Abstract
Although the mathematics of anyon condensation in topological phases has been studied intensively in recent years, a proof of its physical existence is tantamount to constructing an effective Hamiltonian theory. In this paper, we concretely establish the physical foundation of anyon condensation by building the effective Hamiltonian and the Hilbert space, in which we explicitly construct the vacuum of the condensed phase as the coherent states that are the eigenstates of the creation operators creating the condensate anyons. Along with this construction, which is analogous to Laughlin’s construction of wavefunctions of fractional quantum hall states, we generalize the Goldstone theorem in the usual spontaneous symmetry breaking paradigm to the case of anyon condensation. We then prove that the condensed phase is a symmetry enriched (protected) topological phase by directly constructing the corresponding symmetry transformations, which can be considered as a generalization of the Bogoliubov transformation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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] [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:1304.7579v1] [INSPIRE].
M. Barkeshli, C.-M. Jian and X.-L. Qi, Twist defects and projective non-Abelian braiding statistics, Phys. Rev. B 87 (2013) 045130 [arXiv:1208.4834] [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].
I. Cong, M. Cheng and Z. Wang, Defects between gapped boundaries in two-dimensional topological phases of matter, Phys. Rev. B 96 (2017) 195129 [arXiv:1703.03564] [INSPIRE].
Y. Wan and C. Wang, Fermion Condensation and Gapped Domain Walls in Topological Orders, JHEP 03 (2017) 172 [arXiv:1607.01388] [INSPIRE].
J. Maciejko, X.-L. Qi, A. Karch and S.-C. Zhang, Fractional topological insulators in three dimensions, Phys. Rev. Lett. 105 (2010) 246809 [arXiv:1004.3628] [INSPIRE].
B. Swingle, M. Barkeshli, J. McGreevy and T. Senthil, Correlated Topological Insulators and the Fractional Magnetoelectric Effect, Phys. Rev. B 83 (2011) 195139 [arXiv:1005.1076] [INSPIRE].
M. Levin and A. Stern, Classification and analysis of two dimensional Abelian fractional topological insulators, Phys. Rev. B 86 (2012) 115131 [arXiv:1205.1244] [INSPIRE].
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].
L.-Y. Hung and Y. Wan, Symmetry-enriched phases obtained via pseudo anyon condensation, Int. J. Mod. Phys. B 28 (2014) 1450172.
M. Barkeshli, P. Bonderson, M. Cheng and Z. Wang, Symmetry Fractionalization, Defects, and Gauging of Topological Phases, Phys. Rev. B 100 (2019) 115147 [arXiv:1410.4540] [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].
I. Affleck, T. Kennedy, E.H. Lieb and H. Tasaki, Rigorous Results on Valence Bond Ground States in Antiferromagnets, Phys. Rev. Lett. 59 (1987) 799 [INSPIRE].
X. Chen, Z.-C. Gu and X.-G. Wen, Classification of gapped symmetric phases in one-dimensional spin systems, Phys. Rev. B 83 (2011) 035107.
L. Kong, Anyon condensation and tensor categories, Nucl. Phys. B 886 (2014) 436 [arXiv:1307.8244] [INSPIRE].
R.B. Laughlin, Anomalous quantum Hall effect: An Incompressible quantum fluid with fractionallycharged excitations, Phys. Rev. Lett. 50 (1983) 1395 [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].
P. Deligne, Catégories tannakiennes, in Grothendieck Festchrift, Modern Birkhäuser Classics, Birkhäuser, Boston MA U.S.A. (2007), pp. 111–195.
M. Müger, Galois extensions of braided tensor categories and braided crossed G-categories, J. Algebra 277 (2004) 256.
Y. Hu, Y. Wan and Y.-S. Wu, Twisted quantum double model of topological phases in two dimensions, Phys. Rev. B 87 (2013) 125114 [arXiv:1211.3695] [INSPIRE].
A. Coste, T. Gannon and P. Ruelle, Finite group modular data, Nucl. Phys. B 581 (2000) 679 [hep-th/0001158] [INSPIRE].
D. Naidu and D. Nikshych, Lagrangian Subcategories and Braided Tensor Equivalences of Twisted Quantum Doubles of Finite Groups, Commun. Math. Phys. 279 (2008) 845.
M. Müger, On the Structure of Modular Categories, Proc. Lond. Math. Soc. 87 (2003) 291.
B. Uribe, On the classification of pointed fusion categories up to weak Morita equivalence, Pacific J. Math. 290 (2017) 437.
D. Naidu, Crossed pointed categories and their equivariantizations, Pacific J. Math. 247 (2010) 477 [arXiv:1111.5246].
Y. Hu and Y. Wan, Electric-Magnetic duality in twisted quantum double model of topological orders, JHEP 11 (2020) 170 [arXiv:2007.15636] [INSPIRE].
A. Kirillov Jr. and V. Ostrik, On q analog of McKay correspondence and ADE classification of affine sl(2) conformal field theories, math/0101219v3 [INSPIRE].
S. Mac Lane, Categories for the working mathematician, Springer (1998).
J. Fuchs, I. Runkel and C. Schweigert, TFT construction of RCFT correlators 1. Partition functions, Nucl. Phys. B 646 (2002) 353 [hep-th/0204148] [INSPIRE].
P. Roche, V. Pasquier and R. Dijkgraaf, QuasiHopf algebras, group cohomology and orbifold models, Nucl. Phys. B Proc. Suppl. 18 (1990) 60 [INSPIRE].
Author information
Authors and Affiliations
Corresponding authors
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2109.06145
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
Hu, Y., Huang, Z., Hung, LY. et al. Anyon condensation: coherent states, symmetry enriched topological phases, Goldstone theorem, and dynamical rearrangement of symmetry. J. High Energ. Phys. 2022, 26 (2022). https://doi.org/10.1007/JHEP03(2022)026
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2022)026