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Cambridge University Press eBooks

S-matrices, spin chains and vertex models

Cisar Gómez, Martm Ruiz-Altaba, Germán Sierra

Published Apr 18, 1996• 0 citations
Publisher

Abstract

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.

DOI
10.1017/cbo9780511628825.002

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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.
The integrability of some important physical systems with a finite number of degrees of freedom relies on purely algebraic properties. One can solve the Coulomb potential and the harmonic oscillator both classically and quantum mechanically, using the appropriate dynamical algebra only [1]. Integrable systems with an infinite number of degrees of freedom are those with an infinite number of conserved quantities. They exhibit loop algebras, Yang-Baxter algebras and Kac-Moody algebras as symmetries. The investigation of theories has shown in recent years that Yang-Baxter algebras are the basic concepts of integrability [2]. The commutativity of a family of transfer matrices both in field theory and statistical mechanics follows directly from the Yang-Baxter algebra of monodromy matrices. Since the transfer matrix is a generating functional of local and non-local charges (including the hamiltonian) one gets an infinite set of commuting and conserved quantities. So, we can say that the theory is integrable. We are using the word integrable in the large sense explained above. Moreover, theories can be often explicitly solved to a large extent [2, 3]. In this brief report I will concentrate on a specific class of field theories. A complete account can be found in refs. 4-7. This class
From the exact solution of the Ising model by Onsager in 1944 up to that of the hard hexagon model by Baxter in 1980, the statistical mechanics of two-dimensional systems has been enriched by a number of exact results. One speaks (in quick manner) of exact models once a convenient mathematical expression has been obtained for a physical quantity such as the free energy, an order parameter or some correlation, or at the very least once their evaluation is reduced to a problem of classical analysis. Such solutions, often considered as singular curiosities upon their appearance, often have the interest of illustrating the principles and general theorems rigorously established in the framework of definitive theories, and also enabling the control of approximate or perturbative methods applicable to more realistic and complex models. In the theory of phase transitions, the Ising model and the results of Onsager and Yang have eminently played such a reference role. With the various vertex models, the methods of Lieb and Baxter have extended this role and the collection of critical exponents, providing new useful elements of comparison with extrapolation methods, and forcing a refinement of the notion of universality. Intimately linked to two-dimensional classical models (but of less interest for critical phenomena), one-dimensional quantum models such as the linear magnetic chain and Bethe's famous solution have certainly contributed to the understanding of fundamental excitations in many-body systems. One could also mention the physics of one-dimensional conductors.
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.
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.
The objective of this thesis is to study the isotropic XXX-1/2 spin chain model using the Algebraic Bethe Ansatz. To this end, we discuss the concept of integrability as well as the Lax operator and R-matrix, which help generate as many commuting operators in involution as there are degrees of freedom. We establish that the spin chain Hamiltonian belongs to this set and provide a definition of a state vector whose parameters, the Bethe roots, are constrained by a set of equations called the Bethe Ansatz Equations. We show that there is a one-to-one correspondence between the Bethe roots and the eigenfunctions of the system. Next, we proceed to study the nature of the low-lying excitations of both the ferromagnetic and antiferromagnetic model in the thermodynamic limit N → ∞ and show that the Bethe roots can be grouped into complexes or strings, which behave like bound states. We see that integrability is directly related to diffractionless scattering, which is obeyed by systems whose scattering matrices satisfy the Yang-Baxter Equation. In order to provide a more physical interpretation, we calculate the scattering matrix of the two-body problem for a system that satisfies the Yang-Baxter Equation and obtain exchange relations that are identical to those obtained using the Algebraic Bethe Ansatz for the XXX-1/2 spin chain model. Finally, we calculate the scattering matrix for a two-body problem interacting with a delta potential and show that this is the same as what we derived using the Coordinate Bethe Ansatz.