On lattice models of gapped phases with fusion category symmetries
Abstract
We construct topological quantum field theories (TQFTs) and commuting projector Hamiltonians for any 1+1d gapped phases with nonanomalous fusion category symmetries, i.e. finite symmetries that admit SPT phases. The construction is based on twodimensional 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 pseudounitary connected weak Hopf algebra.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.12882
 Bibcode:
 2021arXiv211012882I
 Keywords:

 Condensed Matter  Strongly Correlated Electrons;
 High Energy Physics  Theory
 EPrint:
 35 pages