A unified diagrammatic approach to topological fixed point models

TL;DR

This paper develops a unified diagrammatic tensor network framework for describing topological fixed point models, connecting lattice models, quantum field theories, and topological phases in various dimensions.

Contribution

It introduces a formalism that unifies tensor networks with path integrals for topological phases, extending to fermionic and higher-dimensional models.

Findings

Expresses 2+1D string-net and Kitaev models in general form

Provides a tensor network formalism for fermionic topological phases

Extends the framework to 3+1D topological fixed point models

Abstract

We introduce a systematic mathematical language for describing fixed point models and apply it to the study to topological phases of matter. The framework is reminiscent of state-sum models and lattice topological quantum field theories, but is formalised and unified in terms of tensor networks. In contrast to existing tensor network ansatzes for the study of ground states of topologically ordered phases, the tensor networks in our formalism represent discrete path integrals in Euclidean space-time. This language is more directly related to the Hamiltonian defining the model than other approaches, via a Trotterization of the respective imaginary time evolution. We introduce our formalism by simple examples, and demonstrate its full power by expressing known families of models in 2+1 dimensions in their most general form, namely string-net models and Kitaev quantum doubles based on weak Hopf algebras. To elucidate the versatility of our formalism, we also show how fermionic phases of matter can be described and provide a framework for topological fixed point models in 3+1 dimensions.