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arXiv:1103.2328 · Journal of Physics A Mathematical and Theoretical

Correlation functions of twist fields from Ward identities in the massive Dirac theory

Benjamin Doyon, James Silk

Published Jun 17, 2011• 5 citations
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Abstract

We derive non-linear differential equations for correlation functions of U (1) twist fields in the two-dimensional massive Dirac theory. Primary U (1) twist fields correspond to exponential fields in the sine-Gordon model at the free-fermion point, and it is well-known that their vacuum two-point functions are determined by integrable differential equations. We extend part of this result to more general quantum states (pure or mixed) and to certain descendents, showing that some two-point functions are determined by the sinh-Gordon differential equations whenever there is translation and parity invariance, and the density matrix is the exponential of a bilinear expression in fermions. We use methods involving Ward identities associated to the copy-rotation symmetry in a model with two independent, anti-commuting copies. Such methods were used in the context of the thermally perturbed Ising quantum field theory model. We show that they are applicable to the Dirac theory as well, and we suggest that they are likely to have a much wider applicability to free fermion models in general. Finally, we note that our form-factor study of descendents twist fields combined with a CFT analysis provides a new way of evaluating vacuum expectation values of primary U (1) twist fields: by deriving and solving a recursion relation.

DOI
10.1088/1751-8113/44/29/295402

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We derive non-linear differential equations for correlation functions of U (1) twist fields in the two-dimensional massive Dirac theory. Primary U (1) twist fields correspond to exponential fields in the sine-Gordon model at the free-fermion point, and it is well-known that their vacuum two-point functions are determined by integrable differential equations. We extend part of this result to more general quantum states (pure or mixed) and to certain descendents, showing that some two-point functions are determined by the sinh-Gordon differential equations whenever there is translation and parity invariance, and the density matrix is the exponential of a bilinear expression in fermions. We use methods involving Ward identities associated to the copy-rotation symmetry in a model with two independent, anti-commuting copies. Such methods were used in the context of the thermally perturbed Ising quantum field theory model. We show that they are applicable to the Dirac theory as well, and we suggest that they are likely to have a much wider applicability to free fermion models in general. Finally, we note that our form-factor study of descendents twist fields combined with a CFT analysis provides a new way of evaluating vacuum expectation values of primary U (1) twist fields: by deriving and solving a recursion relation.
The aim of this thesis is to explore correlation functions in two dimensional quantum field theories in two distinct ways. In part I a new method for calculating the differential equations parametrising the correlation functions of twist fields associated with the U (1) symmetry of the Dirac model is presented. While developing this method a new family of descendent twist fields are identified and their form factors calculated. This provides a novel way of calculating the vacuum expectation values of the primary twist fields and is shown to be entirely consistent with known results. The method of calculating the correlation functions of twist fields provides a parametrisation of several other correlation functions for various quantum states. Since this method relies on the Ward identities found in a double copy model it is hoped to have wider applications in other free fermion models. Part II concerns the truncated conformal space approach which has been developed to approximate perturbed conformal field theories. In this part the theory underpinning the approach is discussed and a working algorithm is developed for both bulk and boundary perturbed minimal models. The energy levels, mass gaps and one point functions of various models are computed using the truncated conformal space approach and are shown to be in good agreement with previous calculations. A possible method for using this approach to approximate two point functions in perturbed conformal field theories is discussed.
Following on from previous work we derive the nonlinear differential equations of more general correlators of U (1) twist fields in two-dimensional massive Dirac theory. Using the conserved charges of the double-copy model equations parametrizing the correlators of twist fields with arbitrary twist parameter are found. This method also gives a parametrization of the correlation functions of general, fermionic, descendent twist fields. The equations parametrizing correlators of primary twist fields are compared to those of the literature and evidence is presented to confirm that these equations represent the correct parametrization.
We study U (1) twist fields in a two-dimensional lattice theory of massive Dirac fermions. Factorized formulas for finite-lattice form factors of these fields are derived using elliptic parametrization of the spectral curve of the model, elliptic determinant identities and theta functional interpolation. We also investigate the thermodynamic and infinite-volume scaling limit, where the corresponding expressions reduce to form factors of the exponential fields of the sine-Gordon model at the free-fermion point.
Abstract Using the ‘Liouville space’ (the space of operators) of the massive Dirac theory, we define mixed-state form factors of U (1) twist fields. We consider mixed states with density matrices diagonal in the asymptotic particle basis. This includes the thermal Gibbs state as well as all generalized Gibbs ensembles of the Dirac theory. When the mixed state is specialized to a thermal Gibbs state, using a Riemann–Hilbert problem and low-temperature expansion, we obtain finite-temperature form factors of U (1) twist fields. We then propose the expression for form factors of U (1) twist fields in general diagonal mixed states. We verify that these form factors satisfy a system of nonlinear functional differential equations, which is derived from the trace definition of mixed-state form factors. At last, under weak analytic conditions on the eigenvalues of the density matrix, we write down the large distance form factor expansions of two-point correlation functions of these twist fields. Using the relation between the Dirac and Ising models, this provides the large-distance expansion of the Rényi entropy (for integer Rényi parameter) in the Ising model in diagonal mixed states.
Abstract The evaluation of vacuum expectation values (VEVs) in massive integrable quantum field theory (QFT) is a nontrivial renormalization-group ‘connection problem’—relating large and short distance asymptotics—and is in general unsolved. This is particularly relevant in the context of entanglement entropy, where VEVs of branch-point twist fields give universal saturation predictions. We propose a new method to compute VEVs of twist fields associated to continuous symmetries in QFT. The method is based on a differential equation in the continuous symmetry parameter, and gives VEVs as infinite form-factor series which truncate at two-particle level in free QFT. We verify the method by studying U ( 1 ) twist fields in free models, which are simply related to the branch-point twist fields. We provide the first exact formulae for the VEVs of such fields in the massive uncompactified free boson model, checking against an independent calculation based on angular quantization. We show that logarithmic terms, overlooked in the original work of Callan and Wilczek (1994 Phys. Lett . B 333 55–61), appear both in the massless and in the massive situations. This implies that, in agreement with numerical form-factor observations by Bianchini and Castro-Alvaredo (2016 Nucl. Phys . B 913 879–911), the standard power-law short-distance behavior is corrected by a logarithmic factor. We discuss how this gives universal formulae for the saturation of entanglement entropy of a single interval in near-critical harmonic chains, including log log corrections.
We investigate a particular two-point function of the n -copy Ising model. That is, the correlation function involving the energy field and the branch-point twist field. The latter is associated with the symmetry of the theory under cyclic permutations of its copies. We use a form factor expansion to obtain an exact integral representation of and find its complete short-distance expansion. This allows us to identify all the fields contributing in the short-distance massive OPE of the correlation function under examination, and fix their expectation values, conformal structure constants and massive corrections thereof. Most contributions are given by the composite field and its derivatives. We find all non-vanishing form factors of this latter operator.