Quantum Integrable Systems in One Dimension

The field of integrable systems is a vast one that has grown almost explosively in recent years. It is not remotely possible to review all the developments in the field in one hour, so I am restricting my attention to some particular topics. I shall only consider one dimensional quantum systems, or 1+1 as it is usually said, including the time dimension, but necessarily therefore including some discussion of the statistical mechanics of two dimensional lattice systems (vertex models) which share a common mathematical structure with the 1+1 quantum systems, and have been the key to much of the understanding of the quantum systems.[1] [2] [3] This is still far too big a field to review in any detail in one hour, so I shall focus on some generalizations of the spin one-half chain with nearest-neighbor coupling, for which Bethe first set down his celebrated ansatz[4]. The simplest generalization, to fully anisotropic coupling, the so-called XY Z model, has an amazingly rich mathematical structure, parts of which are still being elucidated. A second generalization, which has been very fashionable lately, is to extend the coupling beyond nearest neighbors, specifically to a long range inverse-square type coupling between spins one-half. This also turns out to be an integrable system, although the wave functions are not of the Bethe ansatz type.[5] [6] How relevant are these models to real physical systems? The simplest isotropic spin one half XXX Heisenberg antiferromagnet is a good Hamiltonian for the quasi-one-dimensional system CPC, and one of the early experimental vindications of the Bethe ansatz was the confirmation by neutron scattering[7] that it correctly predicted the observed elementary excitation spectrum, in contrast to the standard spin wave theory used at the time. An appropriate continuum limit of the XY Z chain gives the sine-Gordon model, and the XXZ chains are good representations of known systems. The closely related onedimensional Hubbard model has been widely used to describe one-dimensional conductors, and may be relevant to some high temperature superconductors. One of the great successes of the Bethe ansatz has been the solution of the Kondo problem[8]. The Luttinger liquid model[9], first used to analyze low