On the form factors of local operators in the lattice sine–Gordon model

Grosjean, Nicolas; Maillet, Jean Michel; Niccoli, Giuliano

doi:10.1088/1742-5468/2012/10/p10006

Cited by 39 publications

(87 citation statements)

References 142 publications

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“…Beyond the light-cone lattice approach of [3], in the literature there exists another integrable lattice regularization 27 for the sine-Gordon model [16,56]. Though, in the framework of this approach local operators [57,58] and their form factors [58] have been computed on the lattice, the continuum results are still missing. It would be also very interesting to see, whether this approach also allows one to compute diagonal matrix elements of local operators of the continuum theory.…”

Section: Discussionmentioning

confidence: 99%

Lattice approach to finite volume form-factors of the Massive Thirring (Sine-Gordon) model

Hegedűs

2017

J. High Energ. Phys.

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Abstract:In this paper we demonstrate, that the light-cone lattice approach for the Massive-Thirring (sine-Gordon) model, through the quantum inverse scattering method, admits an appropriate framework for computing the finite volume form-factors of local operators of the model. In this work we compute the finite volume diagonal matrix elements of the U(1) conserved current in the pure soliton sector of the theory. Based on the systematic large volume expansion of our results, we conjecture an exact expression for the finite volume expectation values of local operators in pure soliton states. At large volume in leading order these expectation values have the same form as in purely elastic scattering theories, but exponentially small corrections differ from previous Thermodynamic Bethe Ansatz conjectures of purely elastic scattering theories.

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Section: Discussionmentioning

confidence: 99%

Lattice approach to finite volume form-factors of the Massive Thirring (Sine-Gordon) model

Hegedűs

2017

J. High Energ. Phys.

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“…This is made possible by considering integrable lattice models with generic representations in each lattice site, meaning the presence of inhomogeneity parameters in generic positions.However, to be fully successful, the program of computing scalar products, form factors and then correlation functions [110,111,[117][118][119][120][121][122][123][124], needs determinant formulas for which the homogeneous and then the large volume limit can be tackled explicitly as in [122]. Unfortunately, it became clear from the results of [94], that although the obtained dressed Vandermonde determinants were easy to derive and appear quite…”

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confidence: 99%

The open XXZ spin chain in the SoV framework: scalar product of separate states

Kitanine¹,

Maillet²,

Niccoli³

et al. 2018

J. Phys. A: Math. Theor.

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In our previous paper [1] we have obtained, for the XXX spin-1/2 Heisenberg open chain, new determinant representations for the scalar products of separate states in the quantum separation of variables (SoV) framework. In this article we perform a similar study in a more complicated case: the XXZ open spin-1/2 chain with the most general integrable boundary terms. To solve this model by means of SoV we use an algebraic Vertex-IRF gauge transformation reducing one of the boundary K-matrices to a diagonal form. As usual within the SoV approach, the scalar products of separate states are computed in terms of dressed Vandermonde determinants having an intricate dependency on the inhomogeneity parameters. We show that these determinants can be transformed into different ones in which the homogeneous limit can be taken straightforwardly. These representations generalize in a non-trivial manner to the trigonometric case the expressions found previously in the rational case. We also show that generically all scalar products can be expressed in a form which is similar to -although more cumbersome than -the well-known Slavnov determinant representation for the scalar products of the Bethe states of the periodic chain. Considering a special choice of the boundary parameters relevant in the thermodynamic limit to describe the half infinite chain with a general boundary, we particularize these representations to the case of one of the two states being an eigenstate. We obtain simplified formulas that should be of direct use to compute the form factors and correlation functions of this model. The resolution of quantum lattice models with the most general boundary conditions preserving integrability properties has become along the years a subject of intense studies . This interest is due, in particular, to their potential relevance for the investigation of the non-equilibrium and transport properties of quantum integrable systems [71][72][73][74][75][76][77][78]. Among several different approaches to this problem, the separation of variables (SoV) method [9,[79][80][81] has proven to produce cutting edge answers to give the possibility to construct the full set of eigenvalues and eigenstates of the Hamiltonians, while providing the first steps towards the computation of the form factors and correlation functions [1,.We would like to mention that numerous other methods were used in this context, including algebraic Bethe ansatz and its modifications, analytic Bethe ansatz, Baxter T -Q equation and q-Onsager algebras. In our recent articles [1, 102, 103] we extensively discussed their connection to our approach and their possible relevance for the computation of form factors and correlation functions.Looking for the exact description of the dynamics of quantum integrable lattice models, the determinant representations for the scalar products of states, including eigenstates of the Hamiltonian, plays a prominent role, as it was clearly demonstrated in the Algebraic Bethe ansatz framework, see e.g., [105][106][107] using the determ...

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“…Expressions for the norms or correlation functions for various models solvable by the quantum separation of variables method have been established, e.g. in the works [9,59,60,82,106,105,139,146]. The expressions obtained there are either directly of the form given above or are amenable to this form (with, possibly, a change of the integration contour from R N to C N , with C a curve in C) upon elementary manipulations.…”

Section: An Opening Discussionmentioning

confidence: 99%

“…The method takes its roots in the 1985 work of Sklyanin [137] and applies to a wide range of lattice quantum integrable models such as spin chains [76,99,122,123], lattice discretisations of quantum field theories in 1+1 dimensions [35,82,126] or multi-particle quantum Hamiltonians [94,95,137]. We will outline the main ideas of the method on the example of the open quantum Toda chain Hamiltonian with (N + 1)-particles [10]:…”

Section: The Quantum Separation Of Variables For the Toda Chainmentioning

confidence: 99%

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Asymptotic Expansion of a Partition Function Related to the Sinh-model

Borot¹,

Guionnet²,

Kozłowski³

2016

Mathematical Physics Studies

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This paper develops a method to carry out the large-N asymptotic analysis of a class of Ndimensional integrals arising in the context of the so-called quantum separation of variables method. We push further ideas developed in the context of random matrices of size N, but in the present problem, two scales 1/N α and 1/N naturally occur. In our case, the equilibrium measure is N α -dependent and characterised by means of the solution to a 2×2 Riemann-Hilbert problem, whose large-N behaviour is analysed in detail. Combining these results with techniques of concentration of measures and an asymptotic analysis of the Schwinger-Dyson equations at the distributional level, we obtain the large-N behaviour of the free energy explicitly up to o(1). The use of distributional Schwinger-Dyson is a novelty that allows us treating sufficiently differentiable interactions and the mixing of scales 1/N α and 1/N, thus waiving the analyticity assumptions often used in random matrix theory.1 gborot@mpim-bonn.mpg.de 2 guionnet@math.mit.edu 3 karol.kozlowski@ens-lyon.fr 2 An opening discussionThe present work develops techniques enabling one to carry out the large-N asymptotic analysis of a class of multiple integrals that arise as representations for the correlation functions in quantum integrable models solvable by the quantum separation of variables. We shall refer to the general class of such integrals as the sinh model: [9,59,60,82,106,105,139,146]. The expressions obtained there are either directly of the form given above or are amenable to this form (with, possibly, a change of the integration contour from R N to C N , with C a curve in C) upon elementary manipulations. Furthermore, a degeneration of z N [W] arises as a multiple integral representation for the partition function of the six-vertex model subject to domain wall boundary conditions [93]. In the context of quantum integrable systems, the number N of integrals defining z N is related to the number of sites in a model (as, e.g. in the case of the compact or non-compact XXZ chains or the lattice regularisations of the Sinh or Sine-Gordon models) or to the number of particles (as, e.g. in the case of the quantum Toda chain). From the point of view of applications, one is mainly interested in the thermodynamic limit of the model, which is attained by sending N to +∞. For instance, in the case of an integrable lattice discretisation of some quantum field theory, one obtains in this way an exact and non-perturbative description of a quantum field theory in 1 + 1 dimensions and in finite volume. This limit, at the level of z N [W], translates itself in the need to extract the large N-asymptotic expansion of ln z N [W] up to o(1). It is, in fact, the constant term in the expansion of ln zwith W ′ some deformation of W that gives rise to the correlation functions of the underlying quantum field theory in finite volume. These applications to physics constitute the first motivation for our analysis. From the purely mathematical side, the motivation of our works stems ...

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On the form factors of local operators in the lattice sine–Gordon model

Cited by 39 publications

References 142 publications

Lattice approach to finite volume form-factors of the Massive Thirring (Sine-Gordon) model

Lattice approach to finite volume form-factors of the Massive Thirring (Sine-Gordon) model

The open XXZ spin chain in the SoV framework: scalar product of separate states

Asymptotic Expansion of a Partition Function Related to the Sinh-model

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