Spin chain

A spin chain is a type of model in statistical physics. Spin chains were originally formulated to model magnetic systems, which typically consist of particles with magnetic spin located at fixed sites on a lattice. A prototypical example is the quantum Heisenberg model. Interactions between the sites are modelled by operators which act on two different sites, often neighboring sites. They can be seen as a quantum version of statistical lattice models, such as the Ising model, in the sense that the parameter describing the spin at each site is promoted from a variable taking values in a discrete set (typically , representing 'spin up' and 'spin down') to a variable taking values in a vector space (typically the spin-1/2 or two-dimensional representation of ). The prototypical example of a spin chain is the Heisenberg model, described by Werner Heisenberg in 1928. This models a one-dimensional lattice of fixed particles with spin 1/2. A simple version (the antiferromagnetic XXX model) was solved, that is, the spectrum of the Hamiltonian of the Heisenberg model was determined, by Hans Bethe using the Bethe ansatz. Now the term Bethe ansatz is used generally to refer to many ansatzes used to solve exactly solvable problems in spin chain theory such as for the other variations of the Heisenberg model (XXZ, XYZ), and even in statistical lattice theory, such as for the six-vertex model. Another spin chain with physical applications is the Hubbard model, introduced by John Hubbard in 1963. This model was shown to be exactly solvable by Elliott Lieb and Fa-Yueh Wu in 1968. Another example of (a class of) spin chains is the Gaudin model, described and solved by Michel Gaudin in 1976 The lattice is described by a graph with vertex set and edge set . The model has an associated Lie algebra . More generally, this Lie algebra can be taken to be any complex, finite-dimensional semi-simple Lie algebra . More generally still it can be taken to be an arbitrary Lie algebra. Each vertex has an associated representation of the Lie algebra , labelled .