scholarly journals Disorder Operators, Quantum Doubles, and Haag Duality in 1 + 1 Dimensions

1997 ◽  
pp. 349-356
Author(s):  
Michael Müger
Keyword(s):  
Haag Duality

scholarly journals ON HAAG DUALITY FOR PURE STATES OF QUANTUM SPIN CHAINS

2008 ◽  
Vol 20 (06) ◽  
pp. 707-724 ◽  
Author(s):  
M. KEYL ◽  
TAKU MATSUI ◽  
D. SCHLINGEMANN ◽  
R. F. WERNER
Keyword(s):  
Von Neumann Algebra ◽  
Quantum Spin ◽  
Spin Chains ◽  
Type I ◽  
Quantum Spin Chains ◽  
Haag Duality ◽  
Von Neumann ◽  
Infinite Intervals ◽  
Pure States

In this note, we consider quantum spin chains and their translationally invariant pure states. We prove Haag duality for quasilocal observables localized in semi-infinite intervals (-∞ , 0] and [1, ∞) when the von Neumann algebra generated by observables localized in [0, ∞) is non-type I.

scholarly journals Haag duality for Kitaev’s quantum double model for abelian groups

2015 ◽  
Vol 27 (09) ◽  
pp. 1550021 ◽  
Author(s):  
Leander Fiedler ◽  
Pieter Naaijkens
Keyword(s):  
Ground State ◽  
Abelian Groups ◽  
Finite Abelian Groups ◽  
Quantum Double ◽  
Haag Duality ◽  
Von Neumann ◽  
Infinite Lattice ◽  
Code Model ◽  
Algebraic Quantum ◽  
Local Observables

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. This property says that if an observable commutes with all observables localized outside the cone region, it actually is an element of the von Neumann algebra generated by the local observables inside the cone. This strengthens locality, which says that observables localized in disjoint regions commute. As an application, we consider the superselection structure of the quantum double model for abelian groups on an infinite lattice in the spirit of the Doplicher–Haag–Roberts program in algebraic quantum field theory. We find that, as is the case for the toric code model on an infinite lattice, the superselection structure is given by the category of irreducible representations of the quantum double.

HAAG DUALITY IN CONFORMAL QUANTUM FIELD THEORY

1990 ◽  
Vol 02 (01) ◽  
pp. 105-125 ◽  
Author(s):  
DETLEV BUCHHOLZ ◽  
HANNS SCHULZ-MIRBACH
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Central Charge ◽  
Algebraic Approach ◽  
Energy Tensor ◽  
Quantum Field ◽  
Haag Duality ◽  
Conformal Fields ◽  
Stress Energy

Haag duality is established in conformal quantum field theory for observable fields on the compactified light ray S1 and Minkowski space S1×S1, respectively. This result provides the foundation for an algebraic approach to the classification of conformal theories. Haag duality can fail, however, for the restriction of conformal fields to the underlying non-compact spaces ℝ, respectively ℝ×ℝ. A prominent example is the stress energy tensor with central charge c>1.

scholarly journals ENTANGLEMENT, HAAG-DUALITY AND TYPE PROPERTIES OF INFINITE QUANTUM SPIN CHAINS

2006 ◽  
Vol 18 (09) ◽  
pp. 935-970 ◽  
Author(s):  
M. KEYL ◽  
T. MATSUI ◽  
D. SCHLINGEMANN ◽  
R. F. WERNER
Keyword(s):  
Hilbert Spaces ◽  
Quantum Spin ◽  
Xy Model ◽  
Type I ◽  
Quantum Spin Chains ◽  
Haag Duality ◽  
Von Neumann ◽  
Neumann Type ◽  
Left And Right ◽  
Normal States

We consider an infinite spin chain as a bipartite system consisting of the left and right half-chains and analyze entanglement properties of pure states with respect to this splitting. In this context, we show that the amount of entanglement contained in a given state is deeply related to the von Neumann type of the observable algebras associated to the half-chains. Only the type I case belongs to the usual entanglement theory which deals with density operators on tensor product Hilbert spaces, and only in this situation separable normal states exist. In all other cases, the corresponding state is infinitely entangled in the sense that one copy of the system in such a state is sufficient to distill an infinite amount of maximally entangled qubit pairs. We apply this results to the critical XY model and show that its unique ground state φS provides a particular example for this type of entanglement.

scholarly journals Continuity of Symplectically Adjoint Maps and the Algebraic Structure of Hadamard Vacuum Representations for Quantum Fields on Curved Spacetime

1997 ◽  
Vol 09 (05) ◽  
pp. 635-674 ◽  
Author(s):  
Rainer Verch
Keyword(s):  
Symplectic Form ◽  
Scalar Product ◽  
Classical System ◽  
Symplectic Space ◽  
Haag Duality ◽  
Globally Hyperbolic ◽  
Classical Energy ◽  
Hyperbolic Spacetime ◽  
Klein Gordon ◽  
Hadamard States

We derive for a pair of operators on a symplectic space which are adjoints of each other with respect to the symplectic form (that is, they are sympletically adjoint) that, if they are bounded for some scalar product on the symplectic space dominating the symplectic form, then they are bounded with respect to a one-parametric family of scalar products canonically associated with the initially given one, among them being its "purification". As a typical example we consider a scalar field on a globally hyperbolic spacetime governed by the Klein–Gordon equation; the classical system is described by a symplectic space and the temporal evolution by symplectomorphisms (which are symplectically adjoint to their inverses). A natural scalar product is that inducing the classical energy norm, and an application of the above result yields that its "purification" induces on the one-particle space of the quantized system a topology which coincides with that given by the two-point functions of quasifree Hadamard states. These findings will be shown to lead to new results concerning the structure of the local (von Neumann) observable-algebras in representations of quasifree Hadamard states of the Klein–Gordon field in an arbitrary globally hyperbolic spacetime, such as local definiteness, local primarity and Haag-duality (and also split- and type III1-properties). A brief review of this circle of notions, as well as of properties of Hadamard states, forms part of the article.

Superselection Structure of Massive Quantum Field Theories in 1+1 Dimensions

1998 ◽  
Vol 10 (08) ◽  
pp. 1147-1170 ◽  
Author(s):  
Michael Müger
Keyword(s):  
Large Class ◽  
Field Theories ◽  
Positive Energy ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Haag Duality ◽  
Conformally Invariant ◽  
Split Property

We show that a large class of massive quantum field theories in 1+1 dimensions, characterized by Haag duality and the split property for wedges, does not admit locally generated superselection sectors in the sense of Doplicher, Haag and Roberts. Thereby the extension of DHR theory to 1+1 dimensions due to Fredenhagen, Rehren and Schroer is vacuous for such theories. Even charged representations which are localizable only in wedge regions are ruled out. Furthermore, Haag duality holds in all locally normal representations. These results are applied to the theory of soliton sectors. Furthermore, the extension of localized representations of a non-Haag dual net to the dual net is reconsidered. It must be emphasized that these statements do not apply to massless theories since they do not satisfy the above split property. In particular, it is known that positive energy representations of conformally invariant theories are DHR representations.

scholarly journals Entropic order parameters for the phases of QFT

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Horacio Casini ◽  
Marina Huerta ◽  
Javier M. Magán ◽  
Diego Pontello
Keyword(s):  
Operator Algebras ◽  
Conformal Symmetry ◽  
Quantization Condition ◽  
Order Parameters ◽  
Haag Duality ◽  
Order And Disorder ◽  
Generalized Symmetries ◽  
Dirac Quantization ◽  
Different Types ◽  
Area Law

Abstract We propose entropic order parameters that capture the physics of generalized symmetries and phases in QFT’s. We do it through an analysis of simple properties (additivity and Haag duality) of the net of operator algebras attached to space-time regions. We observe that different types of symmetries are associated with the breaking of these properties in regions of different non-trivial topologies. When such topologies are connected, we show that the non locally generated operators generate an Abelian symmetry group, and their commutation relations are fixed. The existence of order parameters with area law, like the Wilson loop for the confinement phase, or the ’t Hooft loop for the dual Higgs phase, is shown to imply the existence of more than one possible choice of algebras for the same underlying theory. A natural entropic order parameter arises by this non-uniqueness. We display aspects of the phases of theories with generalized symmetries in terms of these entropic order parameters. In particular, the connection between constant and area laws for dual order and disorder parameters is transparent in this approach, new constraints arising from conformal symmetry are revealed, and the algebraic origin of the Dirac quantization condition (and generalizations thereof) is described. A novel tool in this approach is the entropic certainty relation satisfied by dual relative entropies associated with complementary regions, which quantitatively relates the statistics of order and disorder parameters.

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