From August 2012 to July 2014 I worked on the project *Mathematical structure of anyons in planar quantum spin systems*, funded by a Rubicon fellowship of the Dutch Organisation for Scientific Research NWO. This fellowship allows researchers who recently obtained their PhD at a Dutch university to go work at a foreign institution for up to two years.

# Project output

The following publications can be attributed to the project at least partly:

**P. Naaijkens**, Kosaki-Longo index and classification of charges in 2D quantum spin models,*Journal of Mathematical Physics***54**, 081901 (2013) arXiv:1303.4420- L. Chang, M. Cheng, S.X. Cui, Y. Hu, W. Jin, R. Movassagh,
**P. Naaijkens**, Z. Wang, A. Young, On Enriching the Levin-Wen model with Symmetry,*J. Phys. A: Math. Theor.***48**:12FT01 (2015) arXiv:1412.6589 - L. Fiedler,
**P. Naaijkens**, Haag duality for Kitaev’s quantum double model for abelian groups,*Rev. Math. Phys.***27**:1550021:1–43 (2015) arXiv:1406.1084 - S. Bachmann, W. Dybalski,
**P. Naaijkens**, Lieb-Robinson bounds, Arveson spectrum and Haag-Ruelle scattering theory for gapped quantum spin systems*Ann. Henri Poincaré***17**:1737–1791 (2016) arXiv:1412.2970 **P. Naaijkens**, Kitaev’s quantum double model from a local quantum physics point of view.*In:*R. Brunetti C. Dappiaggi, K. Fredenhagen, J. Yngvason (eds),*Advances in Algebraic Quantum Field Theory*, pp. 365–395, Springer (2015) arXiv:1508.07170**P. Naaijkens**,*Quantum Spin Systems on Infinite Lattices: A Concise Introduction*. Lecture Notes in Physics**933**, Springer International Publishing, arXiv:1311.2717 (2017)

# Project abstract

This project is in mathematical physics. Our research is partly motivated by potential applications in quantum computation. Technically, we propose to study quantum spin systems on infinite two-dimensional lattices. In particular, we plan to develop a framework for describing “charges” or “excitations” in such models. Our main goal is to find - in this quantum spin setting - a suitable analogue of the Doplicher-Haag-Roberts theory of superselection sectors in (algebraic) quantum field theory. This amounts to finding a framework in which single excitations of the spin model can be described by certain linear maps of the observables. The study of such maps should reveal all relevant properties of the excitations, such as what happens if we interchange two excitations (“braiding”) or when two excitations are brought close together (“fusion”). Our focus is primarily on models with anyonic excitations, for example Kitaev’s quantum double model or the Levin-Wen string-net model.

# Funding

This project was funded by Netherlands Organisation for Scientific Research (NWO) project number 680-50-1118.