Pieter Naaijkens' homepagePostdoctoral researcher in mathematical physics. My reserach interests are in using operator algebraic techniques to study topological phases.
https://pieter.naaijkens.nl/
Wed, 23 May 2018 11:58:40 +0200Wed, 23 May 2018 11:58:40 +0200Jekyll v3.7.3Stability of superselection sectors<p>Matthew Cha, Bruno Nachtergaele and I have just uploaded our paper <em><a href="https://arxiv.org/abs/1804.03203">On the stability of charges in infinite quantum spin systems</a></em>. This is a project we have been thinking about for quite some time. In it we study the superselection sectors (or charges, excitations) in 2D topologically ordered quantum spin systems. Typically these charges are <em>anyons</em>: interchanging two of them is a non-trivial operation. This is also called <em>braiding</em>. Another piece of structure are the <em>fusion rules</em>. They say how two charges can be decomposed into simple or irreducible charges. It is analogous to the decomposition of the tensor product of two representations of a group into irreducibles. Mathematically, typically the excitations are described by a <em><a href="https://ncatlab.org/nlab/show/modular+tensor+category">modular tensor category</a></em> (MTC).</p>
<p>A natural question is then how one can obtain this modular tensor category from a physical description of the system. That is, given a local Hamiltonian. It turns out that this can be done rigorously if one considers the thermodynamic limit, with infinitely many sites. In that case, it is possible to formulate a variant of the Doplicher-Haag-Roberts theory from algebraic quantum field theory in the quantum spin setting. In this setting the charges can be identified with equivalence classes of irreducible representations. These representations satisfy certain conditions with respect to a reference ground state. Typically, the reference state is a pure translation invariant state. One then considers only those irreducible representations that “look like” the reference representation outside of a cone, and moreover, this localization region can be moved around. The set of these representations can then be further analysed to recover the full modular tensor category. This has been worked out for the <a href="https://arxiv.org/abs/1012.3857">toric code</a>, and more generally, <a href="https://arxiv.org/abs/1406.1084">abelian quantum double models</a>.</p>
<p>The question we ask ourselves in this paper is: what happens if we perturb the system? Because of its topological nature, the superselection sector structure should not change. At least, as long as we stay in the same gapped ground phase. In our paper we provide a rigorous proof of this statement for abelian quantum double models. As a key tool we develop a superselection theory for <em>almost</em> localized charges. In contrast with the analysis for the unperturbed model, in the perturbed model generally the charges can not be exactly localized (here, in a cone-like region extending to infinity), but only up to an exponentially decaying error term. As a consequence, many of the techniques that work in the unperturbed case break down.</p>
<p>The issue can be explained as follows. We consider the toric code as an example. This model has only one translational invariant ground state, which is pure. Denote this by <script type="math/tex">\omega_0</script>. With help of the GNS construction, we get a representation <script type="math/tex">\pi_0</script>, which we will use as the reference representation. The quasi-local algebra of observables is written <script type="math/tex">\mathcal{A}</script>. Now consider a family of local Hamiltonians <script type="math/tex">s \mapsto H_\Lambda(s)</script>, where <script type="math/tex">s=0</script> corresponds to the toric code. We assume the assignment is <script type="math/tex">C^1</script> and that the gap does not close. Then by the <em><a href="https://arxiv.org/abs/1102.0842">spectral flow</a></em> there is an automorphism <script type="math/tex">\alpha_s \in {Aut}(\mathcal{A})</script> such that <script type="math/tex">\omega_s = \omega_0 \circ \alpha_s</script>, where <script type="math/tex">\omega_s</script> is a ground state of the perturbed model (in fact, more generally we can consider weak<script type="math/tex">*</script>-limits of states whose energy lies below a gap in the spectrum). Hence the new reference representation is <script type="math/tex">\pi_0 \circ \alpha_s</script>.</p>
<p>The problem one has to deal with is that <script type="math/tex">\alpha_s</script> is not strictly local, but rather satisfies a Lieb-Robinson bound. This is an issue because in the DHR theory, we consider representations <script type="math/tex">\pi</script> such that for any cone-like region <script type="math/tex">\Lambda</script>, we have the following relation:</p>
<script type="math/tex; mode=display">\pi | \mathcal{A}(\Lambda^c) \cong \pi_0 | \mathcal{A}(\Lambda^c).</script>
<p>Or in words, when restricted to observables <em>outside</em> the cone <script type="math/tex">\Lambda</script>, the representations look like the reference representation.
The problem is that if we compose both sides with <script type="math/tex">\alpha_s</script>, this does not need to be true any more, since <script type="math/tex">\alpha_s</script> generally will not map observables localized into cone regions onto observables in cone regions.</p>
<p>In the unperturbed models one can show that representations satisfying the criterion above are equivalent to <script type="math/tex">\pi_0 \circ \rho</script> with an endomorphism <script type="math/tex">\rho</script> of <script type="math/tex">\mathcal{A}</script> that acts non-trivially only on <script type="math/tex">\mathcal{A}(\Lambda)</script> for some cone <script type="math/tex">\Lambda</script>. Motivated by the Lieb-Robinson bound for <script type="math/tex">\alpha_s</script>, we relax this condition to allow for endomorphisms that are only approximately localized in a cone. To get the tensor category, we study such endomorphisms by looking at <em>(bi-)asymptopia</em>, a concept introduced by <a href="https://arxiv.org/abs/math-ph/0209038">Buchholz, Doplicher, Morchio, Roberts and Strocchi</a>. Ultimately, this can be used to show that the abelian quantum double models are indeed stable against local perturbations that do not close the gap.</p>
Wed, 11 Apr 2018 00:00:00 +0200
https://pieter.naaijkens.nl/news/research/2018/04/11/stability.html
https://pieter.naaijkens.nl/news/research/2018/04/11/stability.htmlnewsresearchAHP Prize 2016 for our paper<p>Our paper <em><a href="https://dx.doi.org/10.1007/s00023-015-0440-y">Lieb-Robinson bounds, Arveson spectrum and Haag-Ruelle scattering
theory for gapped quantum spin
systems</a></em> has been awarded by the
editorial board of <em>Annales Henri Poincaré</em> with the <strong><a href="http://www.springer.com/birkhauser/mathematics?SGWID=0-40292-6-793469-0">AHP Prize 2016</a></strong>
for the most remarkable paper published in their journal. Many thanks and
gratitude go to my co-authors and Sven Bachmann and Wojciech Dybalski for
collaborating on this very interesting project, and for their hard work. And of
course thanks to the editors for selecting our paper!</p>
Fri, 05 Jan 2018 00:00:00 +0100
https://pieter.naaijkens.nl/news/2018/01/05/ahp-prize.html
https://pieter.naaijkens.nl/news/2018/01/05/ahp-prize.htmlnewsBook on quantum spin systems published<p><img src="/assets/images/book_cover.jpg" class="float_post" /></p>
<p>My new book, <em><a href="https://dx.doi.org/10.1007/978-3-319-51458-1">Quantum Spin Systems on Infinite Lattices: A concise introduction</a></em>, has been published in Springer’s <em>Lecture Notes in Physics</em> series. The book is an expanded version of lecture notes I wrote for a <a href="http://qis.verwaltung.uni-hannover.de/qisserver/rds?state=verpublish&status=init&vmfile=no&moduleCall=webInfo&publishConfFile=webInfo&publishSubDir=veranstaltung&veranstaltung.veranstid=186627">course</a> I taught in the <em>Sommersemester</em> 2013 at Leibniz University Hannover.</p>
<p>The lecture notes are meant as an introduction to the operator algebraic description of quantum spin systems with infinitely many sites. The intended audience are advanced undergraduate students in mathematics or physics, or beginning graduate students. Although most of the important mathematical constructs are recalled briefly in the notes, some mathematical maturity is assumed (e.g., a good understanding of convergence in metric spaces). Moreover, emphasis is placed on developing intuition for what is true, outlining the main ideas (both from physics and mathematics) behind a proof. For the sake of clarity of the exposition, some results are not stated in complete generality, but rather an easier special case is proven.</p>
<p>The book is essentially divided into three parts: the first recalls some basic notions of functional analysis, including Hilbert spaces, bounded linear operators, completion of metric spaces, and an introduction to the theory of <script type="math/tex">C^*</script>-algebras. In the second part this is applied to describe quantum spin systems on infinite lattices, the main subject of the lecture notes. For example, it is explained how to define dynamics on such a system, and how to define ground and thermal states. The fundamental tool of Lieb-Robinson bounds is also discussed, and throughout the lecture notes it will become clear that locality plays an essential role.</p>
<p>Finally, in the last part, recent developments are outlined. Although the treatment here is less detailed, it serves as an introduction to current research topics. Included are applications of Lieb-Robinson bounds to a Haag-Ruelle type of scattering theory and to equivalences of gapped ground state phases, as well as a discussion of the theory of superselection sectors in Kitaev’s toric code.</p>
<p>The book is available directly from <a href="http://www.springer.com/book/9783319514567">Springer</a> or through various other retailers. A preprint (without the final layout and without corrections made in proof) can be found <a href="https://arxiv.org/abs/1311.2717">here</a>.</p>
Sun, 09 Apr 2017 00:00:00 +0200
https://pieter.naaijkens.nl/news/2017/04/09/book.html
https://pieter.naaijkens.nl/news/2017/04/09/book.htmlnewsJones index, data hiding and total quantum dimension<p>The paper <em><a href="http://arxiv.org/abs/1608.02618">Jones index, data hiding and total quantum dimension</a></em>, together with Leander Fiedler and Tobias Osborne is available on the arXiv. We started this project already a few years ago, so I am happy that after changing the scope a few times and finding better ways to explain things, the paper is finaly ready. Below is a high level summary of the paper, explaining why we think it is interesting.</p>
<p>In the paper we study topologically ordered states in the thermodynamic limit. This has
significant advantages over the more prevalent finite dimensional approach,
and opens up the possibility of using deep results only available in infinite
dimensions, as we demonstrate. We illustrate this by a data hiding scheme and
connect these concepts to more familiar quantities such as topological
entanglement entropy. This way we are able to give a new interpretation of
the quantum dimension. We expect our approach to open up new possibilities in
the study of topological phases.</p>
<p>One of the key points of our work is that it forms a bridge between different
fields, from quantum information and topologically ordered condensed matter
systems in physics, to operator algebra and index theory in pure mathematics.
These are quite distinct fields, so a major goal of our paper is to
communicate our results in a way that is accessible to researchers working in
any of these fields. To achieve this, we have included brief discussions of
known results and techniques, along with illustrating how they are related to
our new results. We first illustrate the main intuition behind our results in a finite dimensional
setting, and explain which complications arise there, and how they can be
resolved by going to the thermodynamic limit.</p>
<p>The starting point of our paper is the observation made by one of
us earlier that the total quantum dimension, an invariant related to the
anyonic excitations of the system, can be obtained by considering the Jones
index (or rather, its generalisation) of an inclusion of operator algebras.
What is new in the present paper is that we reinterpret the quantum dimension
(again, in the thermodynamic limit setting) in terms of a secret sharing
scheme. This interpretation can then be made precise using properties of the
index. In particular, it is possible to exactly quantify the amount of
information that can he hidden in this way, and to identify a quantum channel.
This uses deep results on the Jones index, that at the same time can be given
a natural physical interpretation in our context.</p>
<p>To summarise, our work sheds new light on the total quantum dimension. Using
completely different methods than considered before in the literature, we in
the end obtain similar interpretations. We expect that these new methods will
be useful in the further study of topologically ordered systems, for one since
it allows us to tap into a vast wealth of mathematical results. At the same
time, we hope it will be a useful and accessible introduction to the
relatively unknown (but in our opinion extremely useful) operator algebraic
methods.</p>
Mon, 26 Sep 2016 00:00:00 +0200
https://pieter.naaijkens.nl/news/research/2016/09/26/quantum-dimension.html
https://pieter.naaijkens.nl/news/research/2016/09/26/quantum-dimension.htmlnewsresearchGround states of abelian quantum double models<p>A new preprint (with Matthew Cha and Bruno Nachtergaele) is available on the arXiv: <em><a href="http://arxiv.org/abs/1608.04449">The complete set of infinite volume ground states for Kitaev’s abelian quantum double models</a></em>.
In this paper we consider abelian quantum double models, realised as a local quantum spin system by Kitaev’s description for any finite abelian group <script type="math/tex">G</script>.<sup id="fnref:kitaev"><a href="#fn:kitaev" class="footnote">1</a></sup>
This includes the well-known toric code model, which corresponds to a choice of <script type="math/tex">G = \mathbb{Z}_2</script> for the abelian group.</p>
<p>We consider the model on an infinite <script type="math/tex">\mathbb{Z}^2</script> lattice, with edges between nearest neigbours. Each edge contains a quantum spin system of dimension <script type="math/tex">\vert G\vert</script>. Dynamics are implemented by commuting local projectors, acting on stars (the four edged coming out of a vertex), and plaquettes (the four edges making up a square in the lattice). These projectors detect the absence of a charge at a star or plaquette. There is a unique frustration free ground state (in the thermodyamic limit).<sup id="fnref:afh"><a href="#fn:afh" class="footnote">2</a></sup> However, classifying all ground states is much harder: to find all ground states, one has to find all states that satisfy</p>
<script type="math/tex; mode=display">\omega(A^* \delta(A)) \geq 0.</script>
<p>Here <script type="math/tex">\delta(A) = \lim_{\Lambda \to \infty} [H_\Lambda, A]</script>, with <script type="math/tex">A</script> a local observable, is the generator obtained from the local Hamiltonians. There are many results on the existence and structural properties of the set of ground states of quantum spin models, but surprisingly few results find the complete ground state space of concrete models. In fact, as far as we are aware, our result is the first example in two dimensions.</p>
<p>It turns out that the set of ground states is related to the charge structure of the model. It is possible, using local operators, to create pairs of excitations (or charges) from any state. These excitations are labelled by <script type="math/tex">(\chi,c)</script>, where <script type="math/tex">\chi</script> is a character of <script type="math/tex">G</script> and <script type="math/tex">c \in G</script>. To find the total charge, one just has to take the product of all the charges. The local operators create a pair of inverse charges at each end, so they do not change the total charge of the system. One can, however, send of one of the charges to infinity, leaving a state with a <em>single</em> charge. These states lead to inequivalent representations or superselection sectors.<sup id="fnref:fn"><a href="#fn:fn" class="footnote">3</a></sup></p>
<p>It turns out that these states are ground states, and in fact are essentially all possible ground states. Our main result can be summarised as follows:</p>
<p><strong>Theorem</strong> <em>Let <script type="math/tex">G</script> be a finite abelian group. Then each ground state can be written as a convex combination of at most <script type="math/tex">\vert G \vert^2</script> disjoint states. The pure states in each of these sectors are equivalent to a charged stated, whose equivalence classes are labelled by <script type="math/tex">(\chi, c)</script> for <script type="math/tex">\chi</script> a character of <script type="math/tex">G</script>, and <script type="math/tex">c \in G</script>.</em></p>
<p>There are two important ideas behind our proof: (i) we can define “local” charge projections, that project onto all states with a specific total charge in a region <script type="math/tex">\Lambda</script>, and (ii) a specific choice of boundary conditions, so that states with a single excitation in the bulk are ground states of the local Hamiltonian with these boundary conditions.</p>
<p>Then, by taking any ground state, we can make a sequence of finite volume ground states (with the appropriate boundary conditions), by projecting onto the different charged sectors in a box of size <script type="math/tex">L \times L</script>. We show that this limit converges in the weak<script type="math/tex">*</script>-topology to a ground state, and that if we take the right convex combination of these states, we get our original state back. Then, to conclude the proof, we can consider a pure state in one of the sectors. By using some topological properties of the model, in particular that there is some freedom in choosing paths necessary to define the local operators that create a pair of charges, it is possible to show that the expectation values of local operators that are sufficiently far away, are indistinguishable from the expectation value in the charged states that can be constructed explicitly. It follows that the states must be equivalent, in the sense that their GNS representations are unitarily equivalent.</p>
<h4 id="footnotes">Footnotes</h4>
<div class="footnotes">
<ol>
<li id="fn:kitaev">
<p>A. Kitaev, <em>Fault-tolerant quantum computation by anyons</em>, <a href="http://dx.doi.org/10.1016/S0003-4916(02)00018-0">Ann. Physics</a> <strong>303</strong>, 2–30, 2003. [<a href="http://arxiv.org/abs/quant-ph/9707021">arXiv</a>] <a href="#fnref:kitaev" class="reversefootnote">↩</a></p>
</li>
<li id="fn:afh">
<p>R. Alicki, M. Fannes, M. Horodecki, <em>A statistical mechanics view on Kitaev’s proposal for quantum memories</em>, <a href="http://dx.doi.org/10.1088/1751-8113/40/24/012">J. Phys. A.: Math. Theor.</a> <strong>40</strong>, 6451–6467, 2007. <a href="http://arxiv.org/abs/quant-ph/0702102">[arXiv]</a> <a href="#fnref:afh" class="reversefootnote">↩</a></p>
</li>
<li id="fn:fn">
<p>L. Fiedler, P. Naaijkens, <em>Haag duality for Kitaev’s quantum double model for abelian groups</em>, <a href="http://dx.doi.org/10.1142/S0129055X1550021X">Rev. Math. Phys.</a> <strong>27</strong>, 1550021, 2015. <a href="http://arxiv.org/abs/1406.1084">[arXiv]</a> <a href="#fnref:fn" class="reversefootnote">↩</a></p>
</li>
</ol>
</div>
Sun, 21 Aug 2016 00:00:00 +0200
https://pieter.naaijkens.nl/news/research/2016/08/21/ground-states-qdouble.html
https://pieter.naaijkens.nl/news/research/2016/08/21/ground-states-qdouble.htmlnewsresearchHaag duality in Kitaev's quantum double models<p>Our paper on Haag duality in quantum double models has been published in <a href="http://dx.doi.org/10.1142/S0129055X1550021X">Reviews in Mathematical Physics</a>. This is the first paper of my co-author <a href="http://www.leander-fiedler.de">Leander Fiedler</a>, a PhD student in Hannover with whom I have worked closely together while I was a postdoc there, so this is very exciting. Our paper generalises results first obtained for the toric code (where <script type="math/tex">G = \mathbb{Z}_2</script>)<!--_--> to arbitrary abelian groups <script type="math/tex">G</script>. In particular, we prove that Haag duality holds in the translational invariant ground state and uncover the superselection structure of the model.</p>
<p>So what exactly is Haag duality? We consider Kitaev’s quantum double model in two dimensions in the thermodynamic limit. For simplicity, we assume that we have a square lattice, with at each edge between neighbouring vertices a <script type="math/tex">\vert G\vert</script>-dimensional Hilbert space. This leads in a natural way to a <script type="math/tex">C^*</script>-algebra<!--*--> <script type="math/tex">\mathfrak{A}</script> of observables, called the quasi-local algebra.<sup id="fnref:quasilocal"><a href="#fn:quasilocal" class="footnote">1</a></sup> This is very general. The physical input comes from specifying dynamics for the system in terms of local Hamiltonians, in our case those first introduced by Kitaev.<sup id="fnref:kitaev"><a href="#fn:kitaev" class="footnote">2</a></sup> The precise form is not important here: the main point is that this allows us to select ground states for these dynamics. In the present case, there are many ground states, but for our purposes there is a preferred one: there is a unique translational invariant ground state (which is pure).<sup id="fnref:gs"><a href="#fn:gs" class="footnote">3</a></sup></p>
<p>Once we have selected a state <script type="math/tex">\omega_0</script> on <script type="math/tex">\mathfrak{A}</script> we can obtain a representation <script type="math/tex">\pi_0</script> on a Hilbert space <script type="math/tex">\mathcal{H}</script>, such that the state <script type="math/tex">\omega_0</script> is implemented by a vector <script type="math/tex">\Omega \in \mathcal{H}</script>. This can be done via the <a href="https://en.wikipedia.org/wiki/Gelfand–Naimark–Segal_construction">GNS construction</a>. Now consider a “cone region” of the system: take a point in the lattice and two semi-infinite lines emanating from them. We say that, roughly speaking, the sites within the two lines form a cone <script type="math/tex">\Lambda</script>.<sup id="fnref:cone"><a href="#fn:cone" class="footnote">4</a></sup> Associated to the cone is the <script type="math/tex">C^*</script>-<!--*-->algebra <script type="math/tex">\mathfrak{A}(\Lambda)</script> of quasi-local observables localised in that cone. The algebra of all quasi-local observables <em>outside</em> the cone is denoted by <script type="math/tex">\mathfrak{A}(\Lambda)</script>. We can then define two new algebras:</p>
<script type="math/tex; mode=display">\mathcal{R}_\Lambda := \pi_0(\mathfrak{A}(\Lambda))'',
\quad
\mathcal{R}_{\Lambda^c} := \pi_0(\mathfrak{A}(\Lambda^c))'',</script>
<p>where the prime means taking the commutant in <script type="math/tex">\mathfrak{B}(\mathcal{H})</script>. By taking the double commutant we obtain <a href="https://en.wikipedia.org/wiki/Von_Neumann_algebra">von Neumann algebras</a>. It should be noted that these algebras strongly depend on the choice of representation <script type="math/tex">\pi_0</script>. By locality, the property that observables localised in disjoint regions commute, it is easy to see that <script type="math/tex">\mathcal{R}_\Lambda \subset \mathcal{R}_{\Lambda^c}'</script>. <em>Haag duality</em> then is the claim that the reverse inclusion also holds.</p>
<p>The proof relies on a good understanding of how the Hilbert space can be build by acting with local observables on the ground state vector <script type="math/tex">\Omega</script>. It turns out that we can label a convenient orthonormal basis by looking at “charge configurations” of anyons. These configurations can be obtained by acting with local operators associated to paths on the lattice on the ground state vector. A good understanding of the commutation properties of these operators is essential. In this case, these relations have a nice algebraic structure stemming from the representation theory of Hopf algebras. It allows us to look at the subspace of all charge configurations in the cone <script type="math/tex">\Lambda</script> (and trivial charge outside), and see how <script type="math/tex">\mathcal{R}_\Lambda</script> and <script type="math/tex">\mathcal{R}_{\Lambda^c}'</script> act when restricted to this subspace. By proving a density result, it is possible to show that they are each others commutants. This result then extends to the whole Hilbert space.</p>
<p>Haag duality is mainly used as a technical tool in the study of the superselection sectors of the model. For a short introduction of the main ideas behind this construction, which originates from algebraic quantum field theory, see <a href="http://dx.doi.org/10.1007/978-3-319-21353-8_9">here</a> (or the <a href="http://arxiv.org/abs/1508.07170">arXiv version</a>). In essence, it allows us to replace representations of <script type="math/tex">\mathfrak{A}</script>, which satisfy a physically reasonable selection criterion,<sup id="fnref:superselect"><a href="#fn:superselect" class="footnote">5</a></sup> by endomorphisms of <script type="math/tex">\mathfrak{A}</script>. These endomorphisms have a much richer structure than the representations. For example, two endomorphisms can be composed. Physically speaking this is the same as “fusion”: bringing two charges together and see how they combine.<sup id="fnref:fusionrule"><a href="#fn:fusionrule" class="footnote">6</a></sup> By studying these endomorphisms we can uncover all the relevant physical properties of the anyonic charges in the model. As expected, they are described by the representation theory of the quantum double of the group <script type="math/tex">G</script>.</p>
<h4 id="footnotes">Footnotes</h4>
<div class="footnotes">
<ol>
<li id="fn:quasilocal">
<p>This is the algebra of all observables that can be approximated arbitrarily well (in norm) by observables that act only on a finite number of sites. It is the inductive limit of local algebras <script type="math/tex">\mathfrak{A}(\Lambda) = \bigotimes_{x \in \Lambda} M_{\vert G\vert}(\mathbb{C})</script>. <a href="#fnref:quasilocal" class="reversefootnote">↩</a></p>
</li>
<li id="fn:kitaev">
<p>Kitaev, A.: Fault-tolerant quantum computation by anyons. <em>Ann. Physics.</em> <strong>303</strong>, 2–30 (2003). <a href="#fnref:kitaev" class="reversefootnote">↩</a></p>
</li>
<li id="fn:gs">
<p>The current version on the arXiv is a bit imprecise on this point. This is the ground state that we use there, but the claim that this is the only ground state is not correct. See the published version for the correct statement. <a href="#fnref:gs" class="reversefootnote">↩</a></p>
</li>
<li id="fn:cone">
<p>The precise shape is not that important, as long as it is an infinite region without any holes and big enough that any finite set can be translated into it. <a href="#fnref:cone" class="reversefootnote">↩</a></p>
</li>
<li id="fn:superselect">
<p>In particular, we look at representations that look like the ground state representation <em>outside</em> of arbitrary cones <script type="math/tex">\Lambda</script>. Alternatively, this means that the charges we are interested in, which correspond to equivalence classes of such representations, can be localised <em>inside</em> a cone, and can also be moved around to be localised inside any other cone. <a href="#fnref:superselect" class="reversefootnote">↩</a></p>
</li>
<li id="fn:fusionrule">
<p>This is analogous to the representation theory of compact groups, where the tensor product of two irreducible representations can be decomposed into a direct sum of such representations. <a href="#fnref:fusionrule" class="reversefootnote">↩</a></p>
</li>
</ol>
</div>
Mon, 21 Dec 2015 00:00:00 +0100
https://pieter.naaijkens.nl/news/research/2015/12/21/Haag-duality.html
https://pieter.naaijkens.nl/news/research/2015/12/21/Haag-duality.htmlnewsresearchNew website uploaded<p>The website has been updated and redesigned, to reflect my move to UC Davis.</p>
Sun, 11 Oct 2015 20:00:00 +0200
https://pieter.naaijkens.nl/news/2015/10/11/new-website.html
https://pieter.naaijkens.nl/news/2015/10/11/new-website.htmlnewsMarie Sklodowska-Curie project started<p>I have started working on the `Operator-algebraic Approach to Topological Phases’ (OATP) project funded by the European Union. This project will run for three years. See <a href="/mariecurie/">here</a> for more details.</p>
Tue, 01 Sep 2015 11:00:00 +0200
https://pieter.naaijkens.nl/news/2015/09/01/Marie-Curie-started.html
https://pieter.naaijkens.nl/news/2015/09/01/Marie-Curie-started.htmlnews