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arXiv:1412.8442 · Journal of High Energy Physics

Form factor relocalisation and interpolating renormalisation group flows from the staircase model

Patrick Dorey, Guy Siviour, Gábor Takács

Published Mar 1, 2015• 6 citations
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Abstract

We investigate the staircase model, introduced by Aliosha Zamolodchikov through an analytic continuation of the sinh-Gordon S-matrix to describe interpolating flows between minimal models of conformal field theory in two dimensions. Applying the form factor expansion and the c-theorem, we show that the resulting c-function has the same physical content as that found by Zamolodchikov from the thermodynamic Bethe Ansatz. This turns out to be a consequence of a nontrivial underlying mechanism, which leads to an interesting localisation pattern for the spectral integrals giving the multi-particle contributions. We demonstrate several aspects of this form factor relocalisation, which suggests a novel approach to the construction of form factors and spectral sums in integrable renormalisation group flows with non-diagonal scattering.

DOI
10.1007/jhep03(2015)054

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We investigate the staircase model, introduced by Aliosha Zamolodchikov through an analytic continuation of the sinh-Gordon S-matrix to describe interpolating flows between minimal models of conformal field theory in two dimensions. Applying the form factor expansion and the c-theorem, we show that the resulting c-function has the same physical content as that found by Zamolodchikov from the thermodynamic Bethe Ansatz. This turns out to be a consequence of a nontrivial underlying mechanism, which leads to an interesting localisation pattern for the spectral integrals giving the multi-particle contributions. We demonstrate several aspects of this form factor relocalisation, which suggests a novel approach to the construction of form factors and spectral sums in integrable renormalisation group flows with non-diagonal scattering.
We investigate the staircase model, introduced by Aliosha Zamolodchikov through an analytic continuation of the sinh-Gordon S-matrix to describe interpolating flows between minimal models of conformal field theory in two dimensions. Applying the form factor expansion and the c-theorem, we show that the resulting c-function has the same physical content as that found by Zamolodchikov from the thermodynamic Bethe Ansatz. This turns out to be a consequence of a nontrivial underlying mechanism, which leads to an interesting localisation pattern for the spectral integrals giving the multi-particle contributions. We demonstrate several aspects of this form factor relocalisation, which suggests a novel approach to the construction of form factors and spectral sums in integrable renormalisation group flows with non-diagonal scattering.
We study the massless flows described by the staircase model introduced by Al.B. Zamolodchikov through the analytic continuation of the sinh-Gordon S-matrix, focusing on the renormalisation group flow from the tricritical to the critical Ising model. We show that the properly defined roaming limits of certain sinh-Gordon form factors are identical to the form factors of the order and disorder operators for the massless flow. As a by-product, we also construct form factors for a semi-local field in the sinh-Gordon model, which can be associated with the twist field in the ultraviolet limiting free massless bosonic theory.
This thesis explores renormalisation group flows in integrable quantum field theories with boundaries, as described by the g-function. The main focus is on the g-function in the staircase model, the renormalisation group flow of which passes close to the unitary minimal models. This g-function is used to identify flows between boundary conditions both within and between the minimal models. In certain limits the mathcalMAm(+)\\mathcal{M}A_m^{(+)}mathcalMAm(+)​ theories which interpolate between pairs of minimal models emerge from the staircase model, and exact expressions for the g-function in these models are extracted from the staircase g-function. Perturbative tests on the mathcalMA4(+)\\mathcal{M}A_4^{(+)}mathcalMA4(+)​ g-function are discussed, as is initial work on the g-function for the mathcalMA4(−)\\mathcal{M}A_4^{(-)}mathcalMA4(−)​ theory, which describes flows that emerge when the bulk coupling is taken to have the opposite sign to that in mathcalMA4(+)\\mathcal{M}A_4^{(+)}mathcalMA4(+)​. Expressions are also found for excited state versions of the mathcalMAm(+)\\mathcal{M}A_m^{(+)}mathcalMAm(+)​ g-function, and these allow the unique identification of certain boundary flows.
Abstract The staircase model (SM) is a simple generalization of the sinh-Gordon model, obtained by complexifying the coupling constant. This produces a new theory with many interesting features. Chief among them is the fact that scaling functions such as Zamolodchikov’s c -function display ‘roaming’ behaviour, that is, they visit the vicinity of an infinite number of conformal fixed points, the unitary minimal models of conformal field theory. This rich structure also makes the model an interesting candidate for study using the generalized hydrodynamic approach to integrable quantum field theory (IQFT). By studying hydrodynamic quantities such as the effective velocities of quasiparticles we can develop a more physical picture of interaction in the theory, both at and away from equilibrium. Indeed, we find that in the SM the effective velocity displays monotonicity features not found in any other IQFT. These admit a natural interpretation when investigated in the context of the massless theories known as M A k ( + ) models, which provide an effective description of massless flows between consecutive unitary minimal models. Consequently, we find that the effective velocity in the SM can be reconstructed by a ‘cut and paste’ procedure involving the equivalent functions associated to the massless excitations of suitable M A k ( + ) models. We also investigate the average currents and densities of higher spin conserved quantities in the partitioning protocol.
In this paper we compute higher particle form factors of branch point twist fields. These fields were first described in the context of massive (1+1)-dimensional integrable quantum field theories and their correlation functions are related to the bi-partite entanglement entropy. We find analytic expressions for some form factors and check those expressions for consistency, mainly by evaluating the conformal dimension of the corresponding twist field in the underlying conformal field theory. We find that solutions to the form factor equations are not unique so various techniques need to be used to identify those corresponding to the branch point twist field we are interested in. The models for which we carry out our study are characterized by staircase patterns of various physical quantities as functions of the energy scale. As the latter is varied, the β-function associated with these theories comes close to vanishing at several points between the deep infrared and deep ultraviolet regimes. In other words, renormalization group flows approach the vicinity of various critical points before ultimately reaching the ultraviolet fixed point. This feature provides an optimal way of checking the consistency of higher particle form factor solutions, as the changes on the conformal dimension of the twist field at various energy scales can only be accounted for by considering higher particle form factor contributions to the expansion of certain correlation functions.