Lan, Tian; Wang, Juven; Wen, Xiao-Gang. Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy. Physical Review Letters. 18 February 2015, 114 (7): 076402. ISSN 1079-7114. arXiv:1408.6514 . doi:10.1103/PhysRevLett.114.076402. 
  10. ^ Bravyi, S. B.; Kitaev, A. Yu. Quantum codes on a lattice with boundary. 1998. arXiv:quant-ph/9811052 . 
  11. ^ Wang, Juven; Wen, Xiao-Gang; Witten, Edward. Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions. Physical Review X. August 2018, 8 (3): 031048. ISSN 2160-3308. arXiv:1705.06728 . doi:10.1103/PhysRevX.8.031048. 
  12. ^ Read, N.; Green, Dmitry. Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect. Physical Review B. 15 April 2000, 61 (15): 10267–10297. ISSN 0163-1829. arXiv:cond-mat/9906453 . doi:10.1103/physrevb.61.10267. 
  13. ^ Kitaev, A Yu. Unpaired Majorana fermions in quantum wires. Physics-Uspekhi (Uspekhi Fizicheskikh Nauk (UFN) Journal). 1 September 2001, 44 (10S): 131–136. ISSN 1468-4780. arXiv:cond-mat/0010440 . doi:10.1070/1063-7869/44/10s/s29. 
  14. ^ Ivanov, D. A. Non-Abelian Statistics of Half-Quantum Vortices inp-Wave Superconductors. Physical Review Letters. 8 January 2001, 86 (2): 268–271. ISSN 0031-9007. PMID 11177808. arXiv:cond-mat/0005069 . doi:10.1103/physrevlett.86.268. 
  15. ^ Bombin, H. Topological Order with a Twist: Ising Anyons from an Abelian Model. Physical Review Letters. 14 July 2010, 105 (3): 030403. ISSN 0031-9007. PMID 20867748. arXiv:1004.1838 . doi:10.1103/physrevlett.105.030403. 
  16. ^ Barkeshli, Maissam; Qi, Xiao-Liang. Topological Nematic States and Non-Abelian Lattice Dislocations. Physical Review X. 24 August 2012, 2 (3): 031013. ISSN 2160-3308. arXiv:1112.3311 . doi:10.1103/physrevx.2.031013. 
  17. ^ You, Yi-Zhuang; Wen, Xiao-Gang. Projective non-Abelian statistics of dislocation defects in aZNrotor model. Physical Review B (American Physical Society (APS)). 17 October 2012, 86 (16): 161107(R). ISSN 1098-0121. arXiv:1204.0113 . doi:10.1103/physrevb.86.161107.