Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
B. Shi and Y.-M. Lu, Characterizing topological order by the information convex, Phys. Rev. B 99 (2019) 035112 [arXiv:1801.01519] [INSPIRE].
C. Chen, L.-Y. Hung, Y. Li and Y. Wan, Entanglement Entropy of Topological Orders with Boundaries, JHEP 06 (2018) 113 [arXiv:1804.05725] [INSPIRE].
B. Shi, Seeing topological entanglement through the information convex, Phys. Rev. Research. 1 (2019) 033048 [arXiv:1810.01986] [INSPIRE].
J. Lou, C. Shen and L.-Y. Hung, Ishibashi states, topological orders with boundaries and topological entanglement entropy. Part I, JHEP 04 (2019) 017 [arXiv:1901.08238] [INSPIRE].
Y. Hu and Y. Wan, Entanglement Entropy, Quantum Fluctuations and Thermal Entropy in Topological Phases, JHEP 05 (2019) 110 [arXiv:1901.09033] [INSPIRE].
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].
S. Beigi, P.W. Shor and D. Whalen, The Quantum Double Model with Boundary: Condensations and Symmetries, Commun. Math. Phys. 306 (2011) 663.
I. Cong, M. Cheng and Z. Wang, Topological Quantum Computation with Gapped Boundaries, arXiv:1609.02037.
Y. Hu, Z.-X. Luo, R. Pankovich, Y. Wan and Y.-S. Wu, Boundary Hamiltonian theory for gapped topological phases on an open surface, JHEP 01 (2018) 134 [arXiv:1706.03329] [INSPIRE].
Y. Hu, Y. Wan and Y.-S. Wu, From effective Hamiltonian to anomaly inflow in topological orders with boundaries, JHEP 08 (2018) 092 [arXiv:1706.09782] [INSPIRE].
H. Wang, Y. Li, Y. Hu and Y. Wan, Gapped Boundary Theory of the Twisted Gauge Theory Model of Three-Dimensional Topological Orders, JHEP 10 (2018) 114 [arXiv:1807.11083] [INSPIRE].
V.B. Petkova and J.B. Zuber, Generalized twisted partition functions, Phys. Lett. B 504 (2001) 157 [hep-th/0011021] [INSPIRE].
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].
S. Dong, E. Fradkin, R.G. Leigh and S. Nowling, Topological Entanglement Entropy in Chern-Simons Theories and Quantum Hall Fluids, JHEP 05 (2008) 016 [arXiv:0802.3231] [INSPIRE].
J.R. Fliss et al., Interface Contributions to Topological Entanglement in Abelian Chern-Simons Theory, JHEP 09 (2017) 056 [arXiv:1705.09611] [INSPIRE].
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].
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].
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].
R. Blumenhagen and E. Plauschinn, Introduction to conformal field theory, Lect. Notes Phys. 779 (2009) 1 [INSPIRE].
A. Kapustin and N. Saulina, Topological boundary conditions in abelian Chern-Simons theory, Nucl. Phys. B 845 (2011) 393 [arXiv:1008.0654] [INSPIRE].
M. Levin, Protected edge modes without symmetry, Phys. Rev. X 3 (2013) 021009 [arXiv:1301.7355] [INSPIRE].
J.C. Wang and X.-G. Wen, Boundary Degeneracy of Topological Order, Phys. Rev. B 91 (2015) 125124 [arXiv:1212.4863] [INSPIRE].
X.-G. Wen, Topological orders and edge excitations in FQH states, Adv. Phys. 44 (1995) 405 [cond-mat/9506066] [INSPIRE].
F.A. Bais, B.J. Schroers and J.K. Slingerland, Hopf symmetry breaking and confinement in (2 + 1)-dimensional gauge theory, JHEP 05 (2003) 068 [hep-th/0205114] [INSPIRE].
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].
F.A. Bais and J.K. Slingerland, Condensate induced transitions between topologically ordered phases, Phys. Rev. B 79 (2009) 045316 [arXiv:0808.0627] [INSPIRE].
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].
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].
L. Kong, Anyon condensation and tensor categories, Nucl. Phys. B 886 (2014) 436 [arXiv:1307.8244] [INSPIRE].
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].
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].
L.-Y. Hung and Y. Wan, Generalized ADE classification of topological boundaries and anyon condensation, JHEP 07 (2015) 120 [arXiv:1502.02026] [INSPIRE].
J.L. Cardy, Boundary Conditions, Fusion Rules and the Verlinde Formula, Nucl. Phys. B 324 (1989) 581 [INSPIRE].
J.L. Cardy, Boundary conformal field theory, hep-th/0411189 [INSPIRE].
J.P. Serre, Cours d’arithmetique, Presses Universitaires De France (1994).
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1908.07700
Rights and permissions
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.
About this article
Cite this article
Shen, C., Lou, J. & Hung, LY. Ishibashi states, topological orders with boundaries and topological entanglement entropy. Part II. Cutting through the boundary. J. High Energ. Phys. 2019, 168 (2019). https://doi.org/10.1007/JHEP11(2019)168
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2019)168