In classical mechanics, an integrable system is a model that admits a complete set of integrals of motion that are in involution. This notion can be extended to include classical integrable field theories. A natural question is how to lift these notions to the quantum setting. Despite the extensive literature on the subject, these models are still not very well understood. As an example, in the first section of this work, we present a new class of integrable quantum field theories which develop an unusual behaviour that may be interpreted as a Hagedorn transition. The principal reason for the enigmatic nature of these models is the absence of a universal mathematical framework to describe them. One potential solution may be provided by quantum affine Gaudin models, which we examine in greater detail in the second part of this work. In particular, we introduce the first non-trivial Hamiltonian of quartic order for the affine sl2 Gaudin model, as well as the next-to-leading order expression for all higher Hamiltonians. Furthermore, we provide new insights into the double-loop version of the Feigin-Frenkel homomorphism, which is expected to be a crucial component in the construction of the Bethe ansatz for these models.
In classical mechanics, an integrable system is a model that admits a complete set of integrals of motion that are in involution. This notion can be extended to include classical integrable field theories. A natural question is how to lift these notions to the quantum setting. Despite the extensive literature on the subject, these models are still not very well understood. As an example, in the first section of this work, we present a new class of integrable quantum field theories which develop an unusual behaviour that may be interpreted as a Hagedorn transition. The principal reason for the enigmatic nature of these models is the absence of a universal mathematical framework to describe them. One potential solution may be provided by quantum affine Gaudin models, which we examine in greater detail in the second part of this work. In particular, we introduce the first non-trivial Hamiltonian of quartic order for the affine sl2 Gaudin model, as well as the next-to-leading order expression for all higher Hamiltonians. Furthermore, we provide new insights into the double-loop version of the Feigin-Frenkel homomorphism, which is expected to be a crucial component in the construction of the Bethe ansatz for these models.
Various aspects of the theory of quantum integrable systems are reviewed. Basic ideas behind the construction of integrable ultralocal and nonultralocal quantum models are explored by exploiting the underlying algebraic structures related to the Yang-Baxter equations. Physical meaning of abstract mathematical notions like universal R-matrix, quantized algebras, Sklyanin algebra, braided algebra, Hopf algebra etc. and the role played by them in integrable systems are highlighted. Systematic construction of quantum integrable lattice as well as field models and their exact excitation spectra are presented through examples. The coordinate and algebraic formulations of the Bethe ansatz are illustrated with comparison, along with the description of nested and functional Bethe ansatzes. The techniques for deriving quantum Hamiltonians from the Lax operators are demonstrated on concrete models. The exposition of this review is kept in a fairly elementary level with emphasis on the physical contents.
In classical mechanics, two functions over phase space are said to be in involution if their Poisson bracket vanishes. Since Liouville's time, a dynamical system whose phase space is 2 N dimensional is called completely integrable if there are N functions, or “hamiltonians”, or “charges” in involution. A field theoretic system is called integrable if it possesses an infinity of mutually commuting conserved observables. All these mutually commuting conserved charges or hamiltonians allow us to solve the system exactly, without resorting to approximation schemes. Integrability is an unusual wealth of symmetry we might not think of requiring on realistic physical models. Rather, we should expect the complexity of nature not to be exactly solvable. However, integrability is of epistemological importance: exact solutions allow more perfect understanding. Toy models, of which physicists have been very fond since antiquity, often exist just so that exact and complete solutions can be found, in order to grasp the nature of the phenomenon being modeled. Furthermore, and quite surprisingly, physical systems with an infinite symmetry do exist: any non-linear system with soliton solutions is integrable. We shall be interested in discovering under which circumstances certain kinds of physical systems admit complete integrability, what types of systems these are, and in pointing out the physical roots of such a wonderful property. In the process, we shall have the occasion to use some of the most powerful tools elaborated by workers in mathematics. Given our present understanding of two-dimensional models, integrability appears as a consequence of very simple dynamics, characterized by factorized scattering matrices.
Exactly solvable quantum systems represent a fascinating and critically important class of models in theoretical physics, offering exact solutions to non-trivial problems that often shed light on the behavior of more complex, intractable systems. The remarkable feature of these systems lies not only in their solvability but also in the underlying mathematical structures that facilitate such solutions. This paper explores the concept of a universal algebraic structure that underpins exactly solvable quantum systems. We delve into how diverse models, ranging from one-dimensional spin chains to many-body systems and field theories, exhibit common algebraic symmetries, such as Lie algebras, affine Lie algebras, and quantum groups. These algebraic frameworks provide a unified language for understanding integrability, the existence of an infinite number of conserved quantities, and the ability to construct exact eigenstates. We review the historical development of these ideas, from the Bethe ansatz to the development of the inverse scattering method and quantum groups. Our discussion highlights how these mathematical structures are not merely tools for calculation but fundamental descriptors of the physical reality of these systems, offering profound insights into non-perturbative quantum phenomena. The identification of a universal algebraic architecture suggests a deeper coherence in the landscape of quantum mechanics, paving the way for new discoveries and classifications of quantum models.
This thesis presents an introduction to the class of Richardson-Gaudin integrable models, with special focus on the Bethe ansatz wave function, and investigates ways of applying the properties of Richardson-Gaudin models both in and out of integrability. A framework is outlined for the numerical and theoretical treatment of these systems, exposing a duality allowing the Bethe equations to be solved numerically. This is extended to the calculation of inner products and correlation functions. Using this framework, the influence of particle exchange on the Bethe ansatz is discussed, after which it is shown how the Bethe ansatz is able to accurately model wave functions of non-integrable models in two different settings. First, a variational approach is outlined for stationary models where integrability-breaking perturbations are explicitly introduced. Second, an alternative way of breaking integrability is through the introduction of dynamics and periodic driving, where it is shown how integrability can be used to model the resulting Floquet many-body resonances. Throughout this work, it is shown how the clear-cut structure and relatively large freedom in Richardson-Gaudin models makes them ideal for an investigation of the general principles of integrability, as well as being a perfect testing ground for the development of new quantum many-body techniques beyond integrability.
In this work we define a formal notion of a quantum phase crossover for certain Bethe ansatz solvable models. The approach we adopt exploits an exact mapping of the spectrum of a many-body integrable system, which admits an exact Bethe ansatz solution, into the quasi-exactly solvable spectrum of a one-body Schrödinger operator. Bifurcations of the minima for the potential of the Schrödinger operator determine the crossover couplings. By considering the behaviour of particular ground-state correlation functions, these may be identified as quantum phase crossovers in the many-body integrable system with finite particle number. In this approach the existence of the quantum phase crossover is not dependent on the existence of a thermodynamic limit, rendering applications to finite systems feasible. We study two examples of bosonic Hamiltonians which admit second-order crossovers. Keywords: quantum integrability (Bethe ansatz), Bose-Einstein condensation (theory) 1
The properties of exactly integrable two-dimensional quantum systems are reviewed and discussed. The nature of exact integrability as a physical phenomenon and various aspects of the mathematical formalism are explored by discussing several examples, including detailed treatments of the nonlinear Schr\"odinger (delta-function gas) model, the massive Thirring model, and the six-vertex (ice) model. The diagonalization of a Hamiltonian by Bethe's Ansatz is illustrated for the nonlinear Schr\"od\'{\i}nger model, and the integral equation method of Lieb for obtaining the spectrum of the many-body system from periodic boundary conditions is reviewed. Similar methods are applied to the massive Thirring model, where the fermion-antifermion and bound-state spectrum are obtained explicitly by the integral equation method. After a brief review of the classical inverse scattering method, the quantum inverse method for the nonlinear Schr\"odinger model is introduced and shown to be an algebraization of the Bethe Ansatz technique. In the quantum inverse method, an auxiliary linear problem is used to define nonlocal operators which are functionals of the original local field on a fixed-time string of arbitrary length. The particular operators for which the string is infinitely long (free boundary conditions) or forms a closed loop around a cylinder (periodic boundary conditions) correspond to the quantized scattering data and have a special significance. One of them creates the Bethe eigenstates, while the other is the generating function for an infinite number of conservation laws. The analogous operators on a lattice are constructed for the symmetric six-vertex model, where the object which corresponds to a solution of the auxiliary linear problem is a string of vertices contracted over horizontal links (arrows). The relationship between the quantum inverse method and the transfer matrix formalism is exhibited. The inverse Gel'fand-Levitan transform which expresses the local field operator as a functional of the quantized scattering data is formulated for the nonlinear Schr\"odinger equation, and some interesting properties of this transformation are noted, including its reduction to a Jordan-Wigner transformation in the limit of infinitely repulsive coupling.