Journal of High Energy Physics

, 2019:168 | Cite as

Ishibashi states, topological orders with boundaries and topological entanglement entropy. Part II. Cutting through the boundary

  • Ce ShenEmail author
  • Jiaqi Lou
  • Ling-Yan Hung
Open Access
Regular Article - Theoretical Physics


We compute the entanglement entropy in a 2+1 dimensional topological order in the presence of gapped boundaries. Specifically, we consider entanglement cuts that cut through the boundaries. We argue that based on general considerations of the bulk- boundary correspondence, the “twisted characters” feature in the Renyi entropy, and the topological entanglement entropy is controlled by a “half-linking number” in direct analogy to the role played by the S-modular matrix in the absence of boundaries. We also construct a class of boundary states based on the half-linking numbers that provides a “closed-string” picture complementing an “open-string” computation of the entanglement entropy. These boundary states do not correspond to diagonal RCFT’s in general. These are illustrated in specific Abelian Chern-Simons theories with appropriate boundary conditions.


Anyons Conformal Field Theory Chern-Simons Theories Topological Field Theories 


Open Access

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  1. [1]
    J.C. Wang et al., Tunneling Topological Vacua via Extended Operators: (Spin-)TQFT Spectra and Boundary Deconfinement in Various Dimensions, Prog. Theor. Exp. Phys.2018 (2018) 053A01 [arXiv:1801.05416] [INSPIRE].
  2. [2]
    B. Shi and Y.-M. Lu, Characterizing topological order by the information convex, Phys. Rev.B 99 (2019) 035112 [arXiv:1801.01519] [INSPIRE].
  3. [3]
    C. Chen, L.-Y. Hung, Y. Li and Y. Wan, Entanglement Entropy of Topological Orders with Boundaries, JHEP06 (2018) 113 [arXiv:1804.05725] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  4. [4]
    B. Shi, Seeing topological entanglement through the information convex, Phys. Rev. Research.1 (2019) 033048 [arXiv:1810.01986] [INSPIRE].Google Scholar
  5. [5]
    J. Lou, C. Shen and L.-Y. Hung, Ishibashi states, topological orders with boundaries and topological entanglement entropy. Part I, JHEP04 (2019) 017 [arXiv:1901.08238] [INSPIRE].
  6. [6]
    Y. Hu and Y. Wan, Entanglement Entropy, Quantum Fluctuations and Thermal Entropy in Topological Phases, JHEP05 (2019) 110 [arXiv:1901.09033] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    Z.-X. Luo, B.G. Pankovich, Y. Hu and Y.-S. Wu, Correspondence between bulk entanglement and boundary excitation spectra in two-dimensional gapped topological phases, Phys. Rev.B 99 (2019) 205137 [arXiv:1806.07794] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    S. Beigi, P.W. Shor and D. Whalen, The Quantum Double Model with Boundary: Condensations and Symmetries, Commun. Math. Phys.306 (2011) 663.ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    I. Cong, M. Cheng and Z. Wang, Topological Quantum Computation with Gapped Boundaries, arXiv:1609.02037.
  10. [10]
    Y. Hu, Z.-X. Luo, R. Pankovich, Y. Wan and Y.-S. Wu, Boundary Hamiltonian theory for gapped topological phases on an open surface, JHEP01 (2018) 134 [arXiv:1706.03329] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    Y. Hu, Y. Wan and Y.-S. Wu, From effective Hamiltonian to anomaly inflow in topological orders with boundaries, JHEP08 (2018) 092 [arXiv:1706.09782] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    H. Wang, Y. Li, Y. Hu and Y. Wan, Gapped Boundary Theory of the Twisted Gauge Theory Model of Three-Dimensional Topological Orders, JHEP10 (2018) 114 [arXiv:1807.11083] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    V.B. Petkova and J.B. Zuber, Generalized twisted partition functions, Phys. Lett.B 504 (2001) 157 [hep-th/0011021] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    C. Shen and L.-Y. Hung, Defect Verlinde Formula for Edge Excitations in Topological Order, Phys. Rev. Lett.123 (2019) 051602 [arXiv:1901.08285] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    S. Dong, E. Fradkin, R.G. Leigh and S. Nowling, Topological Entanglement Entropy in Chern-Simons Theories and Quantum Hall Fluids, JHEP05 (2008) 016 [arXiv:0802.3231] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    J.R. Fliss et al., Interface Contributions to Topological Entanglement in Abelian Chern-Simons Theory, JHEP09 (2017) 056 [arXiv:1705.09611] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    X. Wen, S. Matsuura and S. Ryu, Edge theory approach to topological entanglement entropy, mutual information and entanglement negativity in Chern-Simons theories, Phys. Rev.B 93 (2016) 245140 [arXiv:1603.08534] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    M. Barkeshli, C.-M. Jian and X.-L. Qi, Theory of defects in Abelian topological states, Phys. Rev.B 88 (2013) 235103 [arXiv:1305.7203] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    T. Lan, J.C. Wang and X.-G. Wen, Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy, Phys. Rev. Lett.114 (2015) 076402 [arXiv:1408.6514] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    R. Blumenhagen and E. Plauschinn, Introduction to conformal field theory, Lect. Notes Phys.779 (2009) 1 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    A. Kapustin and N. Saulina, Topological boundary conditions in abelian Chern-Simons theory, Nucl. Phys.B 845 (2011) 393 [arXiv:1008.0654] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    M. Levin, Protected edge modes without symmetry, Phys. Rev.X 3 (2013) 021009 [arXiv:1301.7355] [INSPIRE].CrossRefGoogle Scholar
  23. [23]
    J.C. Wang and X.-G. Wen, Boundary Degeneracy of Topological Order, Phys. Rev.B 91 (2015) 125124 [arXiv:1212.4863] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    X.-G. Wen, Topological orders and edge excitations in FQH states, Adv. Phys.44 (1995) 405 [cond-mat/9506066] [INSPIRE].
  25. [25]
    F.A. Bais, B.J. Schroers and J.K. Slingerland, Hopf symmetry breaking and confinement in (2 + 1)-dimensional gauge theory, JHEP05 (2003) 068 [hep-th/0205114] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    F.A. Bais, B.J. Schroers and J.K. Slingerland, Broken quantum symmetry and confinement phases in planar physics, Phys. Rev. Lett.89 (2002) 181601 [hep-th/0205117] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    F.A. Bais and J.K. Slingerland, Condensate induced transitions between topologically ordered phases, Phys. Rev.B 79 (2009) 045316 [arXiv:0808.0627] [INSPIRE].
  28. [28]
    F.A. Bais, J.K. Slingerland and S.M. Haaker, A Theory of topological edges and domain walls, Phys. Rev. Lett.102 (2009) 220403 [arXiv:0812.4596] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    M. Barkeshli, C.-M. Jian and X.-L. Qi, Classification of Topological Defects in Abelian Topological States, Phys. Rev.B 88 (2013) 241103 [arXiv:1304.7579] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    L. Kong, Anyon condensation and tensor categories, Nucl. Phys.B 886 (2014) 436 [arXiv:1307.8244] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  31. [31]
    J. Fuchs, C. Schweigert and A. Valentino, Bicategories for boundary conditions and for surface defects in 3D TFT, Commun. Math. Phys.321 (2013) 543 [arXiv:1203.4568] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    L.-Y. Hung and Y. Wan, Ground State Degeneracy of Topological Phases on Open Surfaces, Phys. Rev. Lett.114 (2015) 076401 [arXiv:1408.0014] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    L.-Y. Hung and Y. Wan, Generalized ADE classification of topological boundaries and anyon condensation, JHEP07 (2015) 120 [arXiv:1502.02026] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    J.L. Cardy, Boundary Conditions, Fusion Rules and the Verlinde Formula, Nucl. Phys.B 324 (1989) 581 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    J.L. Cardy, Boundary conformal field theory, hep-th/0411189 [INSPIRE].
  36. [36]
    J.P. Serre, Cours d’arithmetique, Presses Universitaires De France (1994).Google Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.State Key Laboratory of Surface PhysicsFudan UniversityShanghaiChina
  2. 2.Collaborative Innovation Center of Advanced MicrostructuresNanjingChina
  3. 3.Department of Physics and Center for Field Theory and Particle PhysicsFudan UniversityShanghaiChina
  4. 4.Institute for Nanoelectronic Devices and Quantum computingFudan UniversityShanghaiChina

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