Gaudin

Gaudin refers to a family of exactly solvable quantum spin models, commonly known as Gaudin magnets, named after Michel Gaudin who introduced them in the 1970s. The models describe N spins located at distinct sites with long-range interactions whose strength scales inversely with the difference of site parameters z_i. A typical set of Gaudin Hamiltonians is given by H_i = sum_{j≠i} (S_i · S_j)/(z_i - z_j), with the property that all H_i commute, making the system integrable.

The integrability of Gaudin models is underpinned by the Gaudin algebra, which arises from a quasi-classical limit of the inhomogeneous XXX spin chain or from r-matrix formalisms in integrable systems. Variants include rational, trigonometric, and hyperbolic Gaudin models corresponding to different underlying Lie algebra structures, often based on sl(2). The models admit exact solutions via the Bethe ansatz, allowing explicit construction of eigenstates and spectral data.

Connections and applications extend beyond spin chains. The Gaudin model is linked to the Knizhnik-Zamolodchikov equations in conformal field theory and to broader frameworks of quantum integrable systems. In condensed matter and nuclear physics, Richardson-Gaudin models describe exactly solvable pairing interactions, providing insights into finite-size superconductivity and nuclear pairing phenomena.

Origin and usage: the term Gaudin magnets appears in physics literature to denote this class of integrable spin models. The work on Gaudin models has fostered connections between integrable systems, representation theory, and exactly solvable many-body problems.