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2021

On lattice models of gapped phases with fusion category symmetries

Inamura, Kansei

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We construct topological quantum field theories (TQFTs) and commuting projector Hamiltonians for any 1+1d gapped phases with non-anomalous fusion category symmetries, i.e.

finite symmetries that admit SPT phases.
The construction is based on two-dimensional state sum TQFT whose input datum is an $H$-simple left $H$-comodule algebra, where $H$ is a finite dimensional semisimple Hopf algebra.
We show that the actions of fusion category symmetries $\mathcal{C}$ on the boundary conditions of these state sum TQFTs are represented by module categories over $\mathcal{C}$.

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W. Ji, S.-H. Shao and X.-G. Wen, Topological transition on the conformal manifold , Phys. Rev. Research 2
Original
1909
Earlier work this paper cites.
Y.-H. Lin and S.-H. Shao, Duality Defect of the Monster CFT , J. Phys. A 54
Original
1911
Earlier work this paper cites.
R. Thorngren and Y. Wang, Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases , arXiv:1912.02817
Original
1912
Earlier work this paper cites.
W. Ji and X.-G. Wen, Categorical symmetry and noninvertible anomaly in symmetry-breaking and topological phase transitions , Phys. Rev. Research 2
Original
1912
Earlier work this paper cites.
H.A. Kramers and G.H. Wannier, Statistics of the Two-Dimensional Ferromagnet. Part I , Phys. Rev. 60
1941
Earlier work this paper cites.
C.E. Watts, Intrinsic characterizations of some additive functors , Proc. Am. Math. Soc. 11
1960
Earlier work this paper cites.
C. Vafa, Modular invariance and discrete torsion on orbifolds , Nuclear Physics B 273
1986
Earlier work this paper cites.
E. Verlinde, Fusion rules and modular transformations in 2D conformal field theory , Nuclear Physics B 300
1988
Earlier work this paper cites.
R. Dijkgraaf and E. Witten, Topological gauge theories and group cohomology , Communications in Mathematical Physics 129
1990
Earlier work this paper cites.
S. Montgomery, Hopf Algebras and Their Actions on Rings , vol. 82 of CBMS Regional Conference Series in Mathematics , American Mathematical Society, Providence, RI (1993), https://doi.org/10.1090/cbms/082
1993
Earlier work this paper cites.
M. Fukuma, S. Hosono and H. Kawai, Lattice topological field theory in two dimensions , Communications in Mathematical Physics 161
1994
Earlier work this paper cites.
H.-J. Schneider, Lectures on Hopf algebras. Notes by Sonia Natale , Córdoba: Universidad Nacional de Córdoba, Facultad de Matemática, Astronomía y Física (1995)
1995
Earlier work this paper cites.
D. Tambara and S. Yamagami, Tensor Categories with Fusion Rules of Self-Duality for Finite Abelian Groups , Journal of Algebra 209
1998
Earlier work this paper cites.
M. Yan and Y. Zhu, Stabilizer for Hopf algebra actions , Communications in Algebra 26
1998
Earlier work this paper cites.
V. Petkova and J.-B. Zuber, Generalised twisted partition functions , Physics Letters B 504
2001
Earlier work this paper cites.
S. Montgomery, Representation theory of semisimple Hopf algebras , in Algebra – representation theory. Proceedings of the NATO Advanced Study Institute, Constanta, Romania, August 2–12, 2000 , pp. 189–218, Dordrecht: Kluwer Academic Publishers (2001)
2001
Earlier work this paper cites.
J. Fuchs, I. Runkel and C. Schweigert, TFT construction of RCFT correlators I: partition functions , Nuclear Physics B 646
2002
Earlier work this paper cites.
T. Lichtman, R. Thorngren, N.H. Lindner, A. Stern and E. Berg, Bulk anyons as edge symmetries: Boundary phase diagrams of topologically ordered states , Phys. Rev. B 104
Original
2003
Earlier work this paper cites.
T. Johnson-Freyd, On the classification of topological orders , arXiv:2003.06663
Original
2003
Earlier work this paper cites.
V. Ostrik, Module categories, weak Hopf algebras and modular invariants , Transformation Groups 8
2003
Earlier work this paper cites.
D. Nikshych, V. Turaev and L. Vainerman, Invariants of knots and 3-manifolds from quantum groupoids , Topology Appl. 127
2003
Earlier work this paper cites.
J. Fröhlich, J. Fuchs, I. Runkel and C. Schweigert, Kramers-Wannier Duality from Conformal Defects , Phys. Rev. Lett. 93
2004
Earlier work this paper cites.
J. Fuchs, I. Runkel and C. Schweigert, TFT construction of RCFT correlators II: unoriented world sheets , Nuclear Physics B 678
2004
Earlier work this paper cites.
J. Fuchs, I. Runkel and C. Schweigert, TFT construction of RCFT correlators III: Simple currents , Nuclear Physics B 694
2004
Earlier work this paper cites.
P. Etingof and V. Ostrik, Finite tensor categories , Mosc. Math. J. 4
2004
Earlier work this paper cites.
D. Nikshych, Semisimple weak Hopf algebras , Journal of Algebra 275
2004
Earlier work this paper cites.
L. Kong, T. Lan, X.-G. Wen, Z.-H. Zhang and H. Zheng, Algebraic higher symmetry and categorical symmetry: A holographic and entanglement view of symmetry , Phys. Rev. Research 2
Original
2005
Earlier work this paper cites.

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