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arXiv:1002.3387 · Journal of Statistical Mechanics Theory and Experiment

Bethe ansatz matrix elements as non-relativistic limits of form factors of quantum field theory

Márton Kormos, Giuseppe Mussardo, Balázs Pozsgay

Published May 20, 2010• 26 citations
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Abstract

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.

DOI
10.1088/1742-5468/2010/05/p05014

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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 study integrable models solvable by the nested algebraic Bethe ansatz and possessing GL(3)-invariant R-matrix. Assuming that the monodromy matrix of the model can be expanded into series with respect to the inverse spectral parameter, we define zero modes of the monodromy matrix entries as the first nontrivial coefficients of this series. Using these zero modes we establish new relations between form factors of the elements of the monodromy matrix. We prove that all of them can be obtained from the form factor of a diagonal matrix element in special limits of Bethe parameters. As a result we obtain determinant representations for form factors of all the entries of the monodromy matrix.
We investigate form factors of local operators in the multi-component Quantum Non-linear Schrödinger model, a prototype theory solvable by the so-called nested Bethe Ansatz. We determine the analytic properties of the infinite volume form factors using the coordinate Bethe Ansatz solution and we establish a connection with the finite volume matrix elements. In the two-component models we derive a set of recursion relations for the "magnonic form factors", which are the matrix elements on the nested Bethe Ansatz states. In certain simple cases (involving states with only one spin-impurity) we obtain explicit solutions for the recursion relations.
The linear problem associated with the Lax operator of the classical sine-Gordon theory can be recast into the monodromy matrix form that can be extended to quantum theory as well. Product of the quantum monodromy matrices will have contributions from the singularities arising out of the operator product expansions of sine-Gordon field. This enables one to find the star-triangle relations. This is a generalization of the method used by Thacker for the non-relativistic nonlinear Schrödinger field theory. In the infinite volume limit, it leads to an unambiguous description of the algebra involving the scattering data operators. Starting from a vacuum the module of physical states are constructed by the application of chains of the scattering operators and they turn out to have definite eigenvalues of energy and momentum. The mass-spectra and two-particle S matrix elements can be calculated straightforwardly and they coincide with the well-known results of Zamolodchikov and Zamolodchikov. It is also shown that the particle states can be created by a generalized Jordon-Wigner transformation of the free fermionic creation operators — thus establishing a link between the S matrix element and certain co-cycles.
We review the recent results we have obtained in the framework of algebraic Bethe ansatz based on algebras and superalgebras of rank greater than 1 or on their quantum deformation. We present different expressions (explicit, recursive or using the current realization of the algebra) for the Bethe vectors. Then, we provide a general expression (as sum over partitions) for their scalar products. For some particular cases (in the case of gl(3) g l ( 3 ) or its quantum deformation, or of gl(2|1) g l ( 2 | 1 ) ), we provide determinant expressions for the scalar products. We also compute the form factors of the monodromy matrix entries, and give some general methods to relate them. A coproduct formula for Bethe vectors allows to get the form factors of composite models.
The purpose of the "bootstrap program" for integrable quantum field theories in 1+1 dimensions is to construct explicitly a model in terms of its Wightman functions.In this article, this program is mainly illustrated in terms of the sinh-Gordon model and the SU (N ) Gross-Neveu model.The nested off-shell Bethe ansatz for an SU (N ) factorizing S-matrix is constructed.We review some previous results on sinh-Gordon form factors and the quantum operator field equation.The problem of how to sum over intermediate states is considered in the short distance limit of the two point Wightman function for the sinh-Gordon model.