Abstract In this chapter we present the S-matrix theory of two-dimensional integrable models. This leads, in particular, to the exact spectrum of the massive excitations away from the critical point. From a mathematical point of view, the two-dimensional nature of the systems and their integrability are the crucial features that lead to important simplifications of the formalism and its successful application. It is worth mentioning that, initially developed to overcome the obstacles encountered by quantum field theory in dealing with the strong interactions of the hadronic particles1 (such as protons, neutrons, or pions), the S-matrix has achieved its most beautiful intellectual triumph in the analysis of the two-dimensional statistical models away from criticality, particularly when they are described by integrable theories. These significant developments have been pioneered by A.B. Zamolodochikov.
Abstract In this chapter we present the S-matrix theory of two-dimensional integrable models. This leads, in particular, to the exact spectrum of the massive excitations away from the critical point. From a mathematical point of view, the two-dimensional nature of the systems and their integrability are the crucial features that lead to important simplifications of the formalism and its successful application. It is worth mentioning that, initially developed to overcome the obstacles encountered by quantum field theory in dealing with the strong interactions of the hadronic particles1 (such as protons, neutrons, or pions), the S-matrix has achieved its most beautiful intellectual triumph in the analysis of the two-dimensional statistical models away from criticality, particularly when they are described by integrable theories. These significant developments have been pioneered by A.B. Zamolodochikov.
Abstract The integrable quantum field theories analyzed in the previous chapters provide the exact solution of many statistical models away from the critical point. Despite the elegance and the undeniable success of these methods, the generic situation that occurs in statistical physics is that of non-integrable dynamics: many interesting statistical models fall within this class and therefore it would be highly desirable to develop an appropriate formalism to deal with the lack of integrability. This task is notoriously difficult for the rich phenomenology that arises when the dynamics are not integrable: there are decays and production scattering processes, confinement phenomena and nucleation of false vacua, resonance peaks in the cross-sections, etc.
Abstract The Ising model in a magnetic field is one of the most beautiful examples of an integrable model. This chapter presents its exact S-matrix and the exact spectrum of its excitations, which consist of eight particles of different masses. Similarly, it discusses the exact scattering theory behind the thermal deformation of the tricritical Ising model and the unusual features of the exact S-matrix of the non-unitary Yang–Lee model. Other examples are provided by O(n) invariant models, including the important Sine–Gordon model. It also discusses multiple poles, magnetic deformation, the E8 Toda theory, bootstrap fusion rules, non-relativistic limits and quantum group symmetry of the Sine–Gordon model.
The scattering matrix (S-matrix), relating the initial and final states of a physical system undergoing a scattering process, is a fundamental object in quantum mechanics and quantum field theory. The study of factorized S-matrices, in which many-body scattering factorizes into a product of two-body terms to yield an integrable model, has long been considered the domain of mathematical physics. Many beautiful results have been obtained over several decades for integrable models of this kind, with far reaching implications in both mathematics and theoretical physics. The viewpoint that these were only toy models changed dramatically with brilliant experimental advances in realizing low-dimensional quantum many-body systems in the lab. These recent experiments involve both the traditional setting of condensed matter physics and the trapping and cooling of atoms in optical lattices to engineer and study quasi-one-dimensional systems. In some cases the quantum physics of one-dimensional systems is arguably more interesting than their three-dimensional counterparts, because the effect of interactions is more pronounced when atoms are confined to one dimension. This article provides a brief overview of these ongoing developments, which highlight the fundamental importance of integrability.
The scattering matrix (S-matrix), relating the initial and final states of a physical system undergoing a scattering process, is a fundamental object in quantum mechanics and quantum field theory. The study of factorised S-matrices, in which many-body scattering factorizes into a product of two-body terms to yield an integrable model, has long been considered the domain of mathematical physics. Many beautiful results have been obtained over several decades for integrable models of this kind, with far reaching implications in both mathematics and theoretical physics. The viewpoint that these were only toy models changed dramatically with brilliant experimental advances in realizing low-dimensional quantum many-body systems in the lab. These recent experiments involve both the traditional setting of condensed matter physics and the trapping and cooling of atoms in optical lattices to engineer and study quasi-one-dimensional systems. In some cases the quantum physics of one-dimensional systems is arguably more interesting than their three-dimensional counterparts, because the effect of interactions is more pronounced when atoms are confined to one dimension. This article provides a brief overview of these ongoing developments, which highlight the fundamental importance of integrability.
This thesis is devoted to the study of ghost theories out of the critical point, in two dimensions. The first chapter offers a bird's eye view of the most important applications of ghosts to condensed matter physics. After a brief exposition of the basic (and less basic) facts concerning ghosts at the critical point, an outline of the non-perturbative methods, used in Part I and Part II, is furnished. Essentially, they rely on the so-called integrable approach, which is based on the possibility of describing all the states of an integrable quantum field theory in terms of pseudo-particles in a Hilbert space. The scattering properties of such excitations are encoded into a matrix (the S-matrix), which can be exactly determined by imposing a set of stringent constraints, and which allows to specify completely the particle content of the theory (masses, multiplicities, bound states) [38, 39]. The knowledge of the scattering amplitude, then, permits to extract all the thermodynamic quantities of the system (free energy) by means of the Thermodynamic Bethe ansatz (TBA) technique [40] and, at least in principle, to determine the off-critical correlation functions of the local operators of the theory, thanks to the so-called Form Factor bootstrap approach [41-43]. Part I contains the simplest examples of off-critical ghost theories, namely the massive versions of the conformal free bosonic and fermionic ones [44]. Despite their non-interacting nature, still there are non-local sectors of the models, which exhibit a highly interacting behavior. Correlation functions of operators belonging to these sectors are computed exactly and a comparison with the massive ordinary counterparts is performed. Afterwards, the effects;produced by the introduction of impurities are considered [45]. At the moment, such models lack a physical realization, but they are important as 'prototype' systems, shedding light on some crucial basic aspects (e.g. the choice of the most convenient basis for the space of states). Part II deals with a deceptively simple representative of the aforementioned nonlinear sigma models defined on supersymmetric manifolds, where the vector field, with one commuting component and two anticommuting ones, transforms under the global symmetry OSP(ll2). This system has a simple physical realization in terms of a dense loop model, where crossings of loops are allowed [28, 46]. At long wavelength, the theory is gapless and the Goldstone excitations are nothing but free fermionic ghosts [25, 28]. We propose the exact S-matrix for this system and present TBA calculations, supporting such conjecture. The bootstrap form factor approach is outlined, including a detailed discussion about the symmetry properties of the model and the explicit derivation of some basic objects, such as the minimal form factors. Moreover, we compute explicitly the two-point correlation function of a suitably chosen operator of the theory, comparing its large distance limit with the result expected on the basis of conformal field theory considerations. Since the work is still in progress [47], we conclude sketching the main goals and the route we intend to take, in order to pursue them.
This thesis studies exact S-matrices and integrability aspects for several two-dimensional quantum systems. The emphasis is on the algebraic structures that underlie exact solvability in two dimensions. The first project details the construction of an optics model for n-level atoms interacting with an electromagnetic field. In the slowly varying envelope and rotating wave approximations, the equations of motion are shown to have a zero-curvature representation, implying an infinite set of quantum conservation laws. The second system investigated is a disordered field theory of Dirac fermions, which after averaging over the potential can be expressed as an osp(2|2) current- current model. The current-current model in turn can be viewed as the fully renormalized version of an osp(2]2) (1) (marginal) Toda system. Non-local charges satisfying the Uq[osp(2|2)(1)] quantum superalgebra are constructed. The corresponding quantum affine symmetry is used to build the Toda S-matrix for the vector representation. The exact S-matrix for the fundamental “solitons” of the osp(2|2) current-current model is then obtained by taking the marginal limit of the Toda S-matrix. The remaining two projects shift the focus to boundary theories. In the first boundary study, two massive versions of the anisotropic spin 1/2 Kondo model are introduced. The two models have the same bulk sine-Gordon interactions but differ in their boundary terms. At the free fermion point, each of the models can be understood as two decoupled Ising models in boundary magnetic fields. Integrability of the models is discussed and boundary S-matrices at and away from the free fermion point are constructed. The final system studied describes an Anderson impurity in a BCS superconductor. This model can be mapped onto an Ising boundary field theory. The boundary S - matrix is shown to exhibit Andreev scattering and the existence of bound states lying inside the superconducting gap.