# Dr Pieter Naaijkens

Lecturer

- NaaijkensP@cardiff.ac.uk
- +44 29208 74522
- Abacws, Room Room 3.67, Senghennydd Road, Cathays, Cardiff, CF24 4AG

## Overview

#### Research interests

I am interested in quantum spin systems, topological order, and their applications to quantum information theory, with a focus on the use of functional analysis and operator algebraic techniques. One of my focus areas is how one can obtain a full understanding of the (quasi)particle excitations of topologically ordered quantum spin systems. The properties of these excitations can be described by tensor categories, and a substantial part of my work is related to how one can obtain this tensor category by studying certain representations of the *C ^{*}*-algebra of quasi-local observables. An important question we have been working on recently is how stable this structure is with respect to perturbations of the underlying dynamics defining the system, which also is relevant in the classification of topological phases. I have also been working on applying some operator-algebraic techniques to study quantum information.

#### Research group

I am a member of the GAPT (Geometry, Algebra, Mathematical Physics and Topology) Research Group. There also is a GAPT seminar.

## Publication

### 2022

- Naaijkens, P. and Ogata, Y. 2022. The split and approximate split property in 2D systems: stability and absence of superselection sectors. Communications in Mathematical Physics 392, pp. 921-950. (10.1007/s00220-022-04356-3)

### 2020

- Cha, M., Naaijkens, P. and Nachtergaele, B. 2020. On the stability of charges in infinite quantum spin systems. Communications in Mathematical Physics 373, pp. 219-264. (10.1007/s00220-019-03630-1)
- Kato, K. and Naaijkens, P. 2020. An entropic invariant for 2D gapped quantum phases. Journal of Physics A: Mathematical and Theoretical 53(8), article number: 85302. (10.1088/1751-8121/ab63a5)

### 2018

- Cha, M., Naaijkens, P. and Nachtergaele, B. 2018. The complete set of infinite volume ground states for Kitaev's abelian quantum double models. Communications in Mathematical Physics 357(1), pp. 125-157. (10.1007/s00220-017-2989-4)

### 2017

- Fiedler, L., Naaijkens, P. and Osborne, T. J. 2017. Jones index, secret sharing and total quantum dimension. New Journal of Physics 19(2), article number: 23039. (10.1088/1367-2630/aa5c0c)

### 2016

- Bachmann, S., Dybalski, W. and Naaijkens, P. 2016. Lieb-Robinson bounds, Arveson spectrum and Haag-Ruelle scattering theory for gapped quantum spin systems. Annales Henri Poincaré 17, pp. 1737-1791. (10.1007/s00023-015-0440-y)

### 2015

- Naaijkens, P. and Fiedler, L. 2015. Haag duality for Kitaev's quantum double model for abelian groups. Reviews in Mathematical Physics 27(9), article number: 1550021. (10.1142/S0129055X1550021X)
- Chang, L. et al. 2015. On enriching the Levin-Wen model with symmetry. Journal of Physics A: Mathematical and Theoretical 48(12), pp. 12FT01. (10.1088/1751-8113/48/12/12FT01)
- Naaijkens, P. 2015. Kitaev's quantum double model from a local quantum physics point of view. In: Brunetti, R. et al. eds. Advances in Algebraic Quantum Field Theory. Mathematical Physics Studies, pp. 365-395., (10.1007/978-3-319-21353-8_9)

### 2013

- Naaijkens, P. 2013. Kosaki-Longo index and classification of charges in 2D quantum spin models. Journal of Mathematical Physics 54(8), pp. 81901. (10.1063/1.4818272)

### 2012

- Naaijkens, P. 2012. Haag duality and the distal split property for cones in the toric code. Letters in Mathematical Physics 101(3), pp. 341. (10.1007/s11005-012-0572-7)

### 2011

- Naaijkens, P. 2011. On the extension of stringlike localised sectors in 2+1 dimensions. Communications in Mathematical Physics 303(2), pp. 385. (10.1007/s00220-011-1200-6)
- Naaijkens, P. 2011. Localized endomorphisms in Kitaev's toric code on the plane. Reviews in Mathematical Physics 23(4), pp. 347. (10.1142/S0129055X1100431X)

### 2010

- Naaijkens, P. 2010. Topologische kwantumcomputers: rekenen met vlechten. Nieuw Archief voor Wiskunde 11, pp. 187-193.

### 2008

- Berdichevsky, L. and Naaijkens, P. 2008. Four-point functions of different-weight operators in the AdS/CFT correspondence. Journal of High Energy Physics 801, pp. -.

### Articles

- Naaijkens, P. and Ogata, Y. 2022. The split and approximate split property in 2D systems: stability and absence of superselection sectors. Communications in Mathematical Physics 392, pp. 921-950. (10.1007/s00220-022-04356-3)
- Cha, M., Naaijkens, P. and Nachtergaele, B. 2020. On the stability of charges in infinite quantum spin systems. Communications in Mathematical Physics 373, pp. 219-264. (10.1007/s00220-019-03630-1)
- Kato, K. and Naaijkens, P. 2020. An entropic invariant for 2D gapped quantum phases. Journal of Physics A: Mathematical and Theoretical 53(8), article number: 85302. (10.1088/1751-8121/ab63a5)
- Cha, M., Naaijkens, P. and Nachtergaele, B. 2018. The complete set of infinite volume ground states for Kitaev's abelian quantum double models. Communications in Mathematical Physics 357(1), pp. 125-157. (10.1007/s00220-017-2989-4)
- Fiedler, L., Naaijkens, P. and Osborne, T. J. 2017. Jones index, secret sharing and total quantum dimension. New Journal of Physics 19(2), article number: 23039. (10.1088/1367-2630/aa5c0c)
- Bachmann, S., Dybalski, W. and Naaijkens, P. 2016. Lieb-Robinson bounds, Arveson spectrum and Haag-Ruelle scattering theory for gapped quantum spin systems. Annales Henri Poincaré 17, pp. 1737-1791. (10.1007/s00023-015-0440-y)
- Naaijkens, P. and Fiedler, L. 2015. Haag duality for Kitaev's quantum double model for abelian groups. Reviews in Mathematical Physics 27(9), article number: 1550021. (10.1142/S0129055X1550021X)
- Chang, L. et al. 2015. On enriching the Levin-Wen model with symmetry. Journal of Physics A: Mathematical and Theoretical 48(12), pp. 12FT01. (10.1088/1751-8113/48/12/12FT01)
- Naaijkens, P. 2013. Kosaki-Longo index and classification of charges in 2D quantum spin models. Journal of Mathematical Physics 54(8), pp. 81901. (10.1063/1.4818272)
- Naaijkens, P. 2012. Haag duality and the distal split property for cones in the toric code. Letters in Mathematical Physics 101(3), pp. 341. (10.1007/s11005-012-0572-7)
- Naaijkens, P. 2011. On the extension of stringlike localised sectors in 2+1 dimensions. Communications in Mathematical Physics 303(2), pp. 385. (10.1007/s00220-011-1200-6)
- Naaijkens, P. 2011. Localized endomorphisms in Kitaev's toric code on the plane. Reviews in Mathematical Physics 23(4), pp. 347. (10.1142/S0129055X1100431X)
- Naaijkens, P. 2010. Topologische kwantumcomputers: rekenen met vlechten. Nieuw Archief voor Wiskunde 11, pp. 187-193.
- Berdichevsky, L. and Naaijkens, P. 2008. Four-point functions of different-weight operators in the AdS/CFT correspondence. Journal of High Energy Physics 801, pp. -.

### Book sections

- Naaijkens, P. 2015. Kitaev's quantum double model from a local quantum physics point of view. In: Brunetti, R. et al. eds. Advances in Algebraic Quantum Field Theory. Mathematical Physics Studies, pp. 365-395., (10.1007/978-3-319-21353-8_9)

## Research

#### Research interests

I am interested in mathematical physics. That is, the mathematical problems I investigate are motivated by physics. I am particularly interested in studying quantum spin systems with topological order. Such systems lead to anyonic quasi-particle excitations and have provided many examples of new and interesting physical phenomena. I focus on the mathematically rigorous study of such systems using methods from operator algebra theory. Some projects I am thinking about are:

**Stability of superselection sectors.**The anyonic excitations in quantum phases with long range entanglement can be described mathematically using modular tensor categories. Moreover, it is expected that this is an invariant of the gapped quantum phase: if we perturb the system without closing the energy gap, this algebraic structure should not change. With my collaborators I have developed a method to obtain this modular tensor category from first principles, given just the Hamiltonian in the system. In the important example of abelian quantum double models, we can prove that the category one obtains in that way indeed is an invariant of phase.**Scattering theory for anyons.**Scattering processes play a key role in many experiments. It is therefore necessary to have a good theoretical understanding of scattering processes. With my collaborators I have developed a version of Haag-Ruelle scattering theory that can be applied to quantum spin systems. We are presently working on extending this to include scattering of anyons.**Quantum Information Theory.**My interests in topological order was initially motivated by applications to quantum computing. These days, I work on the quantum information side: in particular, I am interested in quantum information theory in systems with infinitely many degrees of freedom, such as quantum field theory. In such systems an operator algebraic approach is very natural, and it turns out that some techniques of the sector theory can be applied to quantum information tasks as well.

#### Funded projects

- PhD studentship, 2020-2024 (EPSRC DTP)
- Marie Sklodowska-Curie Individual Fellowship, 2015-2018 (EU)
- Rubicon fellowship, 2012-2014 (Dutch Research Council)

#### Events organised

- First Virtual LQP Meeting (June 2020)

## Teaching

#### Teaching

In the spring semester I will be teaching the following modules:

- MA2003 Complex Analysis (2020/21)
- MA4016 Quantum Information (2020/21)

Previously I have taught the following modules:

- MA2003 Complex Analysis (2019/20)

#### Previous teaching

Before I moved to Cardiff, I taught:

- Spring 2018: Quantum Information (with David DiVincenzo) (Aachen)
- Spring 2016: MAT 22A (Linear Algebra) (Davis)
- Summer 2014: teaching assistant Lie-Algebren und ihre Darstellungen in der Physik (Hannover)
- Summer 2013: Quantum Spin Systems on Infinite Lattices. Lecture notes can be found here (Hannover)
- Fall 2012: teaching assistant Ergänzungen zur klassischen Physik (Hannover)
- Spring 2011: teaching assistant Symmetry Breaking (Nijmegen)
- Spring 2010: teaching assistant Inleiding Fourieranalyse (Nijmegen)
- Fall 2009: teaching assistant Topologie (Nijmegen)
- Spring 2009: teaching assistant bachelor course Introduction to partial differential equations (Nijmegen)
- Spring 2008: teaching assistant for Analysis I (Nijmegen)

## Biography

### Qualifications

- PhD, Radboud University Nijmegen (2012)
- MSc in Mathematics, Utrecht University (2007)
- MSc in Theoretical Physics, Utrecht University (2007)

### Honours and awards

- Annales Henri Poincaré Prize (2016)
- Marie Skłodowska-Curie Individual Fellowship (2015-2018)
- NWO Rubicon Fellowship (2012-2013)

### Professional memberships

I am a member of the following societies:

- London Mathematical Society
- Institute of Physics
- International Association of Mathematical Physics
- Royal Dutch Mathematics Society (KWG)

### Academic positions

- 2020 - present: Lecturer, Cardiff University
- Nov 2018 - 2019: Postdoc, Universidad Complutense de Madrid
- Sep 2017- Oct 2018: Marie Skłodowska-Curie fellow (return), RTWH Aachen University
- Sep 2015 - Aug 2017: Marie Skłodowska-Curie fellow (outgoing), University of California, Davis
- Apr 2012 - Aug 2015: Postdoc, Leibniz University Hannover
- Jun 2012 - Jun 2014: NWO Rubicon fellow, Leibniz University Hannover
- Oct 2007 - Jan 2012: PhD candidate, Radboud University Nijmegen

### Committees and reviewing

- Institute of Physics
*Mathematical & Theoretical Physics*group committee - School of Mathematics Digital Learning Group

## Supervisions

I am currently not advertising any funded positions. I am however open to supervising self-funded PhD students on topics connected to my research interests:

- quantum information theory
- operator algebras
- topological order & anyons
- quantum spin systems
- tensor categories

Please get into touch with me for more information.

### Current supervision

## Mahdie Hamdan

Research student