Abstract
In this paper, we study gapped edges/interfaces in a 2+1 dimensional bosonic topological order and investigate how the topological entanglement entropy is sensitive to them. We present a detailed analysis of the Ishibashi states describing these edges/interfaces making use of the physics of anyon condensation in the context of Abelian Chern-Simons theory, which is then generalized to more non-Abelian theories whose edge RCFTs are known. Then we apply these results to computing the entanglement entropy of different topological orders. We consider cases where the system resides on a cylinder with gapped boundaries and that the entanglement cut is parallel to the boundary. We also consider cases where the entanglement cut coincides with the interface on a cylinder. In either cases, we find that the topological entanglement entropy is determined by the anyon condensation pattern that characterizes the interface/boundary. We note that conditions are imposed on some non-universal parameters in the edge theory to ensure existence of the conformal interface, analogous to requiring rational ratios of radii of compact bosons.
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References
A. Kitaev and L. Kong, Models for gapped boundaries and domain walls, Commun. Math. Phys. 313 (2012) 351.
M. Levin and X.-G. Wen, Detecting topological order in a ground state wave function, Phys. Rev. Lett. 96 (2006) 110405 [cond-mat/0510613] [INSPIRE].
A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96 (2006) 110404 [hep-th/0510092] [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].
Y.-M. Lu and A. Vishwanath, Theory and classification of interacting ‘integer’ topological phases in two dimensions: a Chern-Simons approach, Phys. Rev. B 86 (2012) 125119 [Erratum ibid. B 89 (2014) 199903] [arXiv:1205.3156] [INSPIRE].
K. Sakai and Y. Satoh, Entanglement through conformal interfaces, JHEP 12 (2008) 001 [arXiv:0809.4548] [INSPIRE].
L.A. Pando Zayas and N. Quiroz, Left-right entanglement entropy of boundary states, JHEP 01 (2015) 110 [arXiv:1407.7057] [INSPIRE].
D. Das and S. Datta, Universal features of left-right entanglement entropy, Phys. Rev. Lett. 115 (2015) 131602 [arXiv:1504.02475] [INSPIRE].
E.M. Brehm and I. Brunner, Entanglement entropy through conformal interfaces in the 2D Ising model, JHEP 09 (2015) 080 [arXiv:1505.02647] [INSPIRE].
E.M. Brehm, I. Brunner, D. Jaud and C. Schmidt-Colinet, Entanglement and topological interfaces, Fortsch. Phys. 64 (2016) 516 [arXiv:1512.05945] [INSPIRE].
M. Gutperle and J.D. Miller, A note on entanglement entropy for topological interfaces in RCFTs, JHEP 04 (2016) 176 [arXiv:1512.07241] [INSPIRE].
J.R. Fliss et al., Interface contributions to topological entanglement in Abelian Chern-Simons theory, JHEP 09 (2017) 056 [arXiv:1705.09611] [INSPIRE].
V.B. Petkova and J.B. Zuber, Generalized twisted partition functions, Phys. Lett. B 504 (2001) 157 [hep-th/0011021] [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 and Y.-M. Lu, Characterizing topological order by the information convex, Phys. Rev. B 99 (2019) 035112 [arXiv:1801.01519] [INSPIRE].
B. Shi, Seeing topological entanglement through the information convex, arXiv:1810.01986 [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].
L.-Y. Hung and Y. Wan, Generalized ADE classification of topological boundaries and anyon condensation, JHEP 07 (2015) 120 [arXiv:1502.02026] [INSPIRE].
A. Kapustin and N. Saulina, Topological boundary conditions in Abelian Chern-Simons theory, Nucl. Phys. B 845 (2011) 393 [arXiv:1008.0654] [INSPIRE].
A. Davydov, M. Mueger, D. Nikshych and V. Ostrik, The Witt group of non-degenerate braided fusion categories, arXiv:1009.2117.
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].
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].
M. Levin, Protected edge modes without symmetry, Phys. Rev. X 3 (2013) 021009 [arXiv:1301.7355] [INSPIRE].
J. Fuchs, Affine Lie algebras and quantum groups: an introduction, with applications in conformal field theory, Cambridge University Press, Cambridge, U.K. (1995).
P. Francesco, P. Mathieu and D. Senechal, Conformal field theory, Springer, New York, NY, U.S.A. (1997) [INSPIRE].
J. Fuchs, M.R. Gaberdiel, I. Runkel and C. Schweigert, Topological defects for the free boson CFT, J. Phys. A 40 (2007) 11403 [arXiv:0705.3129] [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].
J. Fuchs and C. Schweigert, Symmetry breaking boundaries. 1. General theory, Nucl. Phys. B 558 (1999) 419 [hep-th/9902132] [INSPIRE].
A. Yu. Kitaev, Fault tolerant quantum computation by anyons, Annals Phys. 303 (2003) 2 [quant-ph/9707021] [INSPIRE].
H. Bombin and M.A. Martin-Delgado, A family of non-Abelian Kitaev models on a lattice: topological confinement and condensation, Phys. Rev. B 78 (2008) 115421 [arXiv:0712.0190] [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.
A. Bullivant, Y. Hu and Y. Wan, Twisted quantum double model of topological order with boundaries, Phys. Rev. B 96 (2017) 165138 [arXiv:1706.03611] [INSPIRE].
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].
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, 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].
J. Wang and X.-G. Wen, Boundary degeneracy of topological order, Phys. Rev. B 91 (2015) 125124 [arXiv:1212.4863] [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].
C. Bachas and I. Brunner, Fusion of conformal interfaces, JHEP 02 (2008) 085 [arXiv:0712.0076] [INSPIRE].
R. Dijkgraaf, C. Vafa, E.P. Verlinde and H.L. Verlinde, The operator algebra of orbifold models, Commun. Math. Phys. 123 (1989) 485 [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].
Y. Hu and Y. Wan, Entanglement entropy, quantum fluctuations and thermal entropy in topological phases, arXiv:1901.09033 [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].
L.Y. Hung, J.Q. Lou and C. Shen, Ishibashi states, topological orders with boundaries and topological entanglement entropy. Part II. Cutting through the boundary, to appear.
X.-G. Wen, Topological orders and edge excitations in FQH states, Adv. Phys. 44 (1995) 405 [cond-mat/9506066] [INSPIRE].
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Lou, J., Shen, C. & Hung, LY. Ishibashi states, topological orders with boundaries and topological entanglement entropy. Part I. J. High Energ. Phys. 2019, 17 (2019). https://doi.org/10.1007/JHEP04(2019)017
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DOI: https://doi.org/10.1007/JHEP04(2019)017