Beautiful Models: 70 Years of Exactly Solved Quantum Many-Body Problems

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Summary: Gives good insights into quantum integrable models
Review: The main virtue of this book is that it clears up any confusion regarding the notion of integrability in a quantum system. After an historical overview of the theory of exactly solvable systems in chapter 1, the author recalls the notion of integrability in classical mechanics, restricting his discussion to systems that are governed by a Hamiltonian. Using the standard action-angle canonical transformation he shows that the integrability of a Hamiltonian system is, as is well known, indicated by the presence of a finite set of quantities that are in `involution', i.e. they are constants of motion.

This notion of integrability will not work for finite-dimensional quantum systems, as the author shows by using a system that hypothetically has a set L of mutually commuting operators, this set also including the Hamiltonian. He shows that no two commuting operators are algebraically independent, and at most D commuting operators are linearly independent, where D is the dimension of the eigenspace of one of the operators. The author then presents a notion of integrability that is less trivial, in that it will give information on the dynamics of the quantum system.

Since quantum systems are typically systems of particles that are interacting with each other, the dynamical events of interest are the scattering events. Indeed, the scattering theory of quantum systems is highly developed, and has inspired an enormous amount of research in both physics and mathematics. The author justifies the viability of this notion of integrability by considering several elementary systems, starting in one dimension and considering one and then two particles, he shows that energy and momentum conservation are strong enough to force the momenta of the scattered particles to be merely rearrangements of the incoming momenta (the particles essentially "pass through one another").

Things are more complicated when three particles are considered. Energy and momentum conservation can no longer insure that the asymptotic momenta are merely rearrangements of the incoming momenta. The famous `Bethe ansatz' though allows the determination of ratios of amplitudes for scattering in terms of two-body collisions. Genuine 3-body scattering is to be included in the total asymptotic wavefunction, which will deviate from a plane wave as it emerges from a 3-body overlap region. The author call this `diffractive' scattering, and he notes that it will not occur for exactly solvable systems. For generic one-dimensional systems it will occur however, and this prohibits the use of Bethe ansatz. The presence of diffractive scattering will prohibit the occurrence of a third independent conserved quantity, and this gives the author a criterion for defining `non-integrability.' The quantum systems considered in the book are `integrable' and thus do not support diffractive scattering. The consequences of diffractionless scattering in quantum systems are for the author just as interesting as what happens in classical integrable system, and he has written this book to elucidate the properties of these "beautiful" systems.

Early on in the book the author discusses various techniques that will be used in analyzing the systems throughout the book. These techniques arise in the analysis of systems that are in the ground state and just above the ground state, and when they are at finite temperature. The systems that are studies are all integrable and nondiffractive, and thus obey what the author calls the `fundamental equations', which are coupled equations for the momenta of the system. Systems governed by the delta-function, inverse-square, and hyperbolic potentials are a few those considered when discussing these techniques.

An entire chapter of the book is devoted to the Heisenberg-Ising model, which was the first one that was tackled using the Bethe ansatz and is a model for a magnetic chain and a quantum lattice gas. This model is integrable, or "diffractionless" as an application of the Bethe ansatz proves, and the author shows how to obtain a complete set of equations for the spectrum using this ansatz. The ground state energy in zero flux and zero field is calculated, and this calculation is then generalized to the case of non-zero field and flux.

Also considered in the book are exchange models, which are quantum systems that have potentials that allow the exchange of quantum numbers between two particles. The author shows how to take some of the potentials that are considered in the book, such as the hyperbolic potential, and modify them to obtain exchange potentials. The hyperbolic potential in particular is strongly repulsive at the origin, and so the wavefunction of two particles will vanish when they meet, prohibiting mixing of different types of particles. This can be alleviated by the incorporation of permutation operator that exchanges the two particles.

The author also discusses the famous Hubbard model in the last chapter of the book. This model, used in condensed matter physics for modeling systems of strongly interacting electrons, is integrable in one dimension, as the author shows using again the Bethe ansatz.

The best chapter of the book is chapter 7, for therein the author addresses in detail more general questions on the property of integrability and how to prove when a system is integrable and when it is not. Noting that there is no optimal way to show that a system is integrable, he discusses various approaches to showing integrability of systems that support scattering. The discussion in this chapter is very lucid, and consequently readers will gain a lot of insight into the properties of integrable systems, particularly in the role of the Yang-Baxter equations and the subsequent notion of a `transfer operator'. The author's explanation of a transfer operator as corresponding to the scattering of test particle with all the other particles is one that clarifies its role and is an explanation that is not found in the literature on exactly solvable systems. He also gives a very interesting discussion of the physics behind non-integrability, using as examples scattering of light rays off wedges of mirrors. These examples serve to shed more light on the behavior of nonintegrable systems, which is more helpful than mere mathematical calculations.