From September 2015 until September 2018 worked on the Marie Skłodowska-Curie project *Operator-algebraic approach to topological phases*. This is an individual fellowship, the global variant. It allowed me to work for two years with Bruno Nachtergaele at UC Davis, after which I return to Europe to the group of Barbara Terhal at the RWTH Aachen. On this page you can find the scientific output of the project.

#### Project output

The blog links refer to accompanying blog posts.

**Publications and preprints**

- K. Kato, P. Naaijkens,
*An entropic invariant for 2D gapped quantum phases*. Preprint, arXiv:1810.02376 (2018) - M. Cha, P. Naaijkens, B. Nachtergaele,
*On the stability of charges in infinite quantum spin systems*. Preprint, arXiv:1804.03203 (2018) [blog] - P. Naaijkens,
*Subfactors and quantum information theory*. In: F. Bonetto, D. Borthwick, E. Harrell, M. Loss (eds.), “Mathematical Problems in Quantum Physics,” Contemp. Math.**717**, pp. 257–279, Amer. Math. Soc., Providence RI arXiv:1704.05562 (2018) - P. Naaijkens,
*Quantum Spin Systems on Infinite Lattices: A Concise Introduction*. Lecture Notes in Physics**933**, Springer International Publishing, arXiv:1311.2717 (2017) [blog] - M. Cha, P. Naaijkens, B. Nachtergaele,
*The complete set of infinite volume ground states for Kitaev’s abelian quantum double models*. Commun. Math. Phys.**357**:125–157, arXiv:1608.04449 (2018) [blog] - L. Fiedler, P. Naaijkens, T.J. Osborne,
*Jones index, secret sharing and total quantum dimension*. New J. Phys**17**023039, arXiv:1608.02618 (2017) [blog]

**Invited talks**

- Quantum Information and Operator Algebras, INdAM Roma, February 15–16, 2018
- Condensed Matter Theory seminar, Cologne, 18 December 2017
- Algebraic Structures and Quantum Physics, Cardiff, 14–15 December 2017
- Operator Algebra seminar, University of Tokyo, 4 December 2017
- Quantum Algebra and Topology Seminar, UC Santa Barbara, 3 May 2017
- Subfactor seminar, Vanderbilt, Nashville TN, 14 April 2017
- QMath13 (New topics session), Atlanta, GA, 9 October 2016
- Entanglement in Quantum Spin Systems, Simons Center, Stony Brook NY, 3 October 2016
- 34th Western States Meeting, Caltech, 16 February 2016

(Talks at RWTH Aachen and UC Davis are not listed)

#### Research goals

Topologically ordered phases are a new state of matter, discovered only around the late ’80s. In recent years interest in such states has sparked, one of the reasons being applications to topological quantum computing: the topological properties make the state robust against perturbations, making them ideal components in an environment where (thermal or other) noise is one’s biggest enemy. By now there is a plethora of examples of topologically ordered states, whose only unifying feature seems to be that they do not fall into the Landau theory of phases. Although there are many examples, the mathematical framework to rigorously study such systems is less clear, in particular if one wants to consider both so-called long range entangled phases and symmetry protected phases. The goal of this project is to tackle this problem.

The approach that is proposed is to use operator algebraic methods to focus on the algebraic properties of the observables in such systems. This approach has proved successful in algebraic quantum field theory. Using this attack the aim is to find tools to classify the different topological phases, and in particular find methods that are applicable a wide class of models, despite looking very different at first sight. These ideas will be tested on the wide range of topological systems that is available. The focus in this project is on stability properties on the one hand, in particular for invariants of topological phases, and the study of boundary theories on the other hand.

#### Funding

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 657004.