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arXiv:1604.05635 · Journal of High Energy Physics

Roaming form factors for the tricritical to critical Ising flow

Dávid X. Horváth, Patrick Dorey, G. Takács

Published Jul 1, 2016• 9 citations
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Abstract

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.

DOI
10.1007/jhep07(2016)051

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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.
A bstract We study the Rényi entanglement entropies along the massless renormalisation group flow that connects the tricritical and critical Ising field theories. Similarly to the massive integrable field theories, we derive a set of bootstrap equations, from which we can analytically calculate the twist field form factors in a recursive way. Additionally, we also obtain them as a non-trivial ‘roaming limit’ of the sinh-Gordon theory. Then the Rényi entanglement entropies are obtained as expansions in terms of the form factors of these branch point twist fields. We find that the form factor expansion of the entanglement entropy along the flow organises in two different kind of terms. Those that couple particles with the same chirality, and reproduce the entropy of the infrared Ising theory, and those that couple particles with different chirality, which provide the ultraviolet contributions. The massless flow under study possesses a global ℤ 2 spin-flip symmetry. We further consider the composite twist fields associated to this group, which enter in the study of the symmetry resolution of the entanglement. We derive analytical expressions for their form factors both from the bootstrap equations and from the roaming limit of the sinh-Gordon theory.
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.
Abstract We study Euler scale hydrodynamics of massless integrable quantum field theories interpolating between two non-trivial renormalisation group fixed points after inhomogeneous quantum quenches. Using a partitioning protocol with left and right initial thermal states and the recently developed framework of generalised hydrodynamics, we focus on current and density profiles for the energy and momentum as a function of ξ = x/t, where both x and t are sent to infinity. Studying the first few members of the A n and D n massless flows we carry out a systematic treatment of these series and generalise our results to other unitary massless models. In our analysis we find that the profiles exhibit extended plateaux and that non-trivial bounds exist for the energy and momentum densities and currents in the non-equilibrium stationary state, i.e. when ξ = 0. To quantify the magnitude of currents and densities, dynamical central charges are defined and it is shown that the dynamical central charge for the energy current satisfies a certain monotonicity property. We discuss the connection of the Landauer-Büttiker formalism of transport with our results and show that this picture can account for some of the bounds for the currents and for the monotonicity of the dynamical central charge. These properties are shown to be present not only in massless flows but also in the massive sinh-Gordon model suggesting their general validity and the correctness of the Landauer-Büttiker interpretation of transport in integrable field theories. Our results thus imply the existence of a non-equilibrium c-theorem as well, at least in integrable models. Finally we also study the interesting low energy behaviour of the A2 model that corresponds to the massless flow from the tricritical to the critical Ising field theory.