26 research outputs found
Localized endomorphisms in Kitaev's toric code on the plane
We consider various aspects of Kitaev's toric code model on a plane in the
C^*-algebraic approach to quantum spin systems on a lattice. In particular, we
show that elementary excitations of the ground state can be described by
localized endomorphisms of the observable algebra. The structure of these
endomorphisms is analyzed in the spirit of the Doplicher-Haag-Roberts program
(specifically, through its generalization to infinite regions as considered by
Buchholz and Fredenhagen). Most notably, the statistics of excitations can be
calculated in this way. The excitations can equivalently be described by the
representation theory of D(Z_2), i.e., Drinfel'd's quantum double of the group
algebra of Z_2.Comment: 26 pages, 5 figures. v2: proof of Prop. 2.2 fixed, other minor
correction
Kitaev's quantum double model from a local quantum physics point of view
A prominent example of a topologically ordered system is Kitaev's quantum
double model for finite groups (which in particular
includes , the toric code). We will look at these models from
the point of view of local quantum physics. In particular, we will review how
in the abelian case, one can do a Doplicher-Haag-Roberts analysis to study the
different superselection sectors of the model. In this way one finds that the
charges are in one-to-one correspondence with the representations of
, and that they are in fact anyons. Interchanging two of such
anyons gives a non-trivial phase, not just a possible sign change. The case of
non-abelian groups is more complicated. We outline how one could use
amplimorphisms, that is, morphisms to study the superselection
structure in that case. Finally, we give a brief overview of applications of
topologically ordered systems to the field of quantum computation.Comment: Chapter contributed to R. Brunetti, C. Dappiaggi, K. Fredenhagen, J.
Yngvason (eds), Advances in Algebraic Quantum Field Theory (Springer 2015).
Mainly revie
On the extension of stringlike localised sectors in 2+1 dimensions
In the framework of algebraic quantum field theory, we study the category
\Delta_BF^A of stringlike localised representations of a net of observables O
\mapsto A(O) in three dimensions. It is shown that compactly localised (DHR)
representations give rise to a non-trivial centre of \Delta_BF^A with respect
to the braiding. This implies that \Delta_BF^A cannot be modular when
non-trival DHR sectors exist. Modular tensor categories, however, are important
for topological quantum computing. For this reason, we discuss a method to
remove this obstruction to modularity.
Indeed, the obstruction can be removed by passing from the observable net
A(O) to the Doplicher-Roberts field net F(O). It is then shown that sectors of
A can be extended to sectors of the field net that commute with the action of
the corresponding symmetry group. Moreover, all such sectors are extensions of
sectors of A. Finally, the category \Delta_BF^F of sectors of F is studied by
investigating the relation with the categorical crossed product of \Delta_BF^A
by the subcategory of DHR representations. Under appropriate conditions, this
completely determines the category \Delta_BF^F.Comment: 36 pages, 1 eps figure; v2: appendix added, minor corrections and
clarification
Notions of Infinity in Quantum Physics
In this article we will review some notions of infiniteness that appear in
Hilbert space operators and operator algebras. These include proper
infiniteness, Murray von Neumann's classification into type I and type III
factors and the class of F{/o} lner C*-algebras that capture some aspects of
amenability. We will also mention how these notions reappear in the description
of certain mathematical aspects of quantum mechanics, quantum field theory and
the theory of superselection sectors. We also show that the algebra of the
canonical anti-commutation relations (CAR-algebra) is in the class of F{/o}
lner C*-algebras.Comment: 11 page
Onderwijsinnovatie groen onderwijs 2005-2010 : advies Werkgroep Onderwijsvernieuwing .d.d 14 april 2005
Human bone marrow- and adipose-mesenchymal stem cells secrete exosomes enriched in distinctive miRNA and tRNA species
Inspecteur zou weer kritische observator moeten zijn
Met het systeem van risicoanalyse heeft de Onderwijsinspectie zich op te grote afstand gezet van de scholen. Dat zou op termijn wel eens verkeerd uit kunnen pakken, vreest Melanie Ehren, die in 2006 promoveerde op het onderwerp ‘Toezicht en schoolverbetering’. Zij pleit voor het herstel van de inspecteur 'als een kritische observator van de scholen'