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arXiv:1612.02073

Wedge-local observables in the deformed sine-Gordon model

Daniela Cadamuro, Yoh Tanimoto

Published Dec 6, 2016• 2 citations
arXivHTMLPDF

Abstract

In the bootstrap approach to integrable quantum field theories in the (1+1)-dimensional Minkowski space, one conjectures the two-particle S-matrix and tries to study local observables. The massless sine-Gordon model is conjectured to be equivalent to the Thirring model, and its breather-breather S-matrix components (where the first breather corresponds to the scalar field of the sine-Gordon model) are closed under fusion. Yet, the residues of the poles in this breather-breather S-matrix have wrong signs and cannot be considered as a separate model. We find CDD factors which adjust the signs, so that the breather-breather S-matrix alone satisfies reasonable assumptions. Then we propose candidates for observables in wedge-shaped regions and prove their commutativity in the weak sense.

DOI
10.48550/arxiv.1612.02073

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In the bootstrap approach to integrable quantum field theories in the (1+1)-dimensional Minkowski space, one conjectures the two-particle S-matrix and tries to study local observables. The massless sine-Gordon model is conjectured to be equivalent to the Thirring model, and its breather-breather S-matrix components (where the first breather corresponds to the scalar field of the sine-Gordon model) are closed under fusion. Yet, the residues of the poles in this breather-breather S-matrix have wrong signs and cannot be considered as a separate model. We find CDD factors which adjust the signs, so that the breather-breather S-matrix alone satisfies reasonable assumptions. Then we propose candidates for observables in wedge-shaped regions and prove their commutativity in the weak sense.
Recently, large families of two-dimensional quantum field theories with factorizing S-matrices have been constructed by the operator-algebraic methods, by first showing the existence of observables localized in wedge-shaped regions. However, these constructions have been limited to the class of S-matrices whose components are analytic in rapidity in the physical strip. In this work, we construct candidates for observables in wedges for scalar factorizing S-matrices with poles in the physical strip and show that they weakly commute on a certain domain. We discuss some technical issues concerning further developments, especially the self-adjointness of the candidate operators here and strong commutativity between them.
The quantum sine-Gordon model is the simplest massive interacting integrable quantum field theory whose two-particle scattering matrix is generally non-diagonal. As such, it is a model that has been extensively studied, especially in the context of the bootstrap program. In this paper we compute low particle-number form factors of a special local field known as the branch point twist field, whose correlation functions are building blocks for measures of entanglement. We consider the attractive regime where the theory possesses a particle spectrum consisting of a soliton, an antisoliton (of opposite U(1) charges) and several (neutral) breathers. In the breather sector we exploit the fusion procedure to compute form factors of heavier breathers from those of lighter ones. We apply our results to the study of the entanglement dynamics after a small mass quench and for short times. We show that in the presence of two or more breathers the von Neumann and Rényi entropies display undamped oscillations in time, whose frequencies are proportional to the even breather masses and whose amplitudes are proportional to the breather's one-particle form factor.
The purpose of the "bootstrap program" is to construct integrable quantum field theories in 1+1 dimensions in terms of their Wightman functions explicitly. As an input the integrability and general assumptions of local quantum field theories are used. The object is to be achieved in tree steps: 1) The S-matrix is obtained using a qualitative knowledge of the particle spectrum and the Yang-Baxter equations. 2) Matrix elements of local operators are calculated by means of the "form factor program" using the S-matrix as an input. 3) The Wightman functions are calculated by taking sums over intermediate states. The first step has been performed for a large number of models and also the second one for several models. The third step is unsolved up to now. Here the program is illustrated in terms of the sine-Gordon model alias the massive Thirring model. Exploiting the "off-shell" Bethe Ansatz we propose general formulae for form factors. For example the n-particle matrix element for all higher currents are given and in particular all eigenvalues of the higher conserved charges are calculated. Furthermore quantum operator equations are obtained in terms of their matrix elements, in particular the quantum sine-Gordon field equation. Exact expressions for the finite wave function and mass renormalization constants are calculated.
Integrable quantum models (IQMs) are a class of interacting quantum field theories in 1+1 dimensions in which the Hilbert space can be described as a Fock space with generalized statistics. Because of this Fock space description, any observables can be expanded in the creation- and annihilation operators with the coefficients of the expansions being the connected matrix elements. One example of such an observable in IQMs without bound states is the so-called wedge-local field (a free field analog with the interacting creation- and annihilation operators) which is wedge-local in the sense that it strongly commutes with its reflection when smeared with a function localized in the left wedge. In this thesis, we study wedge-local observables in integrable quantum models and the effect of bound states on the properties of the connected matrix elements of these observables. This allows us to shine light on the connection between the localization of an observable and the functional analytic properties of its connected matrix elements in models with critical phenomena, in this case models which exhibit bound states. In scalar integrable quantum models with bound states (e.g. the Bullough-Dodd model), we proof that the connected matrix elements of quadratic forms which satisfy a wedge-locality condition are boundary values of meromorphic functions, so-called form factors, which satisfy a number of properties. Conversely, we show that any expansion in such form factors defines a quadratic form which satisfies the wedge-locality condition. In contrast to models without bound states, such an expansion does not terminate due to a recursion relation between the form factors and the wedge-local field, a wedge-local operator which creates one-particle states from the vacuum, does not fall into the class of the characterization theorem. Further, we conjecture a candidate for the wedge-local field in the Z(4)-Ising model. This model is interesting because it has a ”higher-order” bound state structure and no established wedge-local field is known in such models. However, the conjectured candidate comes with some difficulties. Most notably, it is ill-defined as an unbounded operator.
The purpose of the "bootstrap program" for integrable quantum field theories in 1+1 dimensions is to construct a model in terms of its Wightman functions explicitly. In this article, the program is mainly illustrated in terms of the sine-Gordon and the sinh-Gordon model and (as an exercise) the scaling Ising model. We review some previous results on sine-Gordon breather form factors and quantum operator equations. The problem to sum over intermediate states is attacked in the short distance limit of the two point Wightman function for the sinh-Gordon and the scaling Ising model.
We develop a method for computing form factors of local operators in the framework of Sklyanin's separation of variables (SOV) approach to quantum integrable systems.For that purpose, we consider the sine-Gordon model on a finite lattice and in finite dimensional cyclic representations as our main example.We first build our two central tools for computing matrix elements of local operators, namely, a generic determinant formula for the scalar products of states in the SOV framework and the reconstruction of local fields in terms of the separate variables.The general form factors are then obtained as sums of determinants of finite dimensional matrices, their matrix elements being given as weighted sums running over the separate variables and involving the Baxter Q-operator eigenvalues.