Towards topological fixed-point models beyond gappable boundaries

TL;DR

This paper introduces a more general fixed-point model framework for topological phases of matter that can potentially describe chiral phases and phases without gapped boundaries, expanding beyond traditional models.

Contribution

It proposes a new fixed-point ansatz that overcomes limitations of existing models, enabling the study of topological phases without gapped boundaries or commuting-projector Hamiltonians.

Findings

Proposes a universal fixed-point ansatz for topological phases.

Suggests strategies for constructing models of chiral topological phases.

Argues that the new ansatz can describe phases beyond existing gapped boundary models.

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.