Built on

Jeongwan Haah, “Local stabilizer codes in three dimensions without string logical operators,” Phys. Rev. A 83
Original
1962
Earlier work this paper cites.
Franz J. Wegner, “Duality in generalized Ising models and phase transitions without local order parameters,” J. Math. Phys. 12
1971
Earlier work this paper cites.
D. Sullivan, “Combinatorial invariants of analytic spaces,” in Proceedings of Liverpool Singularities — Symposium , Vol. 192, edited by C. Wall (1971)
1971
Earlier work this paper cites.
Stephen Halperin and Domingo Toledo, “Stiefel-Whitney homology classes,” Annals of Mathematics 96
1972
Earlier work this paper cites.
Richard Z Goldstein and Edward C Turner, “A formula for Stiefel-Whitney homology classes,” Proceedings of the American Mathematical Society 58
1976
Earlier work this paper cites.
John B. Kogut, “An introduction to lattice gauge theory and spin systems,” Rev. Mod. Phys. 51
1979
Earlier work this paper cites.
S. C. Zhang, T. H. Hansson, and S. Kivelson, “Effective-field-theory model for the fractional quantum Hall effect,” Phys. Rev. Lett. 62
1989
Earlier work this paper cites.
Robbert Dijkgraaf and Edward Witten, “Topological gauge theories and group cohomology,” Comm. Math. Phys. 129
1990
Earlier work this paper cites.
Michael A. Levin and Xiao-Gang Wen, “String-net condensation: A physical mechanism for topological phases,” Phys. Rev. B 71
2005
Earlier work this paper cites.
S. Bravyi, M. B. Hastings, and F. Verstraete, “Lieb-Robinson bounds and the generation of correlations and topological quantum order,” Phys. Rev. Lett. 97
2006
Earlier work this paper cites.

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Alexei Kitaev, personal communication (2012)
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Then

Michael H. Freedman and Matthew B. Hastings, “Double semions in arbitrary dimension,” Commun. Math. Phys. 347
Original
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Later among the works it cites.
Daniel S. Freed and Michael J. Hopkins, “Reflection positivity and invertible topological phases,” arXiv e-prints (2016), arXiv:1604.06527
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Jeongwan Haah, “An invariant of topologically ordered states under local unitary transformations,” Communications in Mathematical Physics 342
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2016
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Davide Gaiotto and Theo Johnson-Freyd, “Symmetry Protected Topological phases and Generalized Cohomology,” arXiv e-prints (2017), arXiv:1712.07950 [hep-th]
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2019

Disentangling the Generalized Double Semion Model

Fidkowski, Lukasz, Haah, Jeongwan, Hastings, Matthew B. et al.

Understand

We analyze the class of Generalized Double Semion (GDS) models in arbitrary dimensions from the point of view of lattice Hamiltonians.

We show that on a $d$-dimensional spatial manifold $M$ the dual of the GDS is equivalent, up to constant depth local quantum circuits, to a group cohomology theory tensored with lower dimensional cohomology models that depend on the manifold $M$.
We comment on the space-time topological quantum field theory (TQFT) interpretation of this result.
We also investigate the GDS in the presence of time reversal symmetry, showing that it forms a non-trivial symmetry enriched toric code phase in odd spatial dimensions.

Built on

Jeongwan Haah, “Local stabilizer codes in three dimensions without string logical operators,” Phys. Rev. A 83
Original
1962
Earlier work this paper cites.
Franz J. Wegner, “Duality in generalized Ising models and phase transitions without local order parameters,” J. Math. Phys. 12
1971
Earlier work this paper cites.
D. Sullivan, “Combinatorial invariants of analytic spaces,” in Proceedings of Liverpool Singularities — Symposium , Vol. 192, edited by C. Wall (1971)
1971
Earlier work this paper cites.
Stephen Halperin and Domingo Toledo, “Stiefel-Whitney homology classes,” Annals of Mathematics 96
1972
Earlier work this paper cites.
Richard Z Goldstein and Edward C Turner, “A formula for Stiefel-Whitney homology classes,” Proceedings of the American Mathematical Society 58
1976
Earlier work this paper cites.
John B. Kogut, “An introduction to lattice gauge theory and spin systems,” Rev. Mod. Phys. 51
1979
Earlier work this paper cites.
S. C. Zhang, T. H. Hansson, and S. Kivelson, “Effective-field-theory model for the fractional quantum Hall effect,” Phys. Rev. Lett. 62
1989
Earlier work this paper cites.
Robbert Dijkgraaf and Edward Witten, “Topological gauge theories and group cohomology,” Comm. Math. Phys. 129
1990
Earlier work this paper cites.
Michael A. Levin and Xiao-Gang Wen, “String-net condensation: A physical mechanism for topological phases,” Phys. Rev. B 71
2005
Earlier work this paper cites.
S. Bravyi, M. B. Hastings, and F. Verstraete, “Lieb-Robinson bounds and the generation of correlations and topological quantum order,” Phys. Rev. Lett. 97
2006
Earlier work this paper cites.