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arXiv:1602.05735 · Journal of Physics A Mathematical and Theoretical

A note on four-particle form factors of operators T 2 n T − 2 n in the sinh-Gordon model

Michael Lashkevich, Yaroslav Pugai

Published Jun 24, 2016• 5 citations
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Abstract

Abstract The diagonal matrix elements 〈 θ 1 , θ 2 ∣ T 2 n T − 2 n ∣ θ 1 , θ 2 〉 between two-particle states in the sinh-Gordon model are computed analytically for all integers n > 0 . This confirms the proposal [1] by Smirnov and Zamolodchikov for these matrix elements and demonstrates the effectiveness of the algebraic approach to form factors.

DOI
10.1088/1751-8113/49/30/305401

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Abstract The diagonal matrix elements 〈 θ 1 , θ 2 ∣ T 2 n T − 2 n ∣ θ 1 , θ 2 〉 between two-particle states in the sinh-Gordon model are computed analytically for all integers n > 0 . This confirms the proposal [1] by Smirnov and Zamolodchikov for these matrix elements and demonstrates the effectiveness of the algebraic approach to form factors.
We show that the matrix elements of integrable models computed by the Algebraic Bethe Ansatz can be put in direct correspondence with the Form Factors of integrable relativistic field theories. This happens when the S-matrix of a Bethe Ansatz model can be regarded as a suitable non-relativistic limit of the S-matrix of a field theory, and when there is a well-defined mapping between the Hilbert spaces and operators of the two theories. This correspondence provides an efficient method to compute matrix elements of Bethe Ansatz integrable models, overpassing the technical difficulties of their direct determination. We analyze this correspondence for the simplest example in which it occurs, i.e. the Quantum Non-Linear Schrodinger and the Sinh-Gordon models.
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.
Abstract At the heart of a quantum field theory are the correlation functions of the various fields. In the case of integrable models, the correlators can be expressed in terms of the spectral series based on the matrix elements on the asymptotic states. These matrix elements, also known as form factors, satisfy a set of functional and recursive equations that can exactly solved in many cases of physical interest. Chapter 19 covers general properties of form factors, Faddeev–Zamolodchikov algebra, symmetric polynomials, kinematical and bound state poles, the operator space and kernel functions, the stress-energy tensor and vacuum expectation values and the Ising model in a magnetic field.
We compare two different methods of computing form factors. One is the well established procedure of solving the form factor consistency equations and the other is to represent the field content as well as the particle creation operators in terms of fermionic Fock operators. We compute the corresponding matrix elements for the complex free fermion and the Federbush model. The matrix elements only satisfy the form factor consistency equations involving anyonic factors of local commutativity when the corresponding operators are local. We carry out the ultraviolet limit, analyze the momentum space cluster properties and demonstrate how the Federbush model can be obtained from the SU(3)3-homogeneous sine-Gordon model. We propose a new class of Lagrangians which constitute a generalization of the Federbush model in a Lie algebraic fashion. For these models we evaluate the associated scattering matrices from first principles, which can alternatively also be obtained in a certain limit of the homogeneous sine-Gordon models. Based on talks presented at the conferences: “From QCD to integrable models, old results and new developments”