Gesualdo Delfino

Scattering theory and correlation functions in statistical models with a line of defect

Abstract The scattering theory of the integrable statistical models can be generalized to the case of systems with extended lines of defect. This is done by adding the reflection and transmission amplitudes for the interactions with the line of inhomogeneity to the scattering amplitudes in the bulk. The factorization condition for the new amplitudes gives rise to a set of reflection-transmission equations. The solutions of these equations in the case of a diagonal S -matrix in the bulk are only those with S = ±1. The choice S = −1 corresponds to the Ising model. We compute the exact expressions of the transmission and reflection amplitudes relative to the interaction of the Majorana fermion of the Ising model with the defect. These amplitudes present a weak-strong duality in the coupling constant, the self-dual points being the special values where the defect line acts as a reflecting surface. We also discuss the bosonic case S = 1 which presents instability properties and resonance states. Multi-defect systems which may give rise to a band structure are also considered. The exact expressions of correlation functions is obtained in terms of form factors of the bulk theory and matrix elements of the defect operator.

Interface localization near criticality

A bstractThe theory of interface localization in near-critical planar systems at phase coexistence is formulated from first principles. We show that mutual delocalization of two interfaces, amounting to interfacial wetting, occurs when the bulk correlation length critical exponent ν is larger than or equal to 1. Interaction with a boundary or defect line involves an additional scale and a dependence of the localization strength on the distance from criticality. The implications are particularly rich in the boundary case, where delocalization proceeds through different renormalization patterns sharing the feature that the boundary field becomes irrelevant in the delocalized regime. The boundary delocalization (wetting) transition is shown to be continuous, with surface specific heat and layer thickness exponents which can take values that we determine.

Non-integrable aspects of the multi-frequency sine-Gordon model☆

Abstract We consider the two-dimensional quantum field theory of a scalar field self-interacting via two periodic terms of frequencies α and β. Looking at the theory as a perturbed sine-Gordon model, we use form factor perturbation theory to analyse the evolution of the spectrum of particle excitations. We show how, within this formalism, the non-locality of the perturbation with respect to the solitons is responsible for their confinement in the perturbed theory. The effects of the frequency ratio α/β being a rational or irrational number and the occurrence of massless flows from the gaussian to the Ising fixed point are also discussed. A generalisation of the Ashkin-Teller model and the massive Schwinger model are presented as examples of application of the formalism.

Statistical models with a line of defect

Abstract The factorization condition for the scattering amplitudes of an integrable model with a line of defect gives rise to a set of Reflection-Transmission equations. The solutions of these equations in the case of diagonal S-matrix in the bulk are only those 4with S = ±1. The choice S = −1 corresponds to the Ising model. We compute the transmission and reflection amplitudes relative to the interaction of the Majorana fermion with the defect and we discuss their relevant features.

Integrable field theory and critical phenomena. The Ising model in a magnetic field

The two-dimensional Ising model is the simplest model of statistical mechanics exhibiting a second-order phase transition. While in absence of magnetic field it is known to be solvable on the lattice since Onsagers work of the 1940s, exact results for the magnetic case have been missing until the late 1980s, when A Zamolodchikov solved the model in a field at the critical temperature, directly in the scaling limit, within the framework of integrable quantum field theory. In this paper, we review this field theoretical approach to the Ising universality class, with particular attention to the results obtained starting from Zamolodchikovs scattering solution and to their comparison with the numerical estimates on the lattice. The topics discussed include scattering theory, form factors, correlation functions, universal amplitude ratios and perturbations around integrable directions. Although we restrict our discussion to the Ising model, the emphasis is on the general methods of integrable quantum field theory which can be used in the study of all universality classes of critical behaviour in two dimensions.

Decay of particles above threshold in the Ising field theory with magnetic field

Abstract The two-dimensional scaling Ising model in a magnetic field at critical temperature is integrable and possesses eight stable particles A i ( i = 1 , … , 8 ) with different masses. The heaviest five lie above threshold and owe their stability to integrability. We use form factor perturbation theory to compute the decay widths of the first two particles above threshold when integrability is broken by a small deviation from the critical temperature. The lifetime ratio t 4 / t 5 is found to be 0.233; the particle A 5 decays at 47% in the channel A 1 A 1 and for the remaining fraction in the channel A 1 A 2 . The increase of the lifetime with the mass, a feature which can be expected in two dimensions from phase space considerations, is in this model further enhanced by the dynamics.

Quantum quenches with integrable pre-quench dynamics

We consider the unitary time evolution of a one-dimensional quantum system which is in a stationary state for negative times and then undergoes a sudden change (quench) of a parameter of its Hamiltonian at t = 0. For systems possessing a continuum limit described by a massive quantum field theory we investigate in general perturbative quenches for the case in which the theory is integrable before the quench.

Two-point correlation function in integrable QFT with anti-crossing symmetry

Abstract The two point correlation function of the stress-energy tensor for the φ1,3 massive deformation of the non-unitary model 3,5 is computed. We compare the ultraviolet CFT perturbative expansion of this correlation function with its spectral representation given by a summation over matrix elements of the intermediate asymptotic massive particles. The fast rate of convergence of both approaches provides an explicit example of an accurate interpolation between the infrared and ultraviolet behaviours of a Quantum Field Theory.

The particle spectrum of the three-state Potts field theory: a numerical study

The three-state Potts field theory in two dimensions with thermal and magnetic perturbations provides the simplest model of confinement allowing for both mesons and baryons, as well as for an extended phase with deconfined quarks. We study numerically the evolution of the mass spectrum of this model over its whole parameter range, obtaining a pattern of confinement, particle decay and phase transitions which confirms recent predictions.

Form factors of descendant operators in the massive Lee–Yang model

The form factors of the descendant operators in the massive Lee–Yang model are determined up to level 7. This is first done by exploiting the conserved quantities of the integrable theory to generate the solutions for the descendants starting from the lowest non-trivial solutions in each operator family. We then show that the operator space generated in this way, which is isomorphic to the conformal one, coincides, level by level, with that implied by the S-matrix through the form factor bootstrap. The solutions that we determine satisfy asymptotic conditions carrying the information about the level that we conjecture to hold for all the operators of the model.