Construction of Kink Sectors for Two-Dimensional Quantum Field Theory Models – An Algebraic Approach

Schlingemann, Dirk

doi:10.1142/s0129055x98000288

Cited by 4 publications

(3 citation statements)

References 52 publications

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“…However, there might be additional representations that do not correspond to a single ground state and which define unitarily inequivalent Hilbert spaces, known as superselection sectors, in which to look for excitations. These stringlike excitations appear as kinks in systems with symmetry breaking [12], as electric charges in gauge theories [13], or as anyonic excitations in systems with topological order [14,15].…”

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confidence: 99%

Elementary Excitations in Gapped Quantum Spin Systems

et al. 2013

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For quantum lattice systems with local interactions, the Lieb-Robinson bound serves as an alternative for the strict causality of relativistic systems and allows the proof of many interesting results, in particular, when the energy spectrum exhibits an energy gap. In this Letter, we show that for translation invariant systems, simultaneous eigenstates of energy and momentum with an eigenvalue that is separated from the rest of the spectrum in that momentum sector can be arbitrarily well approximated by building a momentum superposition of a local operator acting on the ground state. The error satisfies an exponential bound in the size of the support of the local operator, with a rate determined by the gap below and above the targeted eigenvalue. We show this explicitly for the Affleck-Kennedy-Lieb-Tasaki model and discuss generalizations and applications of our result.

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confidence: 99%

Elementary Excitations in Gapped Quantum Spin Systems

et al. 2013

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“…These and other models have also been studied from the Euclidean point of view by Fröhlich and Marchetti [76], casting a new perspective on these results. See also Schlingemann [167] for more recent developments.…”

Section: P (φ) 2 Modelsmentioning

confidence: 99%

A Perspective on Constructive Quantum Field Theory

Summers¹

2012

Preprint

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An overview of the accomplishments of constructive quantum field theory is provided. 1 1 Introduction: Background and Motivations Quantum field theory (QFT) is widely viewed as one of the most successful theories in science -it has predicted phenomena before they were observed in nature 2 , and its predictions are believed to be confirmed by experiments to within an extraordinary degree of accuracy 3 . Though it has undergone a long and complex development from its origins 4 in the 1929/30 papers of Heisenberg and Pauli [107,108] and has attained an ever increasing theoretical sophistication, it is still not clear in which sense the physically central quantum field theories such as quantum electrodynamics (QED), quantum chromodynamics (QCD) and the Standard Model (SM) are mathematically well defined theories based upon fundamental physical principles that go beyond the merely ad hoc. Needless to say, there are many physicists working with quantum field theories for whom the question is of little Einstein Causality: 7 For all spacelike separated x, y ∈ R 4 one has φ(x)φ(y) = φ(y)φ(x) in the sense of operator valued distributions on D.The Spectrum Condition (stability of the field system): Restricting one's attention to the translation subgroup R 4 ⊂ P ↑ + , the spectrum of the self-adjoint generators of the group U(R 4 ) is contained in the closed forward lightconeThe reader is referred to [122,184] for a discussion of the physical interpretation and motivation of these conditions. There is an equivalent formulation of these conditions in terms of the Wightman functions [184]which are distributions on S(R dn ). These two sets of conditions are referred to collectively as the Wightman axioms. There are closely related sets of conditions for Fermi fields and higher spin Bose fields [122,184].A more general axiom system which is conceptually closer to the actual operational circumstances of a theory tested by laboratory experiments is constituted by the Haag-Araki-Kastler axioms (HAK axioms), which axiomatize conceptual structures referred to as local quantum physics or algebraic quantum field theory (AQFT). Although more general formulations of AQFT are available, for the purposes of this paper it will suffice to limit our attention to a quadruple 8 ({A(O)} O∈R , H, U, Ω) with H, U and Ω as above and {A(O)} O∈R a net of von Neumann algebras A(O) acting on H, where O ranges through a suitable set R of nonempty open subsets of Minkowski space. The algebra A(O) is interpreted as the algebra generated by all (bounded) observables measurable in the spacetime region O, so the net {A(O)} O∈R is naturally assumed to satisfy isotony: if O 1 ⊂ O 2 , then one must have A(O 1 ) ⊂ A(O 2 ). In this framework the basic principles are formulated as follows. Relativistic Covariance: For every Poincaré element (Λ, a) ∈ P ↑ + and spacetime region O ∈ R one has U(Λ, a)A(O)U(Λ, a) −1 = A(ΛO + a). Einstein Causality: 9 For all spacelike separated regions O 1 , O 2 ∈ R one has AB = BA for all A ∈ A(O 1 ) and all B ∈ A(O 2 ).The Spectrum C...

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“…In [Frö76], Frölich proposed an operator-algebraic formulation of solitons as superselection sectors localized in a half-space. Existence of such solitons has been obtained for a wide class of models [Sch96,Sch98,Müg99], and general structural results have been obtained [Fre93,Reh98]. In two-dimensional conformal field theory, the vacuum is unique due to dilation invariance [Rob74] and translation-invariant states are not always localized in half-space [Tan18], yet solitons appear through α-induction [LR95, BE98,BE99a], and the interrelationship between solitons has led to the operator-algebraic formulation of modular invariant [BE99b].…”

Section: Introductionmentioning

confidence: 99%

Solitons and Nonsmooth Diffeomorphisms in Conformal Nets

Vecchio

Iovieno

Tanimoto

2019

Commun. Math. Phys.

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We show that any solitonic representation of a conformal (diffeomorphism covariant) net on S 1 has positive energy and construct an uncountable family of mutually inequivalent solitonic representations of any conformal net, using nonsmooth diffeomorphisms. On the loop group nets, we show that these representations induce representations of the subgroup of loops compactly supported in S 1 \ {−1} which do not extend to the whole loop group.In the case of the U(1)-current net, we extend the diffeomorphism covariance to the Sobolev diffeomorphisms D s (S 1 ), s > 2, and show that the positive-energy vacuum representations of Diff + (S 1 ) with integer central charges extend to D s (S 1 ). The solitonic representations constructed above for the U(1)-current net and for Virasoro nets with integral central charge are continuously covariant with respect to the stabilizer subgroup of Diff + (S 1 ) of −1 of the circle.

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Construction of Kink Sectors for Two-Dimensional Quantum Field Theory Models – An Algebraic Approach

Cited by 4 publications

References 52 publications

Elementary Excitations in Gapped Quantum Spin Systems

Elementary Excitations in Gapped Quantum Spin Systems

A Perspective on Constructive Quantum Field Theory

Solitons and Nonsmooth Diffeomorphisms in Conformal Nets

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