Quantum symmetry and braid group statistics inG-spin models

  title={Quantum symmetry and braid group statistics inG-spin models},
  author={Korn{\'e}l Szlach{\'a}nyi and P{\'e}ter Vecserny{\'e}s},
  journal={Communications in Mathematical Physics},
In two-dimensional lattice spin systems in which the spins take values in a finite groupG we find a non-Abelian “parafermion” field of the formorder x disorder that carries an action of the Hopf algebra, the double ofG. This field leads to a “quantization” of the Cuntz algebra and allows one to define amplifying homomorphisms on the subalgebra that create the and generalize the endomorphisms in the Doplicher-Haag-Roberts program. The so-obtained category of representations of the observable… 

Order-Disorder Quantum Symmetry in G-Spin Models

Generalizing the Ising model from Z(2) to an arbitrary finite group G we find that the double Hopf algebra D(G) plays the role of the Z(2) × Z(2) symmetry group. Non-Abelian ‘parafermion’ fields are

Quantum Chains of Hopf Algebras with Quantum Double Cosymmetry

Abstract:Given a finite dimensional C*-Hopf algebra H and its dual Ĥ we construct the infinite crossed product and study its superselection sectors in the framework of algebraic quantum field theory.

C*-index of observable algebras in G-spin model

In two-dimensional lattice spin systems in which the spins take values in a finite group G, one can define a field algebra F which carries an action of a Hopf algebra D(G), the double algebra of G

Haag duality for Kitaev's quantum double model for abelian groups

We prove Haag duality for cone-like regions in the ground state representation corresponding to the translational invariant ground state of Kitaev’s quantum double model for finite abelian groups.

Quantum Double Actions on Operator Algebras and Orbifold Quantum Field Theories

Abstract:Starting from a local quantum field theory with an unbroken compact symmetry group G in 1+1-dimensional spacetime we construct disorder fields implementing gauge transformations on the

Quantum spin chains with quantum group symmetry

We consider actions of quantum groups on lattice spin systems. We show that if an action of a quantum group respects the local structure of a lattice system, it has to be an ordinary group. Even

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

The structure of the observable algebra determined by a Hopf *-subalgebra in Hopf spin models

Let H be a finite dimensional Hopf C*-algebra, H1 a Hopf*-subalgebra of H. This paper focuses on the observable algebra AH1 determined by H1 in nonequilibrium Hopf spin models, in which there is a

Conformal field algebras with quantum symmetry from the theory of superselection sectors

According to the theory of superselection sectors of Doplicher, Haag, and Roberts, field operators which make transitions between different superselection sectors—i.e. different irreducible

Braid Group Statistics and their Superselection Rules

We present recent results on the statistics in low-dimensional quantum field theory. They are described by unitary representations of the braid group. We discuss the structure of the “reduced field

Statistics of Fields, the Yang-Baxter Equation, and the Theory of Knots and Links

In these notes we describe an analysis of the statistics problem in quantum field theory. It has been known for some time that in two space-time dimensions there is more to the problem of statistics

Level 1 WZW superselection sectors

AbstractThe superselection structure of the Wess-Zumino-Witten theory based on the affine Lie algebra $$\widehat{so}(N)$$ at level one is investigated for arbitraryN. By making use of the free

Charges in Quantum Field Theory

In quantum field theory one has to deal with superselection sectors, i.e. inequivalent representations πα of the algebra of observables A. It is then desirable to have a unified description, i.e. a

Index of subfactors and statistics of quantum fields. I

We identify the statistical dimension of a superselection sector in a local quantum field theory with the square root of the index of a localized endomorphism of the quasi-local C*-algebra that

Quasitriangular Hopf Algebras and Yang-Baxter Equations

This is an informal introduction to the theory of quasitriangular Hopf algebras and its connections with physics. Basic properties and applications of Hopf algebras and Yang-Baxter equations are

Superselection sectors with braid group statistics and exchange algebras

The theory of superselection sectors is generalized to situations in which normal statistics has to be replaced by braid group statistics. The essential role of the positive Markov trace of algebraic