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 Hsimple left Hcomodule algebra, where H is a finite dimensional semisimple Hopf algebra. We show that the actions of fusion category symmetries C on the boundary conditions of these state sum TQFTs are represented by module categories over C . This agrees with the classification of gapped phases with symmetry 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:

Journal of High Energy Physics
 Pub Date:
 March 2022
 DOI:
 10.1007/JHEP03(2022)036
 arXiv:
 arXiv:2110.12882
 Bibcode:
 2022JHEP...03..036I
 Keywords:

 Discrete Symmetries;
 Global Symmetries;
 Lattice Quantum Field Theory;
 Topological Field Theories;
 Condensed Matter  Strongly Correlated Electrons;
 High Energy Physics  Theory
 EPrint:
 35 pages