Fusion of Low-Entanglement Excitations in 2D Toric Code

TL;DR

This paper investigates how low-entanglement excitations in a 2D toric code can fuse, revealing dependence on fusion circuits and defect ordering, thus enriching understanding of topological phases.

Contribution

It extends the study of low-entanglement excitations to non-invertible topological phases, specifically analyzing fusion properties in the $ obreak{Z}_2$ toric code.

Findings

Fusion outcomes depend on fusion circuit choices.

Fusion results are affected by defect ordering.

LEEs exhibit higher-category structures in topological phases.

Abstract

On top of a $D$-dimensional gapped bulk state, Low Entanglement Excitations (LEE) on $d$($<D$)-dimensional sub-manifolds can have extensive energy but preserves the entanglement area law of the ground state. Due to their multi-dimensional nature, the LEEs embody a higher-category structure in quantum systems. They are the ground state of a modified Hamiltonian and hence capture the notions of `defects' of generalized symmetries. In previous works, we studied the low-entanglement excitations in a trivial phase as well as those in invertible phases. We find that LEEs in these phases have the same structure as lower-dimensional gapped phases and their defects within. In this paper, we study the LEEs inside non-invertible topological phases. We focus on the simple example of $\mathbb{Z}_2$ toric code and discuss how the fusion result of 1d LEEs with 0d morphisms can depend on both the choice of fusion circuit and the ordering of the fused defects.