Integrable Quantum Field Theories Research Articles

Quantization of three-wave equations

The subject of this paper is the revision of discretization and quantization of two similar classical integrable Hamiltonian systems in three-dimensional spacetime: the standard three-wave equations and the ‘modified’ three-wave equations (the Darboux–Manakov–Zakharov system in two distinguished gauges). The quantized systems in discrete spacetime may be understood as the regularized integrable quantum field theories. Integrability of the theories, and in particular the quantum tetrahedron equations for vertex operators, follow from the quantum auxiliary linear problems. The principal object of the lattice field theories is the Heisenberg discrete-time evolution operator constructed with the help of vertex operators.

Form factors in finite volume II: Disconnected terms and finite temperature correlators

Continuing the investigation started in a previous work, we consider form factors of integrable quantum field theories in finite volume, extending our investigation to matrix elements with disconnected pieces. Numerical verification of our results is provided by truncated conformal space approach. Such matrix elements are important in computing finite temperature correlation functions, and we give a new method for generating a low temperature expansion, which we test for the one-point function up to third order.

Non-equilibrium dynamics of spin facilitated glass models

We consider the dynamics of spin facilitated models of glasses in the non-equilibriumageing regime following a sudden quench from high to low temperatures. We briefly reviewknown results obtained for the broad class of kinetically constrained models, and thenpresent new results for the behaviour of the one-spin facilitated Fredrickson–Andersen andEast models in various spatial dimensions. The time evolution of one-time quantities, suchas the energy density, and the detailed properties of two-time correlation and responsefunctions are studied using a combination of theoretical approaches, including exactmappings of master operators and reductions to integrable quantum spin chains,field theory and renormalization group, and independent interval and timescaleseparation methods. The resulting analytical predictions are confirmed by means ofdetailed numerical simulations. The models we consider are characterized by trivialstatic properties, with no finite temperature singularities, but they neverthelessdisplay a surprising variety of dynamic behaviour during ageing, which can bedirectly related to the existence and growth in time of dynamic lengthscales.Well-behaved fluctuation–dissipation ratios can be defined for these models, andwe study their properties in detail. We confirm in particular the existence ofnegative fluctuation–dissipation ratios for a large number of observables. Ourresults suggest that well-defined violations of fluctuation–dissipation relations, of apurely dynamic origin and unrelated to the thermodynamic concept of effectivetemperatures, could in general be present in non-equilibrium glassy materials.

Finite-Temperature Form Factors: a Review

We review the concept of finite-temperature form factor that was introduced re- cently by the author in the context of the Majorana theory. Finite-temperature form factors can be used to obtain spectral decompositions of finite-temperature correlation functions in a way that mimics the form-factor expansion of the zero temperature case. We develop the concept in the general factorised scattering set-up of integrable quantum field theory, list certain expected properties and present the full construction in the case of the massive Majorana theory, including how it can be applied to the calculation of correlation functions in the quantum Ising model. In particular, we include the twisted construction, which was not developed before and which is essential for the application to the quantum Ising model.

The Form Factor Program: a Review and New Results - the Nested SU(N) Off-Shell Bethe Ansatz

The purpose of the bootstrap 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.

Wrapping interactions and a new source of corrections to the spin-chain/string duality

Assuming that the world-sheet sigma-model in the AdS/CFT correspondence is an integrable quantum field theory, we deduce that there might be new corrections to the spin-chain/string Bethe ansatz paradigm. These come from virtual particles propagating around the circumference of the cylinder and render Bethe ansatz quantization conditions only approximate. We determine the nature of these corrections both at weak and at strong coupling in the near-BMN limit, and find that the first corrections behave qualitatively as wrapping interactions at weak coupling.

Exact form factors in integrable quantum field theories: the scaling [formula omitted]-Ising model

A general form factor formula for the scaling Z ( N ) -Ising model is constructed. Exact expressions of all matrix elements are obtained for several local operators. In addition, the commutation rules for order, disorder parameters and para-Fermi fields are derived. Because of the unusual statistics of the fields, the quantum field theory seems not to be related to any classical Lagrangian or field equation.

Beyond the WKB approximation in -symmetric quantum mechanics

The mergings of energy levels associated with the breaking of PT symmetry in the model of Bender and Boettcher, and in its generalisation to incorporate a centrifugal term, are analysed in detail. Even though conventional WKB techniques fail, it is shown how the ODE/IM correspondence can be used to obtain a systematic approximation scheme which captures all previously-observed features. Nonperturbative effects turn out to play a crucial role, governing the behaviour of almost all levels once the symmetry-breaking transition has been passed. In addition, a novel treatment of the radial Schrodinger equation is used to recover the values of local and non-local conserved charges in the related integrable quantum field theories, without any need for resummation even when the angular momentum is nonzero.

Supersymmetric integrable scattering theories with unstable particles

We propose scattering matrices for N=1 supersymmetric integrable quantum field theories in 1+1 dimensions which involve unstable particles in their spectra. By means of the thermodynamic Bethe ansatz we analyze the ultraviolet behaviour of some of these theories and identify the effective Virasoro central charge of the underlying conformal field theories.

Integrable quantum field theory with boundaries: the exact g-function

The g-function was introduced by Affleck and Ludwig in the context of critical quantum systems with boundaries. In the framework of the thermodynamic Bethe ansatz (TBA) method for relativistic scattering theories, all attempts to write an exact integral equation for the off-critical version of this quantity have, up to now, been unsuccessful. We tackle this problem by using an n-particle cluster expansion, close in spirit to form-factor calculations of correlators on the plane. The leading contribution already disagrees with all previous proposals, but a study of this and subsequent terms allows us to deduce an exact infrared expansion for g, written purely in terms of TBA pseudoenergies. Although we only treat the thermally-perturbed Ising and the scaling Lee–Yang models in detail, we propose a general formula for g which should be valid for any model with entirely diagonal scattering.

Form factors of exponential fields for two-parametric family of integrable models

A two-parametric family of integrable models (the SS model) that contains as particular cases several well known integrable quantum field theories is considered. After the quantum group restriction it describes a wide class of integrable perturbed conformal field theories. Exponential fields in the SS model are closely related to the primary fields in these perturbed theories. We use the bosonization approach to derive an integral representation for the form factors of the exponential fields in the SS model. The same representations for the sausage model and the cosine–cosine model are obtained as limiting cases. The results are tested at the special points, where the theory contains free particles.

TOWARDS THE CONSTRUCTION OF WIGHTMAN FUNCTIONS OF INTEGRABLE QUANTUM FIELD THEORIES

The purpose of the "bootstrap program" for integrable quantum field theories in 1+1 dimensions is to construct a model in terms of its Wightman functions explicitly. In this article, the program is mainly illustrated in terms of the sine-Gordon and the sinh-Gordon model and (as an exercise) the scaling Ising model. We review some previous results on sine-Gordon breather form factors and quantum operator equations. The problem to sum over intermediate states is attacked in the short distance limit of the two point Wightman function for the sinh-Gordon and the scaling Ising model.

On vacuum energies and renomalizability in integrable quantum field theories

We compute for various perturbed conformal field theories the vacuum energies by means of the thermodynamic Bethe ansatz. Depending on the infrared and ultraviolet divergencies of the models, governed by the scaling dimensions of the underlying perturbed conformal field theory in the ultraviolet, the vacuum energies exhibit different types of characteristics. In particular, for the homogeneous sine-Gordon models we observe that once the conformal dimension of the perturbing scalar field is smaller or greater than 1/2, the vacuum energies are positive or negative, respectively. This behaviour indicates the need for additional ultraviolet counterterms in the latter case. At the transition points we obtain an infinite vacuum energy, which is partly explainable with the presence of several free fermions in the models studied.

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 Onsager's work of the forties, exact results for the magnetic case have been missing until the late eighties, 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 article we review this field theoretical approach to the Ising universality class, with particular attention to the results obtained starting from Zamolodchikov's 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.

N=1 supersymmetric boundary bootstrap

We investigate the boundary bootstrap programme for finding exact reflection matrices of integrable boundary quantum field theories with N=1 boundary supersymmetry. The bulk S-matrix and the reflection matrix are assumed to take the form S ̂ = S ̃ S , R ̂ = R ̃ R , where S̃ and R̃ are the S-matrix and reflection matrix of some integrable non-supersymmetric boundary theory that is assumed to be known, and S and R describe the mixing of supersymmetric indices. Under the assumption that the bulk particles transform in the kink and boson/fermion representations and the ground state is a singlet we present rules by which the supersymmetry representations and reflection factors for excited boundary bound states can be determined. We apply these rules to the boundary sine-Gordon model, to the boundary a 2 (1) and a 4 (1) affine Toda field theories, to the boundary sinh-Gordon model and to the free particle.

Rational sequences for the conductance in quantum wires from affine Toda field theories

We analyse the expression for the conductance of a quantum wire which is decribed by an integrable quantum field theory. In the high temperature regime we derive a simple formula for the filling fraction. This expression involves only the inverse of a matrix which contains the information of the asymptotic phases of the scattering matrix and the solutions of the constant thermodynamic Bethe ansatz equations. Evaluating these expressions for minimal affine Toda field theory we recover several sequences of rational numbers, which are multiples of the famous Jain sequence for the filling fraction occurring in the context of the fractional quantum Hall effect. For instance we obtain $\nu= 4 m/(2m +1)$ for $A_{4m-1}$-minimal affine Toda field theory. The matrices involved have in general non-rational entries and are not part of previous classification schemes based on integral lattices.

Integrable quantum field theories with supergroup symmetries: the OSP(1/2) case

As a step to understand general patterns of integrability in 1+1 quantum field theories with supergroup symmetry, we study in details the case of OSP(1/2). Our results include the solutions of natural generalizations of models with ordinary group symmetry: the UOSP(1/2) k WZW model with a current–current perturbation, the UOSP(1/2) principal chiral model, and the UOSP(1/2)⊗ UOSP(1/2)/ UOSP(1/2) coset models perturbed by the adjoint. Graded parafermions are also discussed. A pattern peculiar to supergroups is the emergence of another class of models, whose simplest representative is the OSP(1/2)/ OSP(0/2) sigma model, where the (non unitary) orthosymplectic symmetry is realized non-linearly (and can be spontaneously broken). For most models, we provide an integrable lattice realization. We show in particular that integrable osp(1/2) spin chains with integer spin flow to UOSP(1/2) WZW models in the continuum limit, hence providing what is to our knowledge the first physical realization of a super WZW model.

Polarization-Free Quantum Fields and Interaction

A new approach to the inverse scattering problem proposed by Schroer, is applied to two-dimensional integrable quantum field theories. For any two-particle S-matrix S_2 which is analytic in the physical sheet, quantum fields are constructed which are localizable in wedge-shaped regions of Minkowski space and whose two-particle scattering is described by the given S_2. These fields are polarization-free in the sense that they create one-particle states from the vacuum without polarization clouds. Thus they provide examples of temperate polarization-free generators in the presence of non-trivial interaction.

Excited state TBA for the φ2,1 perturbed [formula omitted] model

We examine some excited state energies in the non-unitary integrable quantum field theory (IQFT) obtained from the perturbation of the minimal conformal field theory (CFT) model M 3,5 by its operator φ 2,1. Using the correspondence of this IQFT to the scaling limit of the dilute- A 2 lattice model (in a particular regime) we derive the functional equations for the QFT commuting transfer matrices. These functional equations can be transformed to a closed set of TBA-like integral equations which determine the excited state energies in the finite-size system. In particular, we explicitly construct these equations for the ground state and two lowest excited states. Numerical results for the associated energy gaps are compared with those obtained by the truncated conformal space approach (TCSA).

Finite temperature corrections in 2d integrable models

We study the finite size corrections for the magnetization and the internal energy of the 2d Ising model in a magnetic field by using transfer matrix techniques. We compare these corrections with the functional form recently proposed by Delfino and LeClair–Mussardo for the finite temperature behaviour of one-point functions in integrable 2d quantum field theories. We find a perfect agreement between theoretical expectations and numerical results. Assuming the proposed functional form as an input in our analysis we obtain a relevant improvement in the precision of the continuum limit estimates of both quantities.