Advertisement

Journal of High Energy Physics

, 2019:62 | Cite as

Spontaneous symmetry breaking from anyon condensation

  • Marcel Bischoff
  • Corey Jones
  • Yuan-Ming LuEmail author
  • David Penneys
Open Access
Regular Article - Theoretical Physics

Abstract

In a physical system undergoing a continuous quantum phase transition, spontaneous symmetry breaking occurs when certain symmetries of the Hamiltonian fail to be preserved in the ground state. In the traditional Landau theory, a symmetry group can break down to any subgroup. However, this no longer holds across a continuous phase transition driven by anyon condensation in symmetry enriched topological orders (SETOs). For a SETO described by a G-crossed braided extension \( \mathcal{C}\subseteq {\mathcal{C}}_G^{\times } \), we show that physical considerations require that a connected étale algebra A\( \mathcal{C} \) admit a G-equivariant algebra structure for symmetry to be preserved under condensation of A. Given any categorical action GEqBr(\( \mathcal{C} \)) such that g(A) ≅ A for all gG, we show there is a short exact sequence whose splittings correspond to G-equivariant algebra structures. The non-splitting of this sequence forces spontaneous symmetry breaking under condensation of A, while inequivalent splittings of the sequence correspond to different SETOs resulting from the anyon-condensation transition. Furthermore, we show that if symmetry is preserved, there is a canonically associated SETO of \( {\mathcal{C}}_A^{\mathrm{loc}} \), and gauging this symmetry commutes with anyon condensation.

Keywords

Anyons Spontaneous Symmetry Breaking Topological Field Theories Topological States of Matter 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    L.D. Landau, On the theory of phase transitions, Zh. Eksp. Teor. Fiz. 7 (1937) 19 [Phys. Z. Sowjetunion 11 (1937) 26] [Ukr. J. Phys. 53 (2008) 25] [INSPIRE].
  2. [2]
    L.D. Landau, Theory of phase transformations. II, Phys. Z. Sowjetunion 11 (1937) 545.Google Scholar
  3. [3]
    X.G. Wen, Vacuum degeneracy of chiral spin states in compactified space, Phys. Rev. B 40 (1989) 7387 [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    X.-G. Wen, Quantum field theory of many-body systems: from the origin of sound to an origin of light and electrons, Oxford University Press, New York, NY, U.S.A. (2004).Google Scholar
  5. [5]
    D.C. Tsui, H.L. Stormer and A.C. Gossard, Two-dimensional magnetotransport in the extreme quantum limit, Phys. Rev. Lett. 48 (1982) 1559 [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    F. Wilczek, Fractional statistics and anyon superconductivity, World Scientific Pub Co Inc, Singapore (1990).CrossRefzbMATHGoogle Scholar
  7. [7]
    X.-G. Wen, Quantum orders and symmetric spin liquids, Phys. Rev. B 65 (2002) 165113 [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    X. Chen, Z.-C. Gu, Z.-X. Liu and X.-G. Wen, Symmetry protected topological orders and the group cohomology of their symmetry group, Phys. Rev. B 87 (2013) 155114 [arXiv:1106.4772] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    A.M. Essin and M. Hermele, Classifying fractionalization: symmetry classification of gapped Z 2 spin liquids in two dimensions, Phys. Rev. B 87 (2013) 104406 [arXiv:1212.0593] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    A. Mesaros and Y. Ran, Classification of symmetry enriched topological phases with exactly solvable models, Phys. Rev. B 87 (2013) 155115 [arXiv:1212.0835] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    L.-Y. Hung and X.-G. Wen, Quantized topological terms in weak-coupling gauge theories with a global symmetry and their connection to symmetry-enriched topological phases, Phys. Rev. B 87 (2013) 165107 [arXiv:1212.1827] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    L.-Y. Hung and Y. Wan, Symmetry-enriched phases obtained via pseudo anyon condensation, Int. J. Mod. Phys. B 28 (2014) 1450172.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    M. Barkeshli, P. Bonderson, M. Cheng and Z. Wang, Symmetry, defects and gauging of topological phases, arXiv:1410.4540 [INSPIRE].
  14. [14]
    J.C.Y. Teo, T.L. Hughes and E. Fradkin, Theory of twist liquids: gauging an anyonic symmetry, Annals Phys. 360 (2015) 349 [arXiv:1503.06812] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Y.-M. Lu and A. Vishwanath, Classification and properties of symmetry-enriched topological phases: Chern-Simons approach with applications to Z 2 spin liquids, Phys. Rev. B 93 (2016) 155121 [arXiv:1302.2634] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    N. Tarantino, N.H. Lindner and L. Fidkowski, Symmetry fractionalization and twist defects, New J. Phys. 18 (2016) 035006 [arXiv:1506.06754].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    C.-K. Chiu, J.C. Teo, A.P. Schnyder and S. Ryu, Classification of topological quantum matter with symmetries, Rev. Mod. Phys. 88 (2016) 035005 [arXiv:1505.03535] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    X. Chen, Symmetry fractionalization in two dimensional topological phases, Rev. Phys. 2 (2017) 3 [arXiv:1606.07569] [INSPIRE].CrossRefGoogle Scholar
  19. [19]
    T. Lan, L. Kong and X.-G. Wen, Modular extensions of unitary braided fusion categories and 2 + 1D topological/SPT orders with symmetries, Commun. Math. Phys. 351 (2017) 709 [arXiv:1602.05936] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  20. [20]
    T. Lan, L. Kong and X.-G. Wen, Classification of (2 + 1)-dimensional topological order and symmetry-protected topological order for bosonic and fermionic systems with on-site symmetries, Phys. Rev. B 95 (2017) 235140 [arXiv:1602.05946] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    F.A. Bais and J.K. Slingerland, Condensate induced transitions between topologically ordered phases, Phys. Rev. B 79 (2009) 045316 [arXiv:0808.0627] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    L. Kong, Anyon condensation and tensor categories, Nucl. Phys. B 886 (2014) 436 [arXiv:1307.8244] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    F.J. Burnell, Anyon condensation and its applications, Ann. Rev. Condensed Matter Phys. 9 (2018) 307 [arXiv:1706.04940] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    P. Etingof, S. Gelaki, D. Nikshych and V. Ostrik, Tensor categories, in Math. Surv. Monogr. 205, American Mathematical Society, Providence, RI, U.S.A. (2015).Google Scholar
  25. [25]
    P. Etingof, D. Nikshych and V. Ostrik, Fusion categories and homotopy theory, Quant. Topol. 1 (2010) 209.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    Y. Qi, C.-M. Jian and C. Wang, Folding approach to topological orders enriched by mirror symmetry, arXiv:1710.09391 [INSPIRE].
  27. [27]
    R. Longo and K.-H. Rehren, Nets of subfactors, Rev. Math. Phys. 7 (1995) 567 [hep-th/9411077] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    M. Bischoff, Y. Kawahigashi and R. Longo, Characterization of 2D rational local conformal nets and its boundary conditions: the maximal case, Doc. Math. 20 (2015) 1137 [arXiv:1410.8848] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  29. [29]
    J.C. Baez and M. Shulman, Lectures on N -categories and cohomology, in Towards higher categories, IMA Vol. Math. Appl. 152, Springer, New York, NY, U.S.A. (2010), pg. 1.Google Scholar
  30. [30]
    S.X. Cui, C. Galindo, J.Y. Plavnik and Z. Wang, On gauging symmetry of modular categories, Commun. Math. Phys. 348 (2016) 1043 [arXiv:1510.03475] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    A. Kirillov Jr. and V. Ostrik, On a q-analogue of the McKay correspondence and the ADE classification of \( \mathfrak{s}{\mathfrak{l}}_2 \) conformal field theories, Adv. Math. 171 (2002) 183.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    V. Drinfeld, S. Gelaki, D. Nikshych and V. Ostrik, On braided fusion categories. I, Selecta Math. (N.S.) 16 (2010) 1.Google Scholar
  33. [33]
    A. Davydov, M. Müger, D. Nikshych and V. Ostrik, The Witt group of non-degenerate braided fusion categories, J. Reine Angew. Math. 677 (2013) 135.MathSciNetzbMATHGoogle Scholar
  34. [34]
    C. Galindo, Coherence for monoidal G-categories and braided G-crossed categories, J. Alg. 487 (2017) 118 [arXiv:1604.01679].MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    S.X. Cui, M.S. Zini and Z. Wang, On generalized symmetries and structure of modular categories, arXiv:1809.00245.
  36. [36]
    G.M. Kelly, Doctrinal adjunction, in Category seminar (proc. sem., Sydney, Australia 1972/1973), Lect. Notes Math. 420, Springer, Berlin, Germany (1974).Google Scholar
  37. [37]
    K.S. Brown, Cohomology of groups, Grad. Texts Math. 87, Springer-Verlag, New York, NY, U.S.A. and Berlin, Germany (1982).Google Scholar
  38. [38]
    E. Meir and E. Musicantov, Module categories over graded fusion categories, J. Pure Appl. Alg. 216 (2012) 2449.MathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    A. Henriques, D. Penneys and J. Tener, Categorified trace for module tensor categories over braided tensor categories, Doc. Math. 21 (2016) 1089 [arXiv:1509.02937].MathSciNetzbMATHGoogle Scholar
  40. [40]
    A. Henriques, D. Penneys and J.E. Tener, Planar algebras in braided tensor categories, arXiv:1607.06041.
  41. [41]
    P. Grossman, D. Jordan and N. Snyder, Cyclic extensions of fusion categories via the Brauer-Picard groupoid, Quant. Topol. 6 (2015) 313 [arXiv:1211.6414].MathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    N.D. Mermin, The topological theory of defects in ordered media, Rev. Mod. Phys. 51 (1979) 591 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  43. [43]
    A. Yu. Kitaev, Fault tolerant quantum computation by anyons, Annals Phys. 303 (2003) 2 [quant-ph/9707021] [INSPIRE].
  44. [44]
    S. Jiang and Y. Ran, Anyon condensation and a generic tensor-network construction for symmetry protected topological phases, Phys. Rev. B 95 (2017) 125107 [arXiv:1611.07652] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    S. Gelaki, D. Naidu and D. Nikshych, Centers of graded fusion categories, Alg. Numb. Theor. 3 (2009) 959 [arXiv:0905.3117].MathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    T. Lan, A classification of (2 + 1)D topological phases with symmetries, Ph.D. thesis, University of Waterloo, Waterloo, ON, Canada (2018) [arXiv:1801.01210] [INSPIRE].
  47. [47]
    D. Naidu, Categorical Morita equivalence for group-theoretical categories, Commun. Alg. 35 (2007) 3544.MathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    F.M. Goodman, P. de la Harpe and V.F. Jones, Coxeter graphs and towers of algebras, Math. Sci. Res. Inst. Publ. 14, Springer-Verlag, New York, NY, U.S.A. (1989).Google Scholar
  49. [49]
    J. Böckenhauer, D.E. Evans and Y. Kawahigashi, Longo-Rehren subfactors arising from α-induction, Publ. Res. Inst. Math. Sci. 37 (2001) 1.MathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    M. Izumi, The structure of sectors associated with Longo-Rehren inclusions. II: examples, Rev. Math. Phys. 13 (2001) 603 [INSPIRE].
  51. [51]
    S.-M. Hong, E. Rowell and Z. Wang, On exotic modular tensor categories, Commun. Contemp. Math. 10 (2008) 1049 [arXiv:0710.5761] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    M. Bischoff, A remark on CFT realization of quantum doubles of subfactors: case index < 4, Lett. Math. Phys. 106 (2016) 341 [arXiv:1506.02606] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    E. Ardonne, M. Cheng, E.C. Rowell and Z. Wang, Classification of metaplectic modular categories, J. Alg. 466 (2016) 141 [arXiv:1601.05460].MathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    M. Bischoff and A. Davydov, Hopf algebra actions in tensor categories, arXiv:1811.10528.
  55. [55]
    M. Bischoff, Conformal net realizability of Tambara-Yamagami categories and generalized metaplectic modular categories, arXiv:1803.04949.
  56. [56]
    R. Longo, Conformal subnets and intermediate subfactors, Commun. Math. Phys. 237 (2003) 7 [math.OA/0102196] [INSPIRE].
  57. [57]
    Y. Kawahigashi, R. Longo and M. Müger, Multiinterval subfactors and modularity of representations in conformal field theory, Commun. Math. Phys. 219 (2001) 631 [math.OA/9903104] [INSPIRE].
  58. [58]
    M. Bischoff, Generalized orbifold construction for conformal nets, Rev. Math. Phys. 29 (2016) 1750002 [arXiv:1608.00253] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    M. Bischoff, Y. Kawahigashi, R. Longo and K.-H. Rehren, Tensor categories and endomorphisms of von Neumann algebras — with applications to quantum field theory, SpringerBriefs Math. Phys. 3, Springer, Cham, Switzerland (2015).Google Scholar
  60. [60]
    M. Müger, Conformal orbifold theories and braided crossed G-categories, Commun. Math. Phys. 260 (2005) 727 [Erratum ibid. 260 (2005) 763] [INSPIRE].
  61. [61]
    F. Xu, Algebraic orbifold conformal field theories, Proc. Natl. Acad. Sci. U.S.A. 97 (2000) 14069.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  62. [62]
    D. Buchholz, S. Doplicher, R. Longo and J.E. Roberts, Extensions of automorphisms and gauge symmetries, Commun. Math. Phys. 155 (1993) 123 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Department of MathematicsOhio UniversityAthensU.S.A.
  2. 2.Department of MathematicsThe Ohio State UniversityColumbusU.S.A.
  3. 3.Department of PhysicsThe Ohio State UniversityColumbusU.S.A.

Personalised recommendations