Abstract
In this paper we would like to demonstrate how the known, physically-motivat-ed rules of anyon condensation proposed by Bais et al. can be recovered by the mathematics of twist-free commutative separable Frobenius algebra (CSFA). In some simple cases, those physical rules are also sufficient conditions defining a twist-free CSFA. This allows us to make use of the generalized ADE classification of CSFA’s and modular invariants to classify anyon condensation, characterize the topological boundaries between topological field theories and thus describe all gapped domain walls and gapped boundaries of a large class of topological orders. In fact, this classification is equivalent to the classification we proposed in ref. [1].
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
C. Nayak, S.H. Simon, A. Stern, M. Freedman and S. Das Sarma, Non-Abelian anyons and topological quantum computation, Rev. Mod. Phys. 80 (2008) 1083 [INSPIRE].
L. Kong and X.-G. Wen, Braided fusion categories, gravitational anomalies and the mathematical framework for topological orders in any dimensions, arXiv:1405.5858 [INSPIRE].
A. Kitaev, Anyons in an exactly solved model and beyond, Annals Phys. 321 (2006) 2 [INSPIRE].
E. Rowell, R. Stong and Z. Wang, On classification of modular tensor categories, Commun. Math. Phys. 292 (2009) 343.
J. Wang and X.-G. Wen, Boundary degeneracy of topological order, Phys. Rev. B 91 (2015) 125124 [arXiv:1212.4863] [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].
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].
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].
F.A. Bais and J.K. Slingerland, Condensate induced transitions between topologically ordered phases, Phys. Rev. B 79 (2009) 045316 [arXiv:0808.0627] [INSPIRE].
A. Kitaev and L. Kong, Models for gapped boundaries and domain walls, Commun. Math. Phys. 313 (2012) 351 [arXiv:1104.5047].
L. Kong, Anyon condensation and tensor categories, Nucl. Phys. B 886 (2014) 436 [arXiv:1307.8244] [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].
L.Y. Hung and Y. Wan, Symmetry enriched phases via pseudo anyon condensation, Int. J. Mod. Phys. B 28 (2014) 1450172 [arXiv:1308.4673] [INSPIRE].
M. Barkeshli, P. Bonderson, M. Cheng and Z. Wang, Symmetry, defects and gauging of topological phases, arXiv:1410.4540 [INSPIRE].
P. Di Francesco, M. Pierre and S. David, Conformal field theory, 1st ed., Springer, New York U.S.A. (1999).
J. Böckenhauer, D.E. Evans and Y. Kawahigashi, On α-induction, chiral generators and modular invariants for subfactors, Commun. Math. Phys. 208 (1999) 429 [math.OA/9904109] [INSPIRE].
J. Böckenhauer, D.E. Evans and Y. Kawahigashi, Chiral structure of modular invariants for subfactors, Commun. Math. Phys. 210 (2000) 733 [math.OA/9907149] [INSPIRE].
A. Kirillov Jr. and V. Ostrik, On q-analogue of the McKay correspondence and ADE classification of \( \mathrm{s}\widehat{\mathrm{l}} \) (2) conformal field theories, Adv. Math. 171 (2002) 183 [math.QA/0101219] [INSPIRE].
J. Fuchs, I. Runkel and C. Schweigert, TFT construction of RCFT correlators I: partition functions, Nucl. Phys. B 646 (2002) 353 [hep-th/0204148] [INSPIRE].
J. Fröhlich, J. Fuchs, I. Runkel and C. Schweigert, Correspondences of ribbon categories, Adv. Math. 199 (2006) 192 [math.CT/0309465] [INSPIRE].
D. Gaiotto, Domain walls for two-dimensional renormalization group flows, JHEP 12 (2012) 103 [arXiv:1201.0767] [INSPIRE].
T. Gannon, Modular data: the algebraic combinatorics of conformal field theory, J. Algebr. Comb. 22 (2005) 211 [math.QA/0103044] [INSPIRE].
J.A. Fuchs, Affine Lie algebras and quantum groups, 1st ed., Cambridge University Press, New York U.S.A. (1992).
A.N. Schellekens and S. Yankielowicz, Simple currents, modular invariants and fixed points, Int. J. Mod. Phys. A 5 (1990) 2903 [INSPIRE].
J. Fuchs, I. Runkel and C. Schweigert, TFT construction of RCFT correlators. III: Simple currents, Nucl. Phys. B 694 (2004) 277 [hep-th/0403157] [INSPIRE].
Y. Gu, L.-Y. Hung and Y. Wan, Unified framework of topological phases with symmetry, Phys. Rev. B 90 (2014) 245125 [arXiv:1402.3356] [INSPIRE].
L.-Y. Hung and Y. Wan, K matrix construction of symmetry-enriched phases of matter, Phys. Rev. B 87 (2013) 195103 [arXiv:1302.2951] [INSPIRE].
M. Levin, Protected edge modes without symmetry, Phys. Rev. X 3 (2013) 021009 [arXiv:1301.7355] [INSPIRE].
J. Fuchs, C. Schweigert and A. Valentino, A geometric approach to boundaries and surface defects in Dijkgraaf-Witten theories, Commun. Math. Phys. 332 (2014) 981 [arXiv:1307.3632] [INSPIRE].
D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized global symmetries, JHEP 02 (2015) 172 [arXiv:1412.5148] [INSPIRE].
I.S. Eliëns, J.C. Romers and F.A. Bais, Diagrammatics for Bose condensation in anyon theories, Phys. Rev. B 90 (2014) 195130 [arXiv:1310.6001] [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1502.02026
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Hung, LY., Wan, Y. Generalized ADE classification of topological boundaries and anyon condensation. J. High Energ. Phys. 2015, 120 (2015). https://doi.org/10.1007/JHEP07(2015)120
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2015)120