Abstract
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.
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Notes
Technically, one also needs to keep track of which particle represents the gauge charge.
We conjecture that the beyond cohomology phase is characterized by the following universal property: a pair of identical \({\mathbb {Z}}_2\) symmetry fluxes fuse to an odd number of \(E_8\) states. This is not true for the GDS dual, since it is certainly not true for a twisted Dijkgraaf Witten dual, and the two are equivalent in flat space.
It will be useful for us to consider a simplicial complex purely combinatorially. Given a set of vertices, a k-simplex is defined to be a set of \((k+1)\) vertices. A face of a simplex is a nonempty proper subset of these vertices. A simplicial complex is a collection of simplices whose faces are all contained in the collection.
The link of a simplex \(\sigma \) in a simplicial complex is a collection of simplices \(\tau \) such that \(\tau \cap \sigma = \emptyset \) but \(\tau \cup \sigma \) is a simplex of the complex. For a vertex \(w_j\) to be in the link of \(\Delta ^{d-1}\) there must exists a d-simplex \(\{w_j\} \cup \Delta ^{d-1}\). Heuristically, in a triangulation of a 3-manifold, the link of a point is the 2-sphere that surrounds the point, the link of a 1-simplex (line segment) is the circle that winds about the 1-simplex, and the link of a 2-simplex (triangle) is the set of two points that are opposite to each other with the triangle in the middle.
It is an interesting question whether a Delaunay triangulation of a PL manifold \(M\) must always be a combinatorial manifold but for this paper we require that the triangulation has this property.
All Hamiltonians that we consider have non-negative spectrum (being sums of commuting terms with non-negative eigenvalues). There is a possibility for a ground state to have non-zero energy (when the Hamiltonian is frustrated), but for the most part the ground states we consider will have zero energy. We will hence use the terms ‘ground state’ and ‘zero energy ground state’ interchangeably.
so that the Poincaré dual (Delaunay) is a simplicial complex.
Strictly following this construction, a basis state \(|{g}\rangle \) of a local degree of freedom must be mapped to an orthogonal basis state under any nonidentity symmetry action. This means that the time reversal symmetry should be represented as the global spin flip followed by complex conjugation, which corresponds to the diagonal subgroup of our \({\mathbb {Z}}_2 \times {\mathbb {Z}}_2^T\) later. However, our time reversal symmetry is just the complex conjugation. This should not cause any confusion as our explicit states are always \({\mathbb {Z}}_2 \times {\mathbb {Z}}_2^T\) symmetric.
The cochain can shown to be a cocycle by direct computation; in order for the coboundary of \(\omega \) to assume a nonzero value there should not be any triple repetition in the argument, but then any double repetition in the argument yields two terms that cancel with each other, and alternating arguments gives two nonzero terms which cancel. The chain \(\omega \) is not a coboundary since \(\sum _{\vec {x}} (\delta \lambda )(e,\vec {x}) = 0 \bmod 1\) for any homogeneous \(\lambda :G^{k+1} \rightarrow \{0, 1/2\}\).
Strictly speaking, we do not have a ring homomorphism from R to \({\mathbb {Z}}_2[x]\) that will allow us to speak of the degree in the usual sense. Nonetheless, here we use the term “degree” of a monomial to mean the number of distinct variables in it, and the degree of an element of R is the maximum monomial degree over its all terms.
This is actually a k-sphere. This fact is used, rather obviously, in the proof of Lemma 5.2.
Recall that the kth Stiefel-Whitney homology class is the Poincare dual of \(w_{d-k}\), the \((d-k)\)th Stiefel-Whitney cohomology class.
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Acknowledgements
We thank Michael Freedman for his explanation of Stiefel-Whitney classes and many other useful discussions. We thank Anton Kapustin, Dan Freed, and Arun Debray for useful discussions. N.T. acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC). L.F. acknowledges NSF DMR 1519579.
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Communicated by H. T. Yau.
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Fidkowski, L., Haah, J., Hastings, M.B. et al. Disentangling the Generalized Double Semion Model. Commun. Math. Phys. 380, 1151–1171 (2020). https://doi.org/10.1007/s00220-020-03890-2
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DOI: https://doi.org/10.1007/s00220-020-03890-2