Trimer states with $\mathbb{Z}_3$ topological order in Rydberg atom arrays

TL;DR

This paper introduces and analyzes trimer resonating-valence-bond states in lattice models, revealing their potential to host $ ext{Z}_3$ topological order, and proposes experimental realization using Rydberg atom arrays.

Contribution

The study demonstrates the existence of $ ext{Z}_3$ topological order in trimer RVB states, explores their stability, and connects them to $ ext{Z}_3$ lattice gauge theories, with an experimental proposal for realization.

Findings

$ ext{Z}_3$ topological order can emerge in trimer RVB states.

Topological order remains stable under trimer dilution.

A feasible experimental setup using Rydberg atoms is proposed.

Abstract

Trimers are defined as two adjacent edges on a graph. We study the quantum states obtained as equal-weight superpositions of all trimer coverings of a lattice, with the constraint of having a trimer on each vertex: the so-called trimer resonating-valence-bond (tRVB) states. Exploiting their tensor network representation, we show that these states can host $\mathbb{Z}_3$ topological order or can be gapless liquids with $\mathrm{U}(1) \times \mathrm{U}(1)$ local symmetry. We prove that this continuous symmetry emerges whenever the lattice can be tripartite such that each trimer covers all the three sublattices. In the gapped case, we demonstrate the stability of topological order against dilution of maximal trimer coverings, which is relevant for realistic models where the density of trimers can fluctuate. Furthermore, we clarify the connection between gapped tRVB states and $\mathbb{Z}_3$ lattice gauge theories by smoothly connecting the former to the $\mathbb{Z}_3$ toric code, and discuss the non-local excitations on top of tRVB states. Finally, we analyze via exact diagonalization the zero-temperature phase diagram of a diluted trimer model on the square lattice and demonstrate that the ground state exhibits topological properties in a narrow region in parameter space. We show that a similar model can be implemented in Rydberg atom arrays exploiting the blockade effect. We investigate dynamical preparation schemes in this setup and provide a viable route for probing experimentally $\mathbb{Z}_3$ quantum spin liquids.