Inverse Scattering and Local Observable Algebras in Integrable Quantum Field Theories

We study deformations of 2D Integrable Quantum Field Theories (IQFT) which preserve integrability (the existence of infinitely many local integrals of motion). The IQFT are understood as “effective field theories”, with finite ultraviolet cutoff. We show that for any such IQFT there are infinitely many integrable deformations generated by scalar local fields Xs, which are in one-to-one correspondence with the local integrals of motion; moreover, the scalars Xs are built from the components of the associated conserved currents in a universal way. The first of these scalars, X1, coincides with the composite field (TT¯) built from the components of the energy–momentum tensor. The deformations of quantum field theories generated by X1 are “solvable” in a certain sense, even if the original theory is not integrable. In a massive IQFT the deformations Xs are identified with the deformations of the corresponding factorizable S-matrix via the CDD factor. The situation is illustrated by explicit construction of the form factors of the operators Xs in sine-Gordon theory. We also make some remarks on the problem of UV completeness of such integrable deformations.

We consider scalar two-dimensional quantum field theories with the\nfactorizing S-matrix which has poles in the physical strip. In our previous\nwork, we introduced the bound state operators and constructed candidate\noperators for observables in wedges.\n Under some additional assumptions on the S-matrix, we show that, in order to\nobtain their strong commutativity, it is enough to prove the essential\nself-adjointness of the sum of the left and right bound state operators. This\nessential self-adjointness is shown up to the two-particle component.\n

At half-filling the repulsive Hubbard model is in a Mott insulating phase. The charge degrees of freedom are gapped, whereas the spin degrees of freedom remain gapless. At low energies the spin sector is actually scale invariant (apart from logarithmic corrections) and Conformal Field Theory (CFT) methods may be applied to determine the low-energy behaviour of correlation functions involving only the spin sector. On the other hand, the charge sector is not scale invariant and CFT does not provide any information for correlators involving the charge degrees of freedom. In this chapter we will show that there exists a particular continuum limit of the half filled Hubbard model, in which it is possible to calculate dynamical correlation functions by means of methods of integrable quantum field theory. We first construct a Lorentz invariant scaling limit starting from the results for the excitation spectrum and the S-matrix discussed in Chapter 7. This scaling limit is identified as the SU(2) Thirring model, which is an integrable relativistic quantum field theory. Next we discuss a continuum limit, which is obtained directly from the Hubbard Hamiltonian and describes the vicinity of the scaling limit.

Defects are ubiquitous in nature, for example, in the form of dislocations, shocks, bores, or impurities of various kinds, and their descriptions are an important part of any physical theory. But the following question can be asked. What types of defect are allowed, and what are their properties if maintaining integrability within an integrable field theory in two-dimensional space-time is required? We consider a collection of ideas and questions connected with this problem, including examples of integrable defects and the curiously special roles played by energy-momentum conservation and Bäcklund transformations, solitons scattering on defects, and some interesting effects in the framework of the sine-Gordon model, defects in integrable quantum field theory, and the construction of transmission matrices. In conclusion, we remark on algebraic considerations and future research directions.

Quantum field theories in finite volume: Excited state energies

Integrable quantum field theories and conformal field theories from lattice models in the light-cone approach

The thermodynamic Bethe ansatz approach to the study of integrable quantum field theories was introduced in the early 90s. Since then it has been known that the thermodynamic Bethe ansatz equations can be recast in the form of Y-systems. These Y-systems have a number of interesting properties, notably in the high-temperature limit their solutions are constants from which the central charge of the ultraviolet fixed point can be obtained and they are typically periodic functions, with period proportional to the dimension of the perturbing field. In this letter we discuss the derivation of Y-systems when the standard thermodynamic Bethe ansatz equations are replaced by generalised versions, describing generalised Gibbs ensembles. We shown that for many integrable quantum field theories, there is a large class of distinct generalised Gibbs ensembles which share the same Y-system.

In two recent papers [1, 2] we have proposed a program of study which allows us to compute the correlation functions of local and semi-local fields in generalised textrm{T}overline{textrm{T}} -deformed integrable quantum field theories. This new program, based on the construction of form factors, opens many avenues for future study, one of which we address in this paper: computing entanglement measures employing branch point twist fields. Indeed, over the past 15 years, this has become one the leading methods for the computation of entanglement measures, both in conformal field theory and integrable quantum field theory. Thus the generalisation of this program to textrm{T}overline{textrm{T}} -perturbed theories offers a promising new tool for the study of entanglement measures in the presence of irrelevant perturbations. In this paper, we show that the natural two-particle form factor solution for branch point twist fields in replica theories with diagonal scattering admits a simple generalisation to a solution for textrm{T}overline{textrm{T}} -perturbed theories. Starting with this solution, some of the known properties of entanglement measures in massive integrable quantum field theories can be generalised to the perturbed models. We show this by focusing on the Ising field theory. During the completion of this paper, we became aware of the recent publication [3] where the same problem has been addressed.

Yang-Baxter algebras of monodromy matrices in integrable quantum field theories

Integrable quantum field theories in 1+1 dimensions have recently become amenable to a rigorous construction, but many questions about the structure of their local observables remain open. Our goal is to characterize these local observables in terms of their expansion coefficients in a series expansion by interacting annihilators and creators, similar to form factors. We establish a rigorous one-to-one characterization, where locality of an observable is reflected in analyticity properties of its expansion coefficients; this includes detailed information about the high-energy behaviour of the observable and the growth properties of the analytic functions. Our results hold for generic observables, not only smeared pointlike fields, and the characterizing conditions depend only on the localization region—we consider wedges and double cones—and on the permissible high energy behaviour.

We determine the form factor expansion of the one-point functions in integrable quantum field theory at finite temperature and find that it is simpler than previously conjectured. We show that no singularities are left in the final expression provided that the operator is local with respect to the particles and argue that the divergences arising in the non-local case are related to the absence of spontaneous symmetry breaking on the cylinder. As a specific application, we give the first terms of the low temperature expansion of the one-point functions for the Ising model in a magnetic field.

We study homogeneous quenches in integrable quantum field theory where the initial state contains zero-momentum particles. We demonstrate that the two-particle pair amplitude necessarily has a singularity at the two-particle threshold. Albeit the explicit discussion is carried out for special (integrable) initial states, we argue that the singularity is inevitably present and is a generic feature of homogeneous quenches involving the creation of zero momentum particles. We also identify the singularity in quenches in the Ising model across the quantum critical point, and compute it perturbatively in phase quenches in the quantum sine-Gordon model which are potentially relevant to experiments. We then construct the explicit time dependence of one-point functions using a linked cluster expansion regulated by a finite volume parameter. We find that the secular contribution normally linear in time is modified by a t ln t term. We additionally encounter a novel type of secular contribution which is shown to be related to parametric resonance. It is an interesting open question to resum the new contributions and to establish their consequences directly observable in experiments or numerical simulations.

We present a microscopic approach in the framework of Sklyanin's quantum separation of variables (SOV) for the exact solution of 1+1-dimensional quantum field theories by integrable lattice regularizations. Sklyanin's SOV is the natural quantum analogue of the classical method of separation of variables and it allows a more symmetric description of classical and quantum integrability w.r.t. traditional Bethe ansatz methods. Moreover, it has the advantage to be applicable to a more general class of models for which its implementation gives a characterization of the spectrum complete by construction. Our aim is to introduce a method in this framework which allows at once to derive the spectrum (eigenvalues and eigenvectors) and the dynamics (time dependent correlation functions) of integrable quantum field theories (IQFTs). This approach is presented for a paradigmatic example of relativistic IQFT, the sine-Gordon model.

Chapter 16 covers the general properties of the integrable quantum field theories, including how an integrable quantum field theory is characterized by an infinite number of conserved charges. These theories are illustrated by means of significant examples, such as the Sine–Gordon model or the Toda field theories based on the simple roots of a Lie algebra. For the deformations of a conformal theory, it shown how to set up an efficient counting algorithm to prove the integrability of the corresponding model. The chapter focuses on two-dimensional models, and uses the term ‘two-dimensional’ to denote both a generic two-dimensional quantum field theory as well as its Euclidean version.

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