[Paper Review] On lattice models of gapped phases with fusion category symmetries

Kansei Inamura|arXiv (Cornell University)|Oct 25, 2021

Topological Materials and Phenomena4 citations

TL;DR

This paper constructs topological quantum field theories (TQFTs) and commuting projector Hamiltonians for 1+1d gapped phases with non-anomalous fusion category symmetries using state sum TQFTs based on H-simple left H-comodule algebras, where H is a finite-dimensional semisimple Hopf algebra. The key result is that the actions of fusion category symmetries on boundary conditions are realized via module categories over the fusion category, providing a lattice realization of gapped phases and enabling the study of edge modes in SPT phases.

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.

Motivation & Objective

• To provide a systematic construction of topological quantum field theories (TQFTs) and commuting projector Hamiltonians for 1+1d gapped phases with non-anomalous fusion category symmetries.
• To establish a lattice realization of gapped phases with fusion category symmetries, filling a gap in the existing literature on lattice models for such phases.
• To demonstrate that the actions of fusion category symmetries on boundary conditions are represented by module categories over the fusion category, aligning with the classification of gapped phases with such symmetries.
• To analyze edge modes of SPT phases using the constructed commuting projector Hamiltonians.
• To extend the construction to anomalous fusion category symmetries by replacing semisimple Hopf algebras with semisimple pseudo-unitary connected weak Hopf algebras.

Proposed method

• The construction uses state sum TQFTs whose input is an H-simple left H-comodule algebra, where H is a finite-dimensional semisimple Hopf algebra.
• The paper introduces a pullback procedure of TQFTs via tensor functors F: C → C′, generalizing the pullback of SPT phases by group homomorphisms.
• Boundary conditions of the state sum TQFTs are shown to carry module category structures over the fusion category C, realizing the symmetry actions explicitly.
• Commuting projector Hamiltonians are derived from the state sum TQFTs, which inherit the fusion category symmetry at the lattice level.
• The framework is extended to anomalous symmetries by using semisimple pseudo-unitary connected weak Hopf algebras in place of semisimple Hopf algebras.
• Interfaces and junctions of TQFTs are modeled using bimodules and associated linear maps, with explicit assignments of vector spaces and linear maps for defect configurations.

Experimental results

Research questions

• RQ1 How can one systematically construct TQFTs and commuting projector Hamiltonians for 1+1d gapped phases with non-anomalous fusion category symmetries?
• RQ2 How are the actions of fusion category symmetries on boundary conditions mathematically represented in the TQFT framework?
• RQ3 Can the state sum TQFT construction be generalized to include interfaces and junctions of different TQFTs with fusion category symmetry?
• RQ4 What is the role of module categories in classifying the boundary conditions of TQFTs with fusion category symmetry?
• RQ5 How can the construction be extended to anomalous fusion category symmetries using weak Hopf algebras?

Key findings

• The actions of fusion category symmetries on boundary conditions of the state sum TQFTs are realized through module categories over the fusion category, confirming the classification of gapped phases with such symmetries.
• Commuting projector Hamiltonians constructed from the state sum TQFTs exhibit fusion category symmetries at the lattice level, providing explicit lattice realizations of gapped phases.
• The edge modes of SPT phases with fusion category symmetries are analyzed using the constructed Hamiltonians, revealing their topological nature.
• The pullback construction via tensor functors allows the systematic generation of TQFTs with any non-anomalous fusion category symmetry from a base TQFT.
• The framework is generalized to anomalous fusion category symmetries by replacing semisimple Hopf algebras with semisimple pseudo-unitary connected weak Hopf algebras, extending the construction beyond non-anomalous cases.
• The inclusion of interfaces and junctions in the state sum TQFT is achieved by assigning vector spaces and linear maps to defect configurations, with bimodules and module maps encoding the topological data.