Towards topological fixed-point models beyond gappable boundaries

Abstract

We consider fixed-point models for topological phases of matter formulated as discrete path integrals in the language of tensor networks. Such zero-correlation length models with an exact notion of topological invariance are known in the mathematical community as state-sum constructions or lattice topological quantum field theories. All of the established ansatzes for fixed-point models imply the existence of a gapped boundary as well as a commuting-projector Hamiltonian. Thus, they fail to capture topological phases without a gapped boundary or commuting-projector Hamiltonian, most notably chiral topological phases in $2+1$ dimensions. In this work, we present a more general fixed-point ansatz not affected by the aforementioned restrictions. Thus, our formalism opens up a possible way forward towards a microscopic fixed-point description of chiral phases and we present several strategies that may lead to concrete examples. Furthermore, we argue that our more general ansatz constitutes a universal form of topological fixed-point models, whereas established ansatzes are universal only for fixed-points of phases which admit topological boundaries.