A hierarchy of topological tensor network states

TL;DR

This paper introduces a hierarchical framework of topological tensor network states based on Hopf C*-algebras, revealing new models and mechanisms for topological order and entanglement in quantum many-body systems.

Contribution

It constructs a hierarchy of topological tensor network states, including new models extending Kitaev's quantum double models, and elucidates the role of charge condensation in topological order.

Findings

Identifies ground states of new topological lattice models

Shows the mechanism behind topological entanglement entropy

Relates models through charge condensation

Abstract

We present a hierarchy of quantum many-body states among which many examples of topological order can be identified by construction. We define these states in terms of a general, basis-independent framework of tensor networks based on the algebraic setting of finite-dimensional Hopf C*-algebras. At the top of the hierarchy we identify ground states of new topological lattice models extending Kitaev's quantum double models [26]. For these states we exhibit the mechanism responsible for their non-zero topological entanglement entropy by constructing a renormalization group flow. Furthermore it is shown that those states of the hierarchy associated with Kitaev's original quantum double models are related to each other by the condensation of topological charges. We conjecture that charge condensation is the physical mechanism underlying the hierarchy in general.