On lattice models of gapped phases with fusion category symmetries

TL;DR

The paper develops a lattice-based program for realizing 1+1d gapped phases with fusion-category symmetries by first classifying non-anomalous symmetries as Rep(H), then constructing a canonical ${}_K ext{M}_K$ state-sum TQFT and pulling it back to Rep(H) via a tensor functor. Indecomposable semisimple Rep(H)-module categories correspond to left K-modules for H-simple left H-comodule algebras K, enabling explicit TQFT and commuting-projector Hamiltonian realizations with Rep(H) actions on both bulk and boundary. Ground states on a circle match the TQFT vacua, while edge modes on an interval realize the expected M and M^* modules, reproducing known SPT edge anomalies in the finite-group case as H^2(G,U(1)). The framework extends to anomalous fusion category symmetries by replacing Hopf algebras with semisimple pseudo-unitary connected weak Hopf algebras, outlining a path to a complete lattice realization of gapped phases with general fusion-category symmetries.

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.

Figures (3)

• Figure 1: The Hilbert space on the above spatial circle is given by $Z((x \otimes y) \otimes z)$, where the base point is represented by the cross mark in the above figure. We can also assign a Hilbert space to a circle with an arbitrary number of topological defects in a similar way.
• Figure 2: The building blocks of linear maps: a cylinder amplitude $Z(f)$, a change of the case point $X_{x, y}$, a unit $\eta$, multiplication $M_{x, y}$, a counit $\epsilon$, and comultiplication $\Delta_{x, y}$ (from left to right). Each diagram represents a linear map from the Hilbert space assigned to the bottom to the Hilbert space assigned to the top.
• Figure 3: The boundary state \ref{['eq: MPS']} is represented by the above string diagram where the circle labeled by $M$ corresponds to taking the trace on $M$. This string diagram is invariant under the composition of the cylinder amplitude $Z_T(S^1 \times [0, 1])$ at the bottom.