Hamiltonian and Algebraic Theories of Gapped Boundaries in Topological Phases of Matter

Abstract

We present an exactly solvable lattice Hamiltonian to realize gapped boundaries of Kitaev’s quantum double models for Dijkgraaf-Witten theories. We classify the elementary excitations on the boundary, and systematically describe the bulk-to-boundary condensation procedure. We also present the parallel algebraic/categorical structure of gapped boundaries.

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  • 16 April 2018

    There were two errors in the original publication. First, the term BK in Eq. (2.20) was not well-defined in the case of non-normal subgroups K.

References

  1. 1

    Bakalov, B., Kirillov, A.A.: Lectures on tensor categories and modular functors vol. 21. American Mathematical Society, Providence (2001)

  2. 2

    Bais F.A., Slingerland J.K.: Condensate induced transitions between topologically ordered phases. Phys. Rev. B 79, 045316 (2009)

    ADS  Article  Google Scholar 

  3. 3

    Barkeshli M., Qi X.-L.: Topological nematic states and non-abelian lattice dislocations. Phys. Rev. X 2(3), 031013 (2012)

    Google Scholar 

  4. 4

    Barkeshli M., Bonderson P., Cheng M., Wang Z.: Symmetry, defects, and gauging of topological phases (2014). Arxiv preprint arXiv:1410.4540

  5. 5

    Barkeshli M., Sau J.D.: Physical architecture for a universal topological quantum computer based on a network of Majorana nanowires (2015). Arxiv preprint arxiv:1509.07135

  6. 6

    Barkeshli M., Jian C.-M., Qi X.-L.: Twist defects and projective non-abelian braiding statistics. Phys. Rev. B 87(4), 045130 (2013)

    ADS  Article  Google Scholar 

  7. 7

    Barkeshli M., Jian C.-M., Qi X.-L.: Theory of defects in Abelian topological states. Phys. Rev. B 88(23), 235103 (2013)

    ADS  Article  Google Scholar 

  8. 8

    Barkeshli M., Jian C.-M., Qi X.-L.: Classification of topological defects in abelian topological states. Phys. Rev. B 88(24), 241103 (2013)

    ADS  Article  Google Scholar 

  9. 9

    Beigi S., Shor P.W., Whalen D.: The quantum double model with boundary: condensations and symmetries. Commun. Math. Phys. 306(3), 663–694 (2011)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  10. 10

    Bombin H., Martin-Delgado M.A.: Family of non-abelian Kitaev models on a lattice: topological condensation and confinement. Phys. Rev. B 78, 115421 (2008)

    ADS  Article  Google Scholar 

  11. 11

    Bombin H., Martin-Delgado M.A.: Nested topological order. New J. Phys. 13, 125001 (2011)

    ADS  Article  Google Scholar 

  12. 12

    Bravyi, S.B., Kitaev, A.Y.: Quantum codes on a lattice with boundary (1998). arXiv:quant-ph/9811052

  13. 13

    Chang, L.: Kitaev models based on unitary quantum groupoids. J. Math. Phys. 55(4), 041703, 20 (2014)

  14. 14

    Chang L. et al.: On enriching the Levin-Wen model with symmetry. J. Phys. A Math. Theor. 48(12), 12FT01 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15

    Cheng M.: Superconducting proximity effect on the edge of fractional topological insulators. Phys. Rev. B 86, 195126 (2012)

    ADS  Article  Google Scholar 

  16. 16

    Clarke D.J., Alicea J., Shtengel K.: Exotic non-abelian anyons from conventional fractional quantum Hall states. Nat. Commun. 4, 1348 (2013)

    ADS  Article  Google Scholar 

  17. 17

    Cong, I., Cheng, M., Wang, Z.: Topological quantum computation with gapped boundaries (2016). Arxiv preprint arXiv:1609.02037

  18. 18

    Cong, I., Cheng, M., Wang, Z.: On defects between gapped boundaries in two-dimensional topological phases of matter (2017). Arxiv preprint arXiv:1703.03564

  19. 19

    Cong, I., Cheng, M.,Wang, Z.: Universal quantum computation with gapped boundaries (In preparation)

  20. 20

    Cui S.X., Hong S.-M., Wang Z.: Universal quantum computation with weakly integral anyons. Quantum Inf. Process. 14, 26872727 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21

    Davydov A., Müger M., Nikshych D., Ostrik V.: The Witt group of non-degenerate braided fusion categories. Journal für die reine und angewandte Mathematik 677, 177 (2012). doi:10.1515/crelle.2012.014

    MATH  Google Scholar 

  22. 22

    Davydov A.: Bogomolov multiplier, double class-preserving automorphisms and modular invariants for orbifolds. J. Math. Phys. 55, 092305 (2014) arXiv:1312.7466

    ADS  MathSciNet  Article  MATH  Google Scholar 

  23. 23

    Dennis E., Kitaev A.Y., Landahl A., Preskill J.: Topological quantum memory. J. Math. Phys. 43, 4452 (2002)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  24. 24

    Drinfeld V.: Quasi-Hopf algebras. Leningrad Math. J. 1, 1419–1457 (1989)

    MathSciNet  Google Scholar 

  25. 25

    Eliëns I.S., Romers J.C., Bais F.A.: Diagrammatics for Bose condensation in anyon theories. Phys. Rev. B 90, 195130 (2014)

    ADS  Article  Google Scholar 

  26. 26

    Etingof P., Nikshych D., Ostrik V.: On fusion categories. Ann. Math. 162(2), 581–642 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  27. 27

    Fowler A.G., Mariantoni M., Martinis J.M., Cleland A.N.: Surface codes: towards practical large-scale quantum computation. Phys. Rev. A 86(3), 032324 (2012)

    ADS  Article  Google Scholar 

  28. 28

    Freedman M.H.: P/NP, and the quantum field computer. Proc. Natl. Acad. Sci. 95(1), 98–101 (1998)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  29. 29

    Freedman M., Kitaev A., Larsen M., Wang Z.: Topological quantum computation. Bull. Am. Math. Soc. 40(1), 31–38 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  30. 30

    Fröhlich J., Fuchs J., Runkel I., Schweigert C.: Correspondences of ribbon categories. Adv. Math. 199(1), 192–329 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  31. 31

    Fuchs J., Runkel I., Schweigert C.: TFT construction of RCFT correlators I: partition functions. Nucl. Phys. B 646(3), 353–497 (2002)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  32. 32

    Fuchs J., Runkel I., Schweigert C.: TFT construction of RCFT correlators IV: structure constants and correlation functions. Nucl. Phys. B 715(3), 539–638 (2005)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  33. 33

    Fuchs J., Schweigert C., Valentino A.: Bicategories for boundary conditions and for surface defects in 3-d TFT. Commun. Math. Phys. 321(2), 543–575 (2013)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  34. 34

    Fuchs J., Schweigert C., Valentino A.: A geometric approach to boundaries and surface defects in Dijkgraaf-Witten theories. Commun. Math. Phys. 332, 981 (2014)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  35. 35

    Ganeshan S., Gorshkov A.V., Gurarie V., Galitski V.M.: Exactly soluble model of boundary degeneracy. Phys. Rev. B 95, 045309 (2017)

    ADS  Article  Google Scholar 

  36. 36

    Gelaki S., Naidu D.: Some properties of group-theoretical categories. J. Algebra 322, 2631 (2007) arXiv:0709.4326

    MathSciNet  Article  MATH  Google Scholar 

  37. 37

    Hung L.Y., Wan Y.: Ground-state degeneracy of topological phases on open surfaces. Phys. Rev. Lett. 114(7), 076401 (2015)

    ADS  Article  Google Scholar 

  38. 38

    Kapustin A., Saulina N.: Topological boundary conditions in abelian Chern-Simons theory. Nucl. Phys. B 845, 393 (2011)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  39. 39

    Kapustin A.: Ground-state degeneracy for Abelian anyons in the presence of gapped boundaries. Phys. Rev. B 89, 125307 (2014)

    ADS  Article  Google Scholar 

  40. 40

    Kapustin, A., Saulina, N.: Surface operators in 3d topological field theory and 2d rational conformal field theory. In: Mathematical Foundations of Quantum Field and Perturbative String Theory. AMS (2011). arXiv:1012.0911

  41. 41

    Kirillov A., Ostrik V.: On a q-analogue of the McKay correspondence and the ADE classification of sl 2 conformal field theories. Adv. Math. 171(2), 183–227 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  42. 42

    Kitaev, A.Y.: Fault-tolerant quantum computation by anyons. Ann. Phys. 303(2), 2–30 (2003)

  43. 43

    Kitaev, A.: Bose-condensation and edges of topological quantum phases. Talk at modular categories and applications. Indiana University, 19–22 March 2009

  44. 44

    Kitaev A., Kong L.: Models for gapped boundaries and domain walls. Commun. Math. Phys. 313, 351–373 (2012) . doi:10.1007/s00220-012-1500-5

    ADS  MathSciNet  Article  MATH  Google Scholar 

  45. 45

    Kong, L.: Some universal properties of Levin-Wen models. XVIITH International Congress of Mathematical Physics, World Scientific (2014)

  46. 46

    Kong L.: Anyon condensation and tensor categories. Nucl. Phys. B 886, 436 (2014)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  47. 47

    Kong, L., Wen, X.-G., Hao, Z.: Boundary-bulk relation for topological orders as the functor mapping higher categories to their centers (2015). arXiv:1502.01690

  48. 48

    Lan T., Wang J.C., Wen X.-G.: Gapped domain walls, gapped boundaries, and topological degeneracy. Phys. Rev. Lett. 114(7), 076402 (2015)

    ADS  Article  Google Scholar 

  49. 49

    Levin M.A., Wen X.-G.: String-net condensation: a physical mechanism for topological phases. Phys. Rev. B 71, 045110 (2005)

    ADS  Article  Google Scholar 

  50. 50

    Levin M.: Protected edge modes without symmetry. Phys. Rev. X 3, 021009 (2013)

    Google Scholar 

  51. 51

    Lindner N.H., Berg E., Refael G., Stern A.: Fractionalizing Majorana fermions: non-abelian statistics on the edges of abelian quantum Hall states. Phys. Rev. X 2, 041002 (2012)

    Google Scholar 

  52. 52

    Mac Lane S.: Categories for the working mathematician, vol. 5. Springer Science and Business Media, Berlin (2013)

    Google Scholar 

  53. 53

    Moore C., Rockmore D., Russell A.: Generic quantum Fourier transforms. J. ACM Trans. Algorithms 2(4), 707–723 (2006). doi:10.1145/1198513.1198525

    MathSciNet  Article  MATH  Google Scholar 

  54. 54

    Müger M.: Galois extensions of braided tensor categories and braided crossed G-categories. J. Algebra 277, 256281 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  55. 55

    Müger, M.: Modular categories. In: Heunen, C., Sadrzadeh, M., Grefenstette, E. (eds.) Quantum Physics and Linguistics, Chapter 6. Oxford University Press, Oxford (2013)

  56. 56

    Naidu D., Rowell E.C.: A finiteness property for braided fusion categories. Algebras Represent. Theory 14, 837 (2011). doi:10.1007/s10468-010-9219-5

    MathSciNet  Article  MATH  Google Scholar 

  57. 57

    Nayak C. et al.: Non-abelian anyons and topological quantum computation. Rev. Mod. Phys. 80(3), 1083 (2008)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  58. 58

    Neupert T., He H., von Keyserlingk C., Sierra G., Bernevig B.A.: Boson condensation in topologically ordered quantum liquids. Phys. Rev. B 93, 115103 (2016)

    ADS  Article  Google Scholar 

  59. 59

    Neupert T., He H., von Keyserlingk C., Sierra G., Bernevig B.A.: No-go theorem for boson condensation in topologically ordered quantum liquids. New J. Phys. 18, 123009 (2016)

    ADS  Article  Google Scholar 

  60. 60

    Ostrik, V.: Module categories over the Drinfeld double of a finite group. Int. Math. Res. Not. 27, 1507–1520 (2003)

  61. 61

    Ostrik V.: Module categories, weak Hopf algebras, and modular invariants. Transform. Groups 8(2), 177–206 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  62. 62

    Petkova V.B., Zuber J.B.: The many faces of Ocneanu cells. Nucl. Phys. B 603(3), 449–496 (2001)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  63. 63

    Raussendorf R., Browne D.E., Briegel H.J.: Measurement-based quantum computation on cluster states. Phys. Rev. A 68, 022312 (2003)

    ADS  Article  Google Scholar 

  64. 64

    Schauenburg, P.: Hopf algebra extensions and monoidal categories. New Directions in Hopf Algebras, vol. 43. MSRI Publications (2002)

  65. 65

    Schauenburg P.: Hopf modules and the double of a quasi-Hopf algebra. Trans. Am. Math. Soc. 304(8), 3349–3378 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  66. 66

    Schauenburg P.: Computing higher Frobenius-Schur indicators in fusion categories constructed from inclusions of finite groups. Pac. J. Math. 280(1), 177–201 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  67. 67

    Varona J.: Rational values of the arccosine function. Open Math. 4(2), 319–322 (2006)

    MathSciNet  MATH  Google Scholar 

  68. 68

    Wan Yidun, Wang Chenjie: Fermion condensation and gapped domain walls in topological orders. JHEP 1703, 172 (2017)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  69. 69

    Wang, Z.: Topological Quantum Computation. No. 112. American Mathematical Soc., Providence (2010)

  70. 70

    Wen X.-G.: Vacuum degeneracy of chiral spin states in compactified space. Phys. Rev. B 40(10), 7387 (1989)

    ADS  Article  Google Scholar 

  71. 71

    Wang J.C., Wen X.-G.: Boundary degeneracy of topological order. Phys. Rev. B. 91(12), 125124 (2015)

    ADS  Article  Google Scholar 

  72. 72

    Zhu, Y.: Hecke Algebras and Representation Rings of Hopf Algebras. Studies in Advanced Mathematics, vol. 20. http://hdl.handle.net/1783.1/51261 (2001)

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Correspondence to Zhenghan Wang.

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Communicated by Y. Kawahigashi

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Cong, I., Cheng, M. & Wang, Z. Hamiltonian and Algebraic Theories of Gapped Boundaries in Topological Phases of Matter. Commun. Math. Phys. 355, 645–689 (2017). https://doi.org/10.1007/s00220-017-2960-4

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