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# Four-body ring-exchange interactions and anyonic statistics within a minimal toric-code Hamiltonian

## Abstract

Ring exchange is an elementary interaction for modelling unconventional topological matter. Here, we report the observation of four-body ring-exchange interactions and the topological properties of anyonic excitations within an ultracold atom system. A minimum toric-code Hamiltonian, in which the ring exchange is the dominant term, was implemented in disconnected four-spin plaquette arrays formed by two orthogonal superlattices. The ring-exchange interactions were resolved from the dynamical evolutions of the spin orders in each plaquette, matching well with the predicted energy gaps between two anyonic excitations of the spin system. A braiding operation was applied to the spins in the plaquettes and an induced phase 1.00(3)π in the four-spin state was observed, confirming 1/2 mutual statistics. This work offers new prospects for the quantum simulation of topological phases by engineering many-body interactions.

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## Acknowledgements

We thank A. J. Leggett, P. Zoller and B. Zhao for helpful discussions. This work was supported by the National Key R&D Program of China (2016YFA0301600), National Natural Science Foundation of China (91421305, 11521063), and the Chinese Academy of Sciences.

## Author information

Authors

### Contributions

Y.-A.C., Z.-S.Y. and J.-W.P. initiated and designed this research project. H.-N.D., B.Y., A.R., X.-F.X. and Z.-S.Y. set up the experiment. H.-N.D., B.Y., A.R. and H.S. performed the measurement and analysed the data. All authors contributed to manuscript preparation. Z.-S.Y. and J.-W.P. supervised the whole project.

### Corresponding authors

Correspondence to Zhen-Sheng Yuan or Jian-Wei Pan.

## Ethics declarations

### Competing interests

The authors declare no competing financial interests.

## Supplementary information

### Supplementary information

Supplementary information (PDF 577 kb)

## Rights and permissions

Reprints and Permissions

Dai, HN., Yang, B., Reingruber, A. et al. Four-body ring-exchange interactions and anyonic statistics within a minimal toric-code Hamiltonian. Nature Phys 13, 1195–1200 (2017). https://doi.org/10.1038/nphys4243

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