1313010.1007/13130.1029-8479Journal of High Energy PhysicsJ. High Energ. Phys.1029-8479Springer Berlin HeidelbergBerlin/HeidelbergJHEP01(2025)1062534510.1007/JHEP01(2025)106Regular Article - Theoretical PhysicsEntanglement in BF theory. Part I. Essential topological entanglementhttp://orcid.org/0000-0001-6150-7639FlissJackson R.
jf768@cam.ac.uk
1http://orcid.org/0000-0002-9641-4886VitouladitisStathis
e.vitouladitis@uva.nl
2ahttps://ror.org/013meh722grid.5335.00000 0001 2188 5934Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeWilberforce RoadCB3 0WACambridgeUnited Kingdomhttps://ror.org/04dkp9463grid.7177.60000000084992262Institute for Theoretical PhysicsUniversity of AmsterdamScience Park 9041098 XHAmsterdamNetherlandse.vitouladitis@uva.nl21120251202520251106211202523120244122024161220242112025© The Author(s) 20252025The Author(s)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.A<sc>bstract</sc>

We study the entanglement structure of Abelian topological order described by p-form BF theory in arbitrary dimensions. We do so directly in the low-energy topological quantum field theory by considering the algebra of topological surface operators. We define two appropriate notions of subregion operator algebras which are related by a form of electric-magnetic duality. To each subregion algebra we assign an entanglement entropy which we coin essential topological entanglement. This is a refinement to the traditional topological entanglement entropy. It is intrinsic to the theory, inherently finite, positive, and sensitive to more intricate topological features of the state and the entangling region. This paper is the first in a series of papers investigating entanglement and topological order in higher dimensions.

K<sc>eywords</sc>Chern-Simons TheoriesTopological Field TheoriesTopological States of MatterAnyonsMathematical SciencesPure Mathematicspublisher-imprint-nameSpringervolume-issue-count12issue-article-count124issue-toc-levels0issue-pricelist-year2025issue-copyright-holderSISSA, Trieste, Italyissue-copyright-year2021article-contains-esmNoarticle-numbering-styleContentOnlyarticle-registration-date-year2025article-registration-date-month1article-registration-date-day21article-toc-levels0toc-levels0volume-typeRegularjournal-productArchiveJournalnumbering-styleUnnumberedarticle-grants-typeOpenChoicemetadata-grantOpenAccessabstract-grantOpenAccessbodypdf-grantOpenAccessbodyhtml-grantOpenAccessbibliography-grantOpenAccessesm-grantOpenAccessonline-firstfalsepdf-file-referenceBodyRef/PDF/13130_2025_Article_25345.pdfpdf-typeTypesettarget-typeOnlinePDFissue-online-date-year2025issue-online-date-month1issue-online-date-day15issue-typeRegulararticle-typeOriginalPaperjournal-subject-primaryPhysicsjournal-subject-secondaryElementary Particles, Quantum Field Theoryjournal-subject-secondaryQuantum Field Theories, String Theoryjournal-subject-secondaryClassical and Quantum Gravitation, Relativity Theoryjournal-subject-secondaryQuantum Physicsjournal-subject-collectionPhysics and Astronomyopen-accesstrue

ArXiv ePrint: 2306.06158

Acknowledgments

We thank Sean Hartnoll, Diego Hofman, Diego Liska, Onkar Parrikar, and Ronak Soni for fun and enlightening discussions. JRF thanks Rob Leigh and Matthew Lapa for conversations inspiring this work. JRF thanks the University of Amsterdam for hospitality. SV thanks the University of Cambridge and the Kavli Instintute for Theoretical Physics at UCSB for hospitality. Research at KITP was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958. JRF is supported by STFC consolidated grant ST/T000694/1 and by Simons Foundation Award number 620869. SV is supported by the NWO Spinoza prize awarded to Erik Verlinde.

ReferencesT. Faulkner et al., Snowmass white paper: quantum information in quantum field theory and quantum gravity, in the proceedings of the Snowmass 2021, Seattle, U.S.A., July 17–26 (2022) [arXiv:2203.07117] [INSPIRE].A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett.96 (2006) 110404 [hep-th/0510092] [INSPIRE].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].BravyiSLeemhuisBTerhalBMTopological order in an exactly solvable 3D spin modelAnnals Phys.20113268392011AnPhy.326..839B277172810.1016/j.aop.2010.11.0021221.82023[arXiv:1006.4871] [INSPIRE]S. Bravyi and J. Haah, Quantum Self-Correction in the 3D Cubic Code Model, Phys. Rev. Lett.111 (2013) 200501 [arXiv:1112.3252] [INSPIRE].C. Chamon, Quantum glassiness in strongly correlated clean systems: an example of topological overprotection, Phys. Rev. Lett.94 (2005) 040402 [cond-mat/0404182] [INSPIRE].P. Ye and Z.-C. Gu, Topological quantum field theory of three-dimensional bosonic Abelian-symmetry-protected topological phases, Phys. Rev. B93 (2016) 205157 [arXiv:1508.05689] [INSPIRE].A. Tiwari, X. Chen and S. Ryu, Wilson operator algebras and ground states of coupled BF theories, Phys. Rev. B95 (2017) 245124 [arXiv:1603.08429] [INSPIRE].L. Zou and J. Haah, Spurious long-range entanglement and replica correlation length, Phys. Rev. B94 (2016) 075151 [arXiv:1604.06101] [INSPIRE].I.H. Kim et al., Universal Lower Bound on Topological Entanglement Entropy, Phys. Rev. Lett.131 (2023) 166601 [arXiv:2302.00689] [INSPIRE].BuividovichPVPolikarpovMIEntanglement entropy in gauge theories and the holographic principle for electric stringsPhys. Lett. B20086701412008PhLB..670..141B247863010.1016/j.physletb.2008.10.0321190.81090[arXiv:0806.3376] [INSPIRE]BuividovichPVPolikarpovMINumerical study of entanglement entropy in SU(2) lattice gauge theoryNucl. Phys. B20088024582008NuPhB.802..458B10.1016/j.nuclphysb.2008.04.0241190.81090[arXiv:0802.4247] [INSPIRE]P.V. Buividovich and M.I. Polikarpov, Entanglement entropy in lattice gauge theories, PoSCONFINEMENT8 (2008) 039 [arXiv:0811.3824] [INSPIRE].J. Cano, T.L. Hughes and M. Mulligan, Interactions along an Entanglement Cut in 2+1D Abelian Topological Phases, Phys. Rev. B92 (2015) 075104 [arXiv:1411.5369] [INSPIRE].H. Li and F. Haldane, Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States, Phys. Rev. Lett.101 (2008) 010504 [arXiv:0805.0332] [INSPIRE].A. Chandran, M. Hermanns, N. Regnault and B.A. Bernevig, Bulk-edge correspondence in entanglement spectra, Phys. Rev. B84 (2011) 205136.B. Swingle and T. Senthil, A geometric proof of the equality between entanglement and edge spectra, Phys. Rev. B86 (2012) 045117 [arXiv:1109.1283] [INSPIRE].X.-L. Qi, H. Katsura and A.W.W. Ludwig, General Relationship between the Entanglement Spectrum and the Edge State Spectrum of Topological Quantum States, Phys. Rev. Lett.108 (2012) 196402 [INSPIRE].W. Donnelly, Decomposition of entanglement entropy in lattice gauge theory, Phys. Rev. D85 (2012) 085004 [arXiv:1109.0036] [INSPIRE].H. Casini, M. Huerta and J.A. Rosabal, Remarks on entanglement entropy for gauge fields, Phys. Rev. D89 (2014) 085012 [arXiv:1312.1183] [INSPIRE].W. Donnelly and A.C. Wall, Entanglement entropy of electromagnetic edge modes, Phys. Rev. Lett.114 (2015) 111603 [arXiv:1412.1895] [INSPIRE].SoniRMTrivediSPAspects of Entanglement Entropy for Gauge TheoriesJHEP2016011362016JHEP...01..136S347144410.1007/JHEP01(2016)1361388.81088[arXiv:1510.07455] [INSPIRE]J. Lin and Ð. Radičević, Comments on defining entanglement entropy, Nucl. Phys. B958 (2020) 115118 [arXiv:1808.05939] [INSPIRE].RadičevićÐEntanglement in Weakly Coupled Lattice Gauge TheoriesJHEP2016041632016JHEP...04..163R1388.83140[arXiv:1509.08478] [INSPIRE]T. Grover, A.M. Turner and A. Vishwanath, Entanglement Entropy of Gapped Phases and Topological Order in Three dimensions, Phys. Rev. B84 (2011) 195120 [arXiv:1108.4038] [INSPIRE].BalasubramanianVFlissJRLeighRGParrikarOMulti-Boundary Entanglement in Chern-Simons Theory and Link InvariantsJHEP2017040612017JHEP...04..061B365770710.1007/JHEP04(2017)0611378.81061[arXiv:1611.05460] [INSPIRE]DelcampCDittrichBRielloAOn entanglement entropy in non-Abelian lattice gauge theory and 3D quantum gravityJHEP2016111022016JHEP...11..102D359476510.1007/JHEP11(2016)1021390.83061[arXiv:1609.04806] [INSPIRE]AkersCSoniRMWeiAYMultipartite edge modes and tensor networksSciPost Phys. Core2024707010.21468/SciPostPhysCore.7.4.07007974189[arXiv:2404.03651] [INSPIRE]J.R. Fliss and S. Vitouladitis, Entanglement in BF theory II: Edge-modes, arXiv:2310.18391 [INSPIRE].BergeronMSemenoffGWSzaboRJCanonical bf type topological field theory and fractional statistics of stringsNucl. Phys. B19954376951995NuPhB.437..695B132133610.1016/0550-3213(94)00503-71052.81618[hep-th/9407020] [INSPIRE]BlauMThompsonGTopological Gauge Theories of Antisymmetric Tensor FieldsAnnals Phys.19912051301991AnPhy.205..130B108667610.1016/0003-4916(91)90240-90739.53060[INSPIRE]GegenbergJKunstatterGThe partition function for topological field theoriesAnnals Phys.19942312701994AnPhy.231..270G127386110.1006/aphy.1994.10430794.57010[hep-th/9304016] [INSPIRE]BlauMKakonaMThompsonGMassive Ray-Singer torsion and path integralsJHEP2022082302022JHEP...08..230B447384410.1007/JHEP08(2022)2301522.81549[arXiv:2206.12268] [INSPIRE]K. Van Acoleyen et al., The entanglement of distillation for gauge theories, Phys. Rev. Lett.117 (2016) 131602 [arXiv:1511.04369] [INSPIRE].FlissJRInterface Contributions to Topological Entanglement in Abelian Chern-Simons TheoryJHEP2017090562017JHEP...09..056F371412810.1007/JHEP09(2017)0561382.58019[arXiv:1705.09611] [INSPIRE]FlissJRLeighRGInterfaces and the extended Hilbert space of Chern-Simons theoryJHEP2020070092020JHEP...07..009F413808710.1007/JHEP07(2020)0091451.81351[arXiv:2004.05123] [INSPIRE]D.M. Hofman and S. Vitouladitis, Generalised symmetries and state-operator correspondence for nonlocal operators, arXiv:2406.02662 [INSPIRE].JiangHCWangZBalentsLIdentifying topological order by entanglement entropyNature Phys.201289022012NatPh...8..902J10.1038/nphys24651289.60052[arXiv:1205.4289] [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. B93 (2016) 245140 [arXiv:1603.08534] [INSPIRE].J.M. Magan and D. Pontello, Quantum Complementarity through Entropic Certainty Principles, Phys. Rev. A103 (2021) 012211 [arXiv:2005.01760] [INSPIRE].C.-M. Jian, I.H. Kim and X.-L. Qi, Long-range mutual information and topological uncertainty principle, arXiv:1508.07006 [INSPIRE].MignardMSchauenburgPModular categories are not determined by their modular dataLett. Math. Phys.2021111602021LMaPh.111...60M425406810.1007/s11005-021-01395-01460.18018[arXiv:1708.02796]X. Wen and X.-G. Wen, Distinguish modular categories and 2+1D topological orders beyond modular data: mapping class group of higher genus manifold, arXiv:1908.10381 [INSPIRE].BondersonPOn invariants of Modular categories beyond modular dataJ. Pure Appl. Algebra20192234065394446510.1016/j.jpaa.2018.12.0171439.18021[arXiv:1805.05736] [INSPIRE]A. Kulkarni, M. Mignard and P. Schauenburg, A Topological Invariant for Modular Fusion Categories, arXiv:1806.03158.K. Slagle, Foliated Quantum Field Theory of Fracton Order, Phys. Rev. Lett.126 (2021) 101603 [arXiv:2008.03852] [INSPIRE].N. Tantivasadakarn, R. Verresen and A. Vishwanath, Shortest Route to Non-Abelian Topological Order on a Quantum Processor, Phys. Rev. Lett.131 (2023) 060405 [arXiv:2209.03964] [INSPIRE].M. Iqbal et al., Qutrit Toric Code and Parafermions in Trapped Ions, arXiv:2411.04185 [INSPIRE].A.Y. Kitaev, Fault tolerant quantum computation by anyons, Annals Phys.303 (2003) 2 [quant-ph/9707021] [INSPIRE].DijkgraafRWittenETopological Gauge Theories and Group CohomologyCommun. Math. Phys.19901293931990CMaPh.129..393D104869910.1007/BF020969880703.58011[INSPIRE]GaiottoDKapustinASeibergNWillettBGeneralized Global SymmetriesJHEP2015021722015JHEP...02..172G332128110.1007/JHEP02(2015)1721388.83656[arXiv:1412.5148] [INSPIRE]BhardwajLTachikawaYOn finite symmetries and their gauging in two dimensionsJHEP2018031892018JHEP...03..189B379610310.1007/JHEP03(2018)1891388.81707[arXiv:1704.02330] [INSPIRE]C. Cordova, T.T. Dumitrescu, K. Intriligator and S.-H. Shao, Snowmass White Paper: Generalized Symmetries in Quantum Field Theory and Beyond, in the proceedings of the Snowmass 2021, Seattle, U.S.A., July 17–26 (2022) [arXiv:2205.09545] [INSPIRE].D.S. Freed, G.W. Moore and C. Teleman, Topological symmetry in quantum field theory, arXiv:2209.07471 [INSPIRE].GaiottoDKulpJOrbifold groupoidsJHEP2021021322021JHEP...02..132G426028610.1007/JHEP02(2021)1321460.83094[arXiv:2008.05960] [INSPIRE]ApruzziFSymmetry TFTs from String TheoryCommun. Math. Phys.20234028952023CMaPh.402..895A461669010.1007/s00220-023-04737-21529.81100[arXiv:2112.02092] [INSPIRE]KalloshRLindeADLindeDASusskindLGravity and global symmetriesPhys. Rev. D1995529121995PhRvD..52..912K134416010.1103/PhysRevD.52.9120959.81094[hep-th/9502069] [INSPIRE]T. Banks and N. Seiberg, Symmetries and Strings in Field Theory and Gravity, Phys. Rev. D83 (2011) 084019 [arXiv:1011.5120] [INSPIRE].HarlowDOoguriHSymmetries in quantum field theory and quantum gravityCommun. Math. Phys.202138316692021CMaPh.383.1669H424426110.1007/s00220-021-04040-y1472.81208[arXiv:1810.05338] [INSPIRE]D. Harlow and E. Shaghoulian, Global symmetry, Euclidean gravity, and the black hole information problem, JHEP04 (2021) 175 [arXiv:2010.10539] [INSPIRE].HsiehC-TTachikawaYYonekuraKAnomaly Inflow and p-Form Gauge TheoriesCommun. Math. Phys.20223914952022CMaPh.391..495H439717910.1007/s00220-022-04333-w1487.81131[arXiv:2003.11550] [INSPIRE]A. Amabel, A. Debray and P.J. Haine, Differential Cohomology: Categories, Characteristic Classes, and Connections, arXiv:2109.12250 [INSPIRE].H. Casini, M. Huerta, J.M. Magán and D. Pontello, Entanglement entropy and superselection sectors. Part I. Global symmetries, JHEP02 (2020) 014 [arXiv:1905.10487] [INSPIRE].A. Hatcher, Algebraic topology, Cambridge University Press (2002).Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.