On tensor network representations of the (3+1)d toric code

Clement Delcamp1,2 and Norbert Schuch1,2,3,4

1Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany
2Munich Center for Quantum Science and Technology (MCQST), Schellingstraße 4, 80799 München, Germany
3University of Vienna, Faculty of Physics, Boltzmanngasse 5, 1090 Wien, Austria
4University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

We define two dual tensor network representations of the (3+1)d toric code ground state subspace. These two representations, which are obtained by initially imposing either family of stabilizer constraints, are characterized by different virtual symmetries generated by string-like and membrane-like operators, respectively. We discuss the topological properties of the model from the point of view of these virtual symmetries, emphasizing the differences between both representations. In particular, we argue that, depending on the representation, the phase diagram of boundary entanglement degrees of freedom is naturally associated with that of a (2+1)d Hamiltonian displaying either a global or a gauge $\mathbb Z_2$-symmetry.

► BibTeX data

► References

[1] Norbert Schuch, Ignacio Cirac, and David Pérez-García. Peps as ground states: Degeneracy and topology. Annals of Physics, 325 (10): 2153 – 2192, 2010. ISSN 0003-4916. https:/​/​doi.org/​10.1016/​j.aop.2010.05.008. URL http:/​/​www.sciencedirect.com/​science/​article/​pii/​S0003491610000990.
https:/​/​doi.org/​10.1016/​j.aop.2010.05.008
http:/​/​www.sciencedirect.com/​science/​article/​pii/​S0003491610000990

[2] Norbert Schuch, Didier Poilblanc, J. Ignacio Cirac, and David Pérez-García. Topological order in the projected entangled-pair states formalism: Transfer operator and boundary hamiltonians. Phys. Rev. Lett., 111: 090501, Aug 2013. 10.1103/​PhysRevLett.111.090501. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.111.090501.
https:/​/​doi.org/​10.1103/​PhysRevLett.111.090501

[3] Oliver Buerschaper. Twisted injectivity in projected entangled pair states and the classification of quantum phases. Annals of Physics, 351: 447 – 476, 2014. ISSN 0003-4916. https:/​/​doi.org/​10.1016/​j.aop.2014.09.007. URL http:/​/​www.sciencedirect.com/​science/​article/​pii/​S000349161400267X.
https:/​/​doi.org/​10.1016/​j.aop.2014.09.007
http:/​/​www.sciencedirect.com/​science/​article/​pii/​S000349161400267X

[4] Mehmet Burak Şahinoğlu, Dominic Williamson, Nick Bultinck, Michaël Mariën, Jutho Haegeman, Norbert Schuch, and Frank Verstraete. Characterizing topological order with matrix product operators. Annales Henri Poincaré, 22 (2): 563–592, Jan 2021. ISSN 1424-0661. 10.1007/​s00023-020-00992-4. URL http:/​/​dx.doi.org/​10.1007/​s00023-020-00992-4.
https:/​/​doi.org/​10.1007/​s00023-020-00992-4

[5] N. Bultinck, M. Mariën, D.J. Williamson, M.B. Şahinoğlu, J. Haegeman, and F. Verstraete. Anyons and matrix product operator algebras. Annals of Physics, 378: 183 – 233, 2017a. ISSN 0003-4916. https:/​/​doi.org/​10.1016/​j.aop.2017.01.004. URL http:/​/​www.sciencedirect.com/​science/​article/​pii/​S0003491617300040.
https:/​/​doi.org/​10.1016/​j.aop.2017.01.004
http:/​/​www.sciencedirect.com/​science/​article/​pii/​S0003491617300040

[6] Nick Bultinck, Dominic J Williamson, Jutho Haegeman, and Frank Verstraete. Fermionic projected entangled-pair states and topological phases. Journal of Physics A: Mathematical and Theoretical, 51 (2): 025202, dec 2017b. 10.1088/​1751-8121/​aa99cc. URL https:/​/​doi.org/​10.1088.
https:/​/​doi.org/​10.1088/​1751-8121/​aa99cc

[7] Dominic J. Williamson, Nick Bultinck, and Frank Verstraete. Symmetry-enriched topological order in tensor networks: Defects, gauging and anyon condensation. 2017. URL https:/​/​arxiv.org/​abs/​1711.07982.
arXiv:1711.07982

[8] X.G. Wen. Topological Order in Rigid States. Int. J. Mod. Phys. B, 4: 239, 1990. 10.1142/​S0217979290000139.
https:/​/​doi.org/​10.1142/​S0217979290000139

[9] Xie Chen, Zheng Cheng Gu, and Xiao Gang Wen. Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order. Phys. Rev., B82: 155138, 2010a. 10.1103/​PhysRevB.82.155138.
https:/​/​doi.org/​10.1103/​PhysRevB.82.155138

[10] Alexei Kitaev. Anyons in an exactly solved model and beyond. Annals Phys., 321 (1): 2–111, 2006. 10.1016/​j.aop.2005.10.005.
https:/​/​doi.org/​10.1016/​j.aop.2005.10.005

[11] A. Yu. Kitaev. Fault tolerant quantum computation by anyons. Annals Phys., 303: 2–30, 2003. 10.1016/​S0003-4916(02)00018-0.
https:/​/​doi.org/​10.1016/​S0003-4916(02)00018-0

[12] Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. Topological quantum memory. Journal of Mathematical Physics, 43 (9): 4452–4505, 2002. 10.1063/​1.1499754. URL https:/​/​doi.org/​10.1063/​1.1499754.
https:/​/​doi.org/​10.1063/​1.1499754

[13] Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma. Non-abelian anyons and topological quantum computation. Rev. Mod. Phys., 80: 1083–1159, Sep 2008. 10.1103/​RevModPhys.80.1083. URL https:/​/​link.aps.org/​doi/​10.1103/​RevModPhys.80.1083.
https:/​/​doi.org/​10.1103/​RevModPhys.80.1083

[14] Claudio Castelnovo and Claudio Chamon. Entanglement and topological entropy of the toric code at finite temperature. Phys. Rev. B, 76: 184442, Nov 2007. 10.1103/​PhysRevB.76.184442. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.76.184442.
https:/​/​doi.org/​10.1103/​PhysRevB.76.184442

[15] Zohar Nussinov and Gerardo Ortiz. A symmetry principle for topological quantum order. Annals of Physics, 324 (5): 977 – 1057, 2009. ISSN 0003-4916. https:/​/​doi.org/​10.1016/​j.aop.2008.11.002. URL http:/​/​www.sciencedirect.com/​science/​article/​pii/​S0003491608001711.
https:/​/​doi.org/​10.1016/​j.aop.2008.11.002
http:/​/​www.sciencedirect.com/​science/​article/​pii/​S0003491608001711

[16] F.J. Burnell. Anyon condensation and its applications. Annual Review of Condensed Matter Physics, 9 (1): 307–327, 2018. 10.1146/​annurev-conmatphys-033117-054154. URL https:/​/​doi.org/​10.1146/​annurev-conmatphys-033117-054154.
https:/​/​doi.org/​10.1146/​annurev-conmatphys-033117-054154

[17] F. A. Bais and J. K. Slingerland. Condensate-induced transitions between topologically ordered phases. Phys. Rev. B, 79: 045316, Jan 2009. 10.1103/​PhysRevB.79.045316. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.79.045316.
https:/​/​doi.org/​10.1103/​PhysRevB.79.045316

[18] J. Haegeman, V. Zauner, N. Schuch, and F. Verstraete. Shadows of anyons and the entanglement structure of topological phases. Nature Communications, 6 (1), Oct 2015. ISSN 2041-1723. 10.1038/​ncomms9284. URL http:/​/​dx.doi.org/​10.1038/​ncomms9284.
https:/​/​doi.org/​10.1038/​ncomms9284

[19] Kasper Duivenvoorden, Mohsin Iqbal, Jutho Haegeman, Frank Verstraete, and Norbert Schuch. Entanglement phases as holographic duals of anyon condensates. Phys. Rev. B, 95: 235119, Jun 2017. 10.1103/​PhysRevB.95.235119. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.95.235119.
https:/​/​doi.org/​10.1103/​PhysRevB.95.235119

[20] Mohsin Iqbal, Kasper Duivenvoorden, and Norbert Schuch. Study of anyon condensation and topological phase transitions from a $\mathbb{Z}_{4}$ topological phase using the projected entangled pair states approach. Phys. Rev. B, 97: 195124, May 2018. 10.1103/​PhysRevB.97.195124. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.97.195124.
https:/​/​doi.org/​10.1103/​PhysRevB.97.195124

[21] Alexis Schotte, Jose Carrasco, Bram Vanhecke, Laurens Vanderstraeten, Jutho Haegeman, Frank Verstraete, and Julien Vidal. Tensor-network approach to phase transitions in string-net models. Phys. Rev. B, 100: 245125, Dec 2019. 10.1103/​PhysRevB.100.245125. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.100.245125.
https:/​/​doi.org/​10.1103/​PhysRevB.100.245125

[22] Xie Chen, Bei Zeng, Zheng-Cheng Gu, Isaac L. Chuang, and Xiao-Gang Wen. Tensor product representation of a topological ordered phase: Necessary symmetry conditions. Phys. Rev. B, 82: 165119, Oct 2010b. 10.1103/​PhysRevB.82.165119. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.82.165119.
https:/​/​doi.org/​10.1103/​PhysRevB.82.165119

[23] Leon Balents. Energy density of variational states. Phys. Rev. B, 90: 245116, Dec 2014. 10.1103/​PhysRevB.90.245116. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.90.245116.
https:/​/​doi.org/​10.1103/​PhysRevB.90.245116

[24] Sujeet K. Shukla, M. Burak Şahinoğlu, Frank Pollmann, and Xie Chen. Boson condensation and instability in the tensor network representation of string-net states. Phys. Rev. B, 98: 125112, Sep 2018. 10.1103/​PhysRevB.98.125112. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.98.125112.
https:/​/​doi.org/​10.1103/​PhysRevB.98.125112

[25] S. Elitzur. Impossibility of spontaneously breaking local symmetries. Phys. Rev. D, 12: 3978–3982, Dec 1975. 10.1103/​PhysRevD.12.3978. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevD.12.3978.
https:/​/​doi.org/​10.1103/​PhysRevD.12.3978

[26] Dominic J. Williamson, Clement Delcamp, Frank Verstraete, and Norbert Schuch. On the stability of topological order in tensor network states. 2020. URL https:/​/​arxiv.org/​abs/​2012.15346.
arXiv:2012.15346

[27] Michael H. Freedman and David A. Meyer. Projective plane and planar quantum codes. 1998. URL https:/​/​arxiv.org/​abs/​quant-ph/​9810055.
arXiv:quant-ph/9810055

[28] S. Yang, L. Lehman, D. Poilblanc, K. Van Acoleyen, F. Verstraete, J. I. Cirac, and N. Schuch. Edge theories in projected entangled pair state models. Phys. Rev. Lett., 112: 036402, Jan 2014. 10.1103/​PhysRevLett.112.036402. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.112.036402.
https:/​/​doi.org/​10.1103/​PhysRevLett.112.036402

[29] J. Ignacio Cirac, Didier Poilblanc, Norbert Schuch, and Frank Verstraete. Entanglement spectrum and boundary theories with projected entangled-pair states. Phys. Rev. B, 83: 245134, Jun 2011. 10.1103/​PhysRevB.83.245134. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.83.245134.
https:/​/​doi.org/​10.1103/​PhysRevB.83.245134

[30] Ling-Yan Hung and Yidun Wan. Ground-state degeneracy of topological phases on open surfaces. Phys. Rev. Lett., 114: 076401, Feb 2015. 10.1103/​PhysRevLett.114.076401. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.114.076401.
https:/​/​doi.org/​10.1103/​PhysRevLett.114.076401

[31] Wenjie Ji and Xiao-Gang Wen. Noninvertible anomalies and mapping-class-group transformation of anomalous partition functions. Phys. Rev. Research, 1: 033054, Oct 2019. 10.1103/​PhysRevResearch.1.033054. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevResearch.1.033054.
https:/​/​doi.org/​10.1103/​PhysRevResearch.1.033054

[32] Wei-Qiang Chen, Chao-Ming Jian, Liang Kong, Yi-Zhuang You, and Hao Zheng. Topological phase transition on the edge of two-dimensional z2 topological order. Physical Review B, 102 (4), Jul 2020. ISSN 2469-9969. 10.1103/​physrevb.102.045139. URL http:/​/​dx.doi.org/​10.1103/​PhysRevB.102.045139.
https:/​/​doi.org/​10.1103/​physrevb.102.045139

[33] Tsuf Lichtman, Ryan Thorngren, Netanel H. Lindner, Ady Stern, and Erez Berg. Bulk anyons as edge symmetries: Boundary phase diagrams of topologically ordered states. Physical Review B, 104 (7), Aug 2021. ISSN 2469-9969. 10.1103/​physrevb.104.075141. URL http:/​/​dx.doi.org/​10.1103/​PhysRevB.104.075141.
https:/​/​doi.org/​10.1103/​physrevb.104.075141

[34] Jacob C. Bridgeman, Stephen D. Bartlett, and Andrew C. Doherty. Tensor networks with a twist: Anyon-permuting domain walls and defects in projected entangled pair states. Phys. Rev. B, 96: 245122, Dec 2017. 10.1103/​PhysRevB.96.245122.
https:/​/​doi.org/​10.1103/​PhysRevB.96.245122

[35] Claudio Castelnovo and Claudio Chamon. Topological order in a three-dimensional toric code at finite temperature. Phys. Rev. B, 78: 155120, Oct 2008. 10.1103/​PhysRevB.78.155120. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.78.155120.
https:/​/​doi.org/​10.1103/​PhysRevB.78.155120

[36] John C. Baez, Derek K. Wise, and Alissa S. Crans. Exotic statistics for strings in 4d BF theory. Adv. Theor. Math. Phys., 11 (5): 707–749, 2007. 10.4310/​ATMP.2007.v11.n5.a1.
https:/​/​doi.org/​10.4310/​ATMP.2007.v11.n5.a1

[37] Juven Wang and Xiao-Gang Wen. Non-Abelian string and particle braiding in topological order: Modular SL(3,$\mathbb{Z}$) representation and (3+1) -dimensional twisted gauge theory. Phys. Rev., B91 (3): 035134, 2015. 10.1103/​PhysRevB.91.035134.
https:/​/​doi.org/​10.1103/​PhysRevB.91.035134

[38] Pavel Putrov, Juven Wang, and Shing-Tung Yau. Braiding Statistics and Link Invariants of Bosonic/​Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions. Annals Phys., 384: 254–287, 2017. 10.1016/​j.aop.2017.06.019.
https:/​/​doi.org/​10.1016/​j.aop.2017.06.019

[39] Chenjie Wang and Michael Levin. Braiding statistics of loop excitations in three dimensions. Phys. Rev. Lett., 113 (8): 080403, 2014. 10.1103/​PhysRevLett.113.080403.
https:/​/​doi.org/​10.1103/​PhysRevLett.113.080403

[40] AtMa P. O. Chan, Peng Ye, and Shinsei Ryu. Braiding with Borromean Rings in (3+1)-Dimensional Spacetime. Physical Review Letters, 121 (6), Aug 2018. ISSN 1079-7114. 10.1103/​physrevlett.121.061601. URL http:/​/​dx.doi.org/​10.1103/​PhysRevLett.121.061601.
https:/​/​doi.org/​10.1103/​physrevlett.121.061601

[41] Alex Bullivant, João Faria Martins, and Paul Martin. Representations of the loop braid group and aharonov–bohm like effects in discrete $(3+1)$-dimensional higher gauge theory. Advances in Theoretical and Mathematical Physics, 23 (7): 1685–1769, 2019. ISSN 1095-0753. 10.4310/​atmp.2019.v23.n7.a1. URL http:/​/​dx.doi.org/​10.4310/​ATMP.2019.v23.n7.a1.
https:/​/​doi.org/​10.4310/​atmp.2019.v23.n7.a1

[42] Zheyan Wan, Juven Wang, and Yunqin Zheng. Quantum 4d yang-mills theory and time-reversal symmetric 5d higher-gauge topological field theory. Physical Review D, 100 (8), Oct 2019. ISSN 2470-0029. 10.1103/​physrevd.100.085012. URL http:/​/​dx.doi.org/​10.1103/​PhysRevD.100.085012.
https:/​/​doi.org/​10.1103/​physrevd.100.085012

[43] Heidar Moradi and Xiao-Gang Wen. Universal topological data for gapped quantum liquids in three dimensions and fusion algebra for non-abelian string excitations. Phys. Rev. B, 91: 075114, Feb 2015. 10.1103/​PhysRevB.91.075114. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.91.075114.
https:/​/​doi.org/​10.1103/​PhysRevB.91.075114

[44] Clement Delcamp. Excitation basis for (3+1)d topological phases. JHEP, 12: 128, 2017. 10.1007/​JHEP12(2017)128.
https:/​/​doi.org/​10.1007/​JHEP12(2017)128

[45] Alex Bullivant and Clement Delcamp. Tube algebras, excitations statistics and compactification in gauge models of topological phases. Journal of High Energy Physics, 2019 (10), Oct 2019. ISSN 1029-8479. 10.1007/​jhep10(2019)216. URL http:/​/​dx.doi.org/​10.1007/​JHEP10(2019)216.
https:/​/​doi.org/​10.1007/​jhep10(2019)216

[46] Davide Gaiotto, Anton Kapustin, Nathan Seiberg, and Brian Willett. Generalized Global Symmetries. JHEP, 02: 172, 2015. 10.1007/​JHEP02(2015)172.
https:/​/​doi.org/​10.1007/​JHEP02(2015)172

[47] Wenjie Ji and Xiao-Gang Wen. Categorical symmetry and noninvertible anomaly in symmetry-breaking and topological phase transitions. Physical Review Research, 2 (3), Sep 2020. ISSN 2643-1564. 10.1103/​physrevresearch.2.033417. URL http:/​/​dx.doi.org/​10.1103/​PhysRevResearch.2.033417.
https:/​/​doi.org/​10.1103/​physrevresearch.2.033417

[48] Juven Wang, Xiao-Gang Wen, and Edward Witten. Symmetric gapped interfaces of spt and set states: Systematic constructions. Physical Review X, 8 (3), Aug 2018a. ISSN 2160-3308. 10.1103/​physrevx.8.031048. URL http:/​/​dx.doi.org/​10.1103/​PhysRevX.8.031048.
https:/​/​doi.org/​10.1103/​physrevx.8.031048

[49] Hongyu Wang, Yingcheng Li, Yuting Hu, and Yidun Wan. Gapped boundary theory of the twisted gauge theory model of three-dimensional topological orders. Journal of High Energy Physics, 2018 (10), Oct 2018b. ISSN 1029-8479. 10.1007/​jhep10(2018)114. URL http:/​/​dx.doi.org/​10.1007/​JHEP10(2018)114.
https:/​/​doi.org/​10.1007/​jhep10(2018)114

[50] Alex Bullivant and Clement Delcamp. Gapped boundaries and string-like excitations in (3+1)d gauge models of topological phases. Journal of High Energy Physics, 2021 (7), Jul 2021. ISSN 1029-8479. 10.1007/​jhep07(2021)025. URL http:/​/​dx.doi.org/​10.1007/​JHEP07(2021)025.
https:/​/​doi.org/​10.1007/​jhep07(2021)025

[51] Liang Kong, Tian Lan, Xiao-Gang Wen, Zhi-Hao Zhang, and Hao Zheng. Algebraic higher symmetry and categorical symmetry: A holographic and entanglement view of symmetry. Physical Review Research, 2 (4), Oct 2020. ISSN 2643-1564. 10.1103/​physrevresearch.2.043086. URL http:/​/​dx.doi.org/​10.1103/​PhysRevResearch.2.043086.
https:/​/​doi.org/​10.1103/​physrevresearch.2.043086

[52] Subir Sachdev. Topological order, emergent gauge fields, and fermi surface reconstruction. Reports on Progress in Physics, 82 (1): 014001, nov 2018. 10.1088/​1361-6633/​aae110. URL https:/​/​doi.org/​10.1088.
https:/​/​doi.org/​10.1088/​1361-6633/​aae110

[53] F.J. Wegner. Duality in Generalized Ising Models and Phase Transitions Without Local Order Parameters. J. Math. Phys., 12: 2259–2272, 1971a. 10.1063/​1.1665530.
https:/​/​doi.org/​10.1063/​1.1665530

[54] John B. Kogut. An introduction to lattice gauge theory and spin systems. Rev. Mod. Phys., 51: 659–713, Oct 1979. 10.1103/​RevModPhys.51.659. URL https:/​/​link.aps.org/​doi/​10.1103/​RevModPhys.51.659.
https:/​/​doi.org/​10.1103/​RevModPhys.51.659

[55] Matthew P. A. Fisher. Duality in low dimensional quantum field theories, pages 419–438. Springer Netherlands, Dordrecht, 2004. https:/​/​doi.org/​10.1007/​978-1-4020-3463-3_13.
https:/​/​doi.org/​10.1007/​978-1-4020-3463-3_13

[56] Franz J. Wegner. Duality in generalized ising models and phase transitions without local order parameters. Journal of Mathematical Physics, 12 (10): 2259–2272, 1971b. 10.1063/​1.1665530. URL https:/​/​doi.org/​10.1063/​1.1665530.
https:/​/​doi.org/​10.1063/​1.1665530

[57] Eduardo Fradkin. Field Theories of Condensed Matter Physics. Cambridge University Press, 2 edition, 2013. 10.1017/​CBO9781139015509.
https:/​/​doi.org/​10.1017/​CBO9781139015509

[58] Robert Savit. Duality in field theory and statistical systems. Rev. Mod. Phys., 52: 453–487, Apr 1980. 10.1103/​RevModPhys.52.453. URL https:/​/​link.aps.org/​doi/​10.1103/​RevModPhys.52.453.
https:/​/​doi.org/​10.1103/​RevModPhys.52.453

[59] Laurens Lootens, Jürgen Fuchs, Jutho Haegeman, Christoph Schweigert, and Frank Verstraete. Matrix product operator symmetries and intertwiners in string-nets with domain walls. SciPost Physics, 10 (3), Mar 2021. ISSN 2542-4653. 10.21468/​scipostphys.10.3.053. URL http:/​/​dx.doi.org/​10.21468/​SciPostPhys.10.3.053.
https:/​/​doi.org/​10.21468/​scipostphys.10.3.053

[60] V. G. Turaev and O. Y. Viro. State sum invariants of 3 manifolds and quantum 6j symbols. Topology, 31: 865–902, 1992. 10.1016/​0040-9383(92)90015-A.
https:/​/​doi.org/​10.1016/​0040-9383(92)90015-A

[61] John W. Barrett and Bruce W. Westbury. Invariants of piecewise linear three manifolds. Trans. Am. Math. Soc., 348: 3997–4022, 1996. 10.1090/​S0002-9947-96-01660-1.
https:/​/​doi.org/​10.1090/​S0002-9947-96-01660-1

[62] Michael A. Levin and Xiao-Gang Wen. String net condensation: A Physical mechanism for topological phases. Phys. Rev., B71: 045110, 2005. 10.1103/​PhysRevB.71.045110.
https:/​/​doi.org/​10.1103/​PhysRevB.71.045110

[63] Christopher L. Douglas and David J. Reutter. Fusion 2-categories and a state-sum invariant for 4-manifolds, 2018. URL https:/​/​arxiv.org/​abs/​1812.11933.
arXiv:1812.11933

[64] Yi-Zhuang You and Xiao-Gang Wen. Projective non-Abelian statistics of dislocation defects in a $\mathbb{Z}_{N}$ rotor model. Phys. Rev. B, 86: 161107, Oct 2012. 10.1103/​PhysRevB.86.161107. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.86.161107.
https:/​/​doi.org/​10.1103/​PhysRevB.86.161107

[65] Alexei Kitaev and Liang Kong. Models for gapped boundaries and domain walls. Communications in Mathematical Physics, 313 (2): 351–373, Jun 2012. ISSN 1432-0916. 10.1007/​s00220-012-1500-5. URL http:/​/​dx.doi.org/​10.1007/​s00220-012-1500-5.
https:/​/​doi.org/​10.1007/​s00220-012-1500-5

[66] Anton Kapustin and Nathan Seiberg. Coupling a QFT to a TQFT and Duality. JHEP, 04: 001, 2014. 10.1007/​JHEP04(2014)001.
https:/​/​doi.org/​10.1007/​JHEP04(2014)001

[67] Clement Delcamp and Apoorv Tiwari. On 2-form gauge models of topological phases. Journal of High Energy Physics, 2019 (5): 64, May 2019. ISSN 1029-8479. 10.1007/​JHEP05(2019)064. URL https:/​/​doi.org/​10.1007/​JHEP05(2019)064.
https:/​/​doi.org/​10.1007/​JHEP05(2019)064

Cited by

[1] Dominic J. Williamson, Clement Delcamp, Frank Verstraete, and Norbert Schuch, "On the stability of topological order in tensor network states", Physical Review B 104 23, 235151 (2021).

[2] Clement Delcamp, "Tensor network approach to electromagnetic duality in (3+1)d topological gauge models", arXiv:2112.08324.

The above citations are from Crossref's cited-by service (last updated successfully 2022-07-05 17:11:47) and SAO/NASA ADS (last updated successfully 2022-07-05 17:11:48). The list may be incomplete as not all publishers provide suitable and complete citation data.