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arXiv:1705.00319 · Journal of High Energy Physics

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

Árpád Hegedűs

Published Aug 1, 2017• 6 citations
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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.

DOI
10.1007/jhep08(2017)059

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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.
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.
In this paper we present sets of linear integral equations which make possible to compute the finite volume expectation values of the trace of the stress energy tensor (Θ) and the U ( 1 ) current ( J μ ) in any eigenstate of the Hamiltonian of the sine-Gordon model. The solution of these equations in the large volume limit allows one to get exact analytical formulas for the expectation values in the Bethe-Yang limit. These analytical formulas are used to test an earlier conjecture for the Bethe-Yang limit of expectation values in non-diagonally scattering theories. The analytical tests have been carried out upto three particle states and gave agreement with the conjectured formula, provided the definition of polarized symmetric diagonal form-factors is modified appropriately. Nevertheless, we point out that our results provide only a partial confirmation of the conjecture, and further investigations are necessary to confirm all of its subtleties. The most important missing piece in the confirmation is the mathematical proof of the finiteness of the symmetric diagonal limit of form-factors in a non-diagonally scattering theory.
The 6-vertex model with appropriately chosen alternating inhomogeneities gives the so-called light-cone lattice regularization of the sine-Gordon (Massive-Thirring) model. In this integrable lattice model we consider pure hole states above the antiferromagnetic vacuum and express the norm of Bethe-wave functions in terms of the hole's positions and the counting-function of the state under consideration. In the light-cone regularized picture pure hole states correspond to pure soliton (fermion) states of the sine-Gordon (massive Thirring) model. Hence, we analyze the continuum limit of our new formula for the norm of the Bethe-wave functions. We show, that the physically most relevant determinant part of our formula can be expanded in the large volume limit and turns out to be proportional to the Gaudin-determinant of pure soliton states in the sine-Gordon model defined in finite volume.
Using the methods of the "form factor program" exact expressions of all matrix elements are obtained for several operators of the quantum sine Gordon model: all powers of the fundamental bose field, general exponentials of it, the energy momentum tensor and all higher currents. It is found that the quantum sineGordon field equation holds with an exact relation between the "bare" mass and the renormalized mass. Also a relation for the trace of the energy momentum is obtained. The eigenvalue problem for all higher conserved charges is solved. All results are compared with Feynman graph expansions and full agreement is found. PACS: 11.10.-z; 11.10.Kk; 11.55.Ds Keywords: Integrable quantum field theory, Form factors The classical sine-Gordon model is given by the wave equation 2'(t; x) + ff fi sin fi'(t; x) = 0: Since Coleman [1] found the wonderful duality between the quantum sine-Gordon and the massive Thirring model a lot of effort has been made to understand this quantum field theo...
We consider diagonal matrix elements of local operators between multisoliton states in finite volume in the sine-Gordon model and formulate a conjecture regarding their finite size dependence which is valid up to corrections exponential in the volume. This conjecture extends the results of Pozsgay and Tak\'acs which were only valid for diagonal scattering. In order to test the conjecture, we implement a numerical renormalization group improved truncated conformal space approach. The numerical comparisons confirm the conjecture, which is expected to be valid for general integrable field theories. The conjectured formula can be used to evaluate finite temperature one-point and two-point functions using recently developed methods.
The light-cone approach is reviewed. This method allows to find the underlying quantum field theory for any integrable lattice model in its gapless regime. The relativistic spectrum and S-matrix follows straightforwardly in this way through the Bethe Ansatz. We show here how to derive the infinite number of local commuting and non-local and non-commuting conserved charges in integrable QFT, taking the massive Thirring model (sine-Gordon) as an example. They are generated by quantum monodromy operators and provide a representation of q−q-q−deformed affine Lie algebras $U_q({\hat\G})$. Based on lectures delivered at the XXXqXXX_qXXXq​ Karpacz Winter School, Poland, February 14-26, 1994.