Enriched string-net models and their excitations

David Green1, Peter Huston2, Kyle Kawagoe1, David Penneys1, Anup Poudel1, and Sean Sanford1

1The Ohio State University
2Vanderbilt University

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Abstract

Boundaries of Walker-Wang models have been used to construct commuting projector models which realize chiral unitary modular tensor categories (UMTCs) as boundary excitations. Given a UMTC $\mathcal{A}$ representing the Witt class of an anomaly, the article [10] gave a commuting projector model associated to an $\mathcal{A}$-enriched unitary fusion category $\mathcal{X}$ on a 2D boundary of the 3D Walker-Wang model associated to $\mathcal{A}$. That article claimed that the boundary excitations were given by the enriched center/Müger centralizer $Z^\mathcal{A}(\mathcal{X})$ of $\mathcal{A}$ in $Z(\mathcal{X})$.
In this article, we give a rigorous treatment of this 2D boundary model, and we verify this assertion using topological quantum field theory (TQFT) techniques, including skein modules and a certain semisimple algebra whose representation category describes boundary excitations. We also use TQFT techniques to show the 3D bulk point excitations of the Walker-Wang bulk are given by the Müger center $Z_2(\mathcal{A})$, and we construct bulk-to-boundary hopping operators $Z_2(\mathcal{A})\to Z^{\mathcal{A}}(\mathcal{X})$ reflecting how the UMTC of boundary excitations $Z^{\mathcal{A}}(\mathcal{X})$ is symmetric-braided enriched in $Z_2(\mathcal{A})$.
This article also includes a self-contained comprehensive review of the Levin-Wen string net model from a unitary tensor category viewpoint, as opposed to the skeletal $6j$ symbol viewpoint.

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► References

[1] F. J. Burnell, Xie Chen, Lukasz Fidkowski, and Ashvin Vishwanath. Exactly soluble model of a three-dimensional symmetry-protected topological phase of bosons with surface topological order. Phys. Rev. B, 90:245122, Dec 2014. 10.1103/​PhysRevB.90.245122 arXiv:1302.7072.
https:/​/​doi.org/​10.1103/​PhysRevB.90.245122
arXiv:1302.7072

[2] Adrien Brochier, David Jordan, Pavel Safronov, and Noah Snyder. Invertible braided tensor categories. Algebr. Geom. Topol., 21(4):2107–2140, 2021. MR4302495 10.2140/​agt.2021.21.2107 arXiv:2003.13812.
https:/​/​doi.org/​10.2140/​agt.2021.21.2107
arXiv:2003.13812
https:/​/​www.ams.org/​mathscinet-getitem?mr=MR4302495

[3] Jessica Christian, David Green, Peter Huston, and David Penneys. A lattice model for condensation in Levin-Wen systems. J. High Energy Phys., 2023(55):Paper No. 55, 55, 2023. MR4642306 10.1007/​jhep09(2023)055 arXiv:2303.04711.
https:/​/​doi.org/​10.1007/​jhep09(2023)055
arXiv:2303.04711
https:/​/​www.ams.org/​mathscinet-getitem?mr=MR4642306

[4] Thibault D. Décoppet. Rigid and separable algebras in fusion 2-categories. Adv. Math., 419:Paper No. 108967, 53, 2023. 10.1016/​j.aim.2023.108967.
https:/​/​doi.org/​10.1016/​j.aim.2023.108967

[5] Alexei Davydov, Michael Müger, Dmitri Nikshych, and Victor Ostrik. The Witt group of non-degenerate braided fusion categories. J. Reine Angew. Math., 677:135–177, 2013. 10.1515/​crelle.2012.014 MR3039775 arXiv:1009.2117.
https:/​/​doi.org/​10.1515/​crelle.2012.014
arXiv:1009.2117
https:/​/​www.ams.org/​mathscinet-getitem?mr=MR3039775

[6] Alexei Davydov, Dmitri Nikshych, and Victor Ostrik. On the structure of the Witt group of braided fusion categories. Selecta Math. (N.S.), 19(1):237–269, 2013. MR3022755 10.1007/​s00029-012-0093-3 arXiv:1109.5558.
https:/​/​doi.org/​10.1007/​s00029-012-0093-3
arXiv:1109.5558
https:/​/​www.ams.org/​mathscinet-getitem?mr=MR3022755

[7] Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik. Tensor categories, volume 205 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2015. MR3242743 10.1090/​surv/​205.
https:/​/​doi.org/​10.1090/​surv/​205
https:/​/​www.ams.org/​mathscinet-getitem?mr=MR3242743

[8] Daniel S. Freed and Constantin Teleman. Gapped Boundary Theories in Three Dimensions. Comm. Math. Phys., 388(2):845–892, 2021. MR4334249 10.1007/​s00220-021-04192-x arXiv:2006.10200.
https:/​/​doi.org/​10.1007/​s00220-021-04192-x
arXiv:2006.10200
https:/​/​www.ams.org/​mathscinet-getitem?mr=MR4334249

[9] Davide Gaiotto and Theo Johnson-Freyd. Condensations in higher categories, 2019. 10.48550/​arXiv.1905.09566.
https:/​/​doi.org/​10.48550/​arXiv.1905.09566

[10] Peter Huston, Fiona Burnell, Corey Jones, and David Penneys. Composing topological domain walls and anyon mobility. SciPost Phys., 15(3):Paper No. 076, 85, 2023. 10.21468/​scipostphys.15.3.076.
https:/​/​doi.org/​10.21468/​scipostphys.15.3.076

[11] Yuting Hu, Nathan Geer, and Yong-Shi Wu. Full dyon excitation spectrum in extended Levin-Wen models. Phys. Rev. B, 97:195154, May 2018. 10.1103/​PhysRevB.97.195154 arXiv:1502.03433.
https:/​/​doi.org/​10.1103/​PhysRevB.97.195154
arXiv:1502.03433

[12] Seung-Moon Hong. On symmetrization of 6j-symbols and Levin-Wen Hamiltonian, July 2009. 10.48550/​arXiv.0907.2204.
https:/​/​doi.org/​10.48550/​arXiv.0907.2204

[13] André Henriques and David Penneys. Bicommutant categories from fusion categories. Selecta Math. (N.S.), 23(3):1669–1708, 2017. MR3663592 10.1007/​s00029-016-0251-0 arXiv:1511.05226.
https:/​/​doi.org/​10.1007/​s00029-016-0251-0
arXiv:1511.05226
https:/​/​www.ams.org/​mathscinet-getitem?mr=MR3663592

[14] André Henriques, David Penneys, and James Tener. Categorified trace for module tensor categories over braided tensor categories. Doc. Math., 21:1089–1149, 2016. MR3578212 10.48550/​arXiv.1509.02937.
https:/​/​doi.org/​10.48550/​arXiv.1509.02937
https:/​/​www.ams.org/​mathscinet-getitem?mr=MR3578212

[15] André Henriques, David Penneys, and James Tener. Planar Algebras in Braided Tensor Categories. Mem. Amer. Math. Soc., 282(1392), 2023. MR4528312 10.1090/​memo/​1392 arXiv:1607.06041.
https:/​/​doi.org/​10.1090/​memo/​1392
arXiv:1607.06041
https:/​/​www.ams.org/​mathscinet-getitem?mr=MR4528312

[16] André Henriques, David Penneys, and James Tener. Unitary anchored planar algebras, 2023. 10.48550/​arXiv.2301.11114.
https:/​/​doi.org/​10.48550/​arXiv.2301.11114

[17] Masaki Izumi. The structure of sectors associated with Longo-Rehren inclusions. II. Examples. Rev. Math. Phys., 13(5):603–674, 2001. MR1832764 10.1142/​S0129055X01000818.
https:/​/​doi.org/​10.1142/​S0129055X01000818
https:/​/​www.ams.org/​mathscinet-getitem?mr=MR1832764

[18] Theo Johnson-Freyd. On the classification of topological orders. Comm. Math. Phys., 393(2):989–1033, 2022. MR4444089 10.1007/​s00220-022-04380-3 arXiv:2003.06663.
https:/​/​doi.org/​10.1007/​s00220-022-04380-3
arXiv:2003.06663
https:/​/​www.ams.org/​mathscinet-getitem?mr=MR4444089

[19] Theo Johnson-Freyd and David Reutter. Minimal nondegenerate extensions. J. Amer. Math. Soc., 37(1):81–150, 2024. 10.1090/​jams/​1023.
https:/​/​doi.org/​10.1090/​jams/​1023

[20] Alexander Kirillov Jr. String-net model of Turaev-Viro invariants, 2011. 10.48550/​arXiv.1106.6033.
https:/​/​doi.org/​10.48550/​arXiv.1106.6033

[21] Robert Koenig, Greg Kuperberg, and Ben W. Reichardt. Quantum computation with Turaev-Viro codes. Ann. Physics, 325(12):2707–2749, 2010. MR2726654 10.1016/​j.aop.2010.08.001 arXiv:1002.2816.
https:/​/​doi.org/​10.1016/​j.aop.2010.08.001
arXiv:1002.2816
https:/​/​www.ams.org/​mathscinet-getitem?mr=MR2726654

[22] L. Kong. Some universal properties of Levin-Wen models. In XVIIth International Congress on Mathematical Physics, pages 444–455. World Sci. Publ., Hackensack, NJ, 2014. MR3204497 10.1142/​9789814449243_0042 arXiv:1211.4644.
https:/​/​doi.org/​10.1142/​9789814449243_0042
arXiv:1211.4644
https:/​/​www.ams.org/​mathscinet-getitem?mr=MR3204497

[23] Anton Kapustin and Ryan Thorngren. Higher symmetry and gapped phases of gauge theories. In Algebra, geometry, and physics in the 21st century, volume 324 of Progr. Math., pages 177–202. Birkhäuser/​Springer, Cham, 2017. 10.1007/​978-3-319-59939-7_5 MR3702386 arXiv:1309.4721.
https:/​/​doi.org/​10.1007/​978-3-319-59939-7_
arXiv:1309.4721
https:/​/​www.ams.org/​mathscinet-getitem?mr=MR3702386

[24] Liang Kong, Xiao-Gang Wen, and Hao Zheng. Boundary-bulk relation in topological orders. Nuclear Physics B, 922:62–76, 2017. 10.1016/​j.nuclphysb.2017.06.023 arXiv:1702.00673.
https:/​/​doi.org/​10.1016/​j.nuclphysb.2017.06.023
arXiv:1702.00673

[25] Liang Kong and Hao Zheng. Drinfeld center of enriched monoidal categories. Adv. Math., 323:411–426, 2018. 10.1016/​j.aim.2017.10.038 arXiv:1704.01447.
https:/​/​doi.org/​10.1016/​j.aim.2017.10.038
arXiv:1704.01447

[26] R. B. Laughlin. Anomalous quantum hall effect: An incompressible quantum fluid with fractionally charged excitations. Phys. Rev. Lett., 50:1395–1398, May 1983. 10.1103/​PhysRevLett.50.1395.
https:/​/​doi.org/​10.1103/​PhysRevLett.50.1395

[27] Michael Levin. Protected edge modes without symmetry. Phys. Rev. X, 3:021009, May 2013. 10.1103/​PhysRevX.3.021009 arXiv:1301.7355.
https:/​/​doi.org/​10.1103/​PhysRevX.3.021009
arXiv:1301.7355

[28] Chien-Hung Lin, Michael Levin, and Fiona J. Burnell. Generalized string-net models: A thorough exposition. Phys. Rev. B, 103:195155, May 2021. 10.1103/​PhysRevB.103.195155 arXiv:2012.14424.
https:/​/​doi.org/​10.1103/​PhysRevB.103.195155
arXiv:2012.14424

[29] Michael A. Levin and Xiao-Gang Wen. String-net condensation: A physical mechanism for topological phases. Phys. Rev. B, 71:045110, Jan 2005. 10.1103/​PhysRevB.71.045110 arXiv:cond-mat/​0404617.
https:/​/​doi.org/​10.1103/​PhysRevB.71.045110
arXiv:cond-mat/0404617

[30] Michael Müger. From subfactors to categories and topology. II. The quantum double of tensor categories and subfactors. J. Pure Appl. Algebra, 180(1-2):159–219, 2003. MR1966525 10.1016/​S0022-4049(02)00248-7 arXiv:math.CT/​0111205.
https:/​/​doi.org/​10.1016/​S0022-4049(02)00248-7
arXiv:math.CT/0111205
https:/​/​www.ams.org/​mathscinet-getitem?mr=MR1966525

[31] Vincentas Mulevičius. Condensation inversion and Witt equivalence via generalised orbifolds, 2022. 10.48550/​arXiv.2206.02611.
https:/​/​doi.org/​10.48550/​arXiv.2206.02611

[32] Pieter Naaijkens. Quantum spin systems on infinite lattices, volume 933 of Lecture Notes in Physics. Springer, Cham, 2017. A concise introduction. MR3617688 10.1007/​978-3-319-51458-1.
https:/​/​doi.org/​10.1007/​978-3-319-51458-1
https:/​/​www.ams.org/​mathscinet-getitem?mr=MR3617688

[33] David Penneys. Unitary dual functors for unitary multitensor categories. High. Struct., 4(2):22–56, 2020. 10.48550/​arXiv.1808.00323 MR4133163 arXiv:1808.00323.
https:/​/​doi.org/​10.48550/​arXiv.1808.00323
arXiv:1808.00323
https:/​/​www.ams.org/​mathscinet-getitem?mr=MR4133163

[34] Alexis Virelizier. Kirby elements and quantum invariants. Proc. London Math. Soc. (3), 93(2):474–514, 2006. MR2251160 10.1112/​S0024611506015905 arXiv:math/​0312337.
https:/​/​doi.org/​10.1112/​S0024611506015905
arXiv:math/0312337
https:/​/​www.ams.org/​mathscinet-getitem?mr=MR2251160

[35] C. W. von Keyserlingk, F. J. Burnell, and S. H. Simon. Three-dimensional topological lattice models with surface anyons. Phys. Rev. B, 87:045107, Jan 2013. 10.1103/​PhysRevB.87.045107 arXiv:1208.5128.
https:/​/​doi.org/​10.1103/​PhysRevB.87.045107
arXiv:1208.5128

[36] X. G. Wen. Topological orders in rigid states. International Journal of Modern Physics B, 04(02):239–271, 1990. 10.1142/​S0217979290000139.
https:/​/​doi.org/​10.1142/​S0217979290000139

[37] Xiao-Gang Wen. Topological orders and edge excitations in fractional quantum hall states. Advances in Physics, 44(5):405–473, 1995. 10.1007/​BFb0113370 arXiv:cond-mat/​9506066.
https:/​/​doi.org/​10.1007/​BFb0113370
arXiv:cond-mat/9506066

[38] Xiao-Gang Wen. Classifying gauge anomalies through symmetry-protected trivial orders and classifying gravitational anomalies through topological orders. Phys. Rev. D, 88:045013, Aug 2013. 10.1103/​PhysRevD.88.045013 arXiv:1303.1803.
https:/​/​doi.org/​10.1103/​PhysRevD.88.045013
arXiv:1303.1803

[39] Xiao-Gang Wen. Colloquium: Zoo of quantum-topological phases of matter. Rev. Mod. Phys., 89:041004, Dec 2017. 10.1103/​RevModPhys.89.041004 arXiv:1610.03911.
https:/​/​doi.org/​10.1103/​RevModPhys.89.041004
arXiv:1610.03911

[40] X. G. Wen and Q. Niu. Ground-state degeneracy of the fractional quantum hall states in the presence of a random potential and on high-genus riemann surfaces. Phys. Rev. B, 41:9377–9396, May 1990. 10.1103/​PhysRevB.41.9377.
https:/​/​doi.org/​10.1103/​PhysRevB.41.9377

[41] Kevin Walker and Zhenghan Wang. (3+1)-tqfts and topological insulators. Frontiers of Physics, 7(2):150–159, 2012. 10.1007/​s11467-011-0194-z arXiv:1104.2632.
https:/​/​doi.org/​10.1007/​s11467-011-0194-z
arXiv:1104.2632

[42] Yanbai Zhang. From the Temperley-Lieb categories to toric code, 2017. Undergraduate honors thesis, available at https:/​/​tqft.net/​web/​research/​students/​YanbaiZhang/​thesis.pdf.
https:/​/​tqft.net/​web/​research/​students/​YanbaiZhang/​thesis.pdf

Cited by

[1] Corey Jones, Pieter Naaijkens, David Penneys, and Daniel Wallick, "Local topological order and boundary algebras", arXiv:2307.12552, (2023).

[2] Kyle Kawagoe, Corey Jones, Sean Sanford, David Green, and David Penneys, "Levin-Wen is a gauge theory: entanglement from topology", arXiv:2401.13838, (2024).

[3] Mario Tomba, Shuqi Wei, Brett Hungar, Daniel Wallick, Kyle Kawagoe, Chian Yeong Chuah, and David Penneys, "Boundary algebras of the Kitaev Quantum Double model", arXiv:2309.13440, (2023).

[4] Ying Chan, Tian Lan, and Linqian Wu, "Torus algebra and logical operators at low energy", arXiv:2403.01577, (2024).

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