Abstract
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} . This agrees with the classification of gapped phases with symmetry mathcal{C} . We also find that the commuting projector Hamiltonians for these state sum TQFTs have fusion category symmetries at the level of the lattice models and hence provide lattice realizations of gapped phases with fusion category symmetries. As an application, we discuss the edge modes of SPT phases based on these commuting projector Hamiltonians. Finally, we mention that we can extend the construction of topological field theories to the case of anomalous fusion category symmetries by replacing a semisimple Hopf algebra with a semisimple pseudo-unitary connected weak Hopf algebra.
Highlights
The invertible Z2 spin-flip defect.1 More generally, any diagonal RCFTs have fusion category symmetries generated by the Verlinde lines [48]
We show that the actions of fusion category symmetries C on the boundary conditions of these state sum topological quantum field theories (TQFTs) are represented by module categories over C
We find that the commuting projector Hamiltonians for these state sum TQFTs have fusion category symmetries at the level of the lattice models and provide lattice realizations of gapped phases with fusion category symmetries
Summary
We begin with a brief review of unitary fusion categories, tensor functors, and module categories [66]. These isomorphisms satisfy the following commutative diagram:. Y ∈ C and M ∈ M, we have a natural isomorphism mx,y,M : (x⊗y)⊗M → x⊗(y⊗M ) called a module associativity constraint that satisfies the following commutative diagram: mx⊗y,z,M ((x ⊗ y) ⊗ z)⊗M αx,y,z ⊗idM (x ⊗ (y ⊗ z))⊗M (x ⊗ y)⊗(z⊗M ). The tensor product of morphisms f ∈ HomKK (Y1, Y1) and g ∈ HomKK (Y2, Y2) is defined in terms of the splitting maps as f ⊗K g := πY1,Y2 ◦ (f ⊗ g) ◦ ιY1,Y2, where HomKK (Y, Y ) denotes the space of K-K bimodule maps from Y to Y.
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