[1] They are exactly solvable models, and are also examples of quantum spin chains.
The simplest case was first described by Michel Gaudin in 1976,[1] with the associated Lie algebra taken to be
be a semi-simple Lie algebra of finite dimension
One problem of interest in the theory of Gaudin models is finding simultaneous eigenvectors and eigenvalues of these operators.
Instead of working with the multiple Gaudin Hamiltonians, there is another operator
, and also on the quadratic Casimir, which is an element of the universal enveloping algebra
by multiplying by a number dependent on the representation, denoted
Similarly to the Harish-Chandra isomorphism, these commuting elements have associated degrees, and in particular the Gaudin Hamiltonians form the degree 2 part of the algebra.
an eigenvector of the Gaudin algebra, one obtains a linear functional
The spectral problem, that is, determining eigenvalues and simultaneous eigenvectors of the Gaudin algebra, then becomes a matter of determining characters on the Gaudin algebra.
, define the vacuum vector to be the tensor product of the highest weight states from each representation:
It can be shown that a Bethe vector is an eigenvector of the Gaudin Hamiltonians if the set of equations
These are the Bethe ansatz equations for spin deviation
In theory, the Bethe ansatz equations can be solved to give the eigenvectors and eigenvalues of the Gaudin Hamiltonian.
In practice, if the equations are to completely solve the spectral problem, one must also check If, for a specific configuration of sites and weights, the Bethe ansatz generates all eigenvectors, then it is said to be complete for that configuration of Gaudin model.
It is possible to construct examples of Gaudin models which are incomplete.
One problem in the theory of Gaudin models is then to determine when a given configuration is complete or not, or at least characterize the 'space of models' for which the Bethe ansatz is complete.
in general position the Bethe ansatz is known to be complete.
[4] Even when the Bethe ansatz is not complete, in this case it is due to the multiplicity of a root being greater than one in the Bethe ansatz equations, and it is possible to find a complete basis by defining generalized Bethe vectors.
, there exist specific configurations for which completeness fails due to the Bethe ansatz equations having no solutions.
[6] Analogues of the Bethe ansatz equation can be derived for Lie algebras of higher rank.
, there are higher Gaudin Hamiltonians, for which it is unknown how to generalize the Bethe ansatz.
There is an ODE/IM isomorphism between the Gaudin algebra (or the universal Feigin–Frenkel center), which are the 'integrals of motion' for the theory, and opers, which are ordinary differential operators, in this case on
There exist generalizations arising from weakening the restriction on
A different way to generalize is to pick out a preferred automorphism of a particular Lie algebra
One can then define Hamiltonians which transform nicely under the action of the automorphism.
Historically, the quantum Gaudin model was defined and studied first, unlike most physical systems.
Certain classical integrable field theories can be viewed as classical dihedral affine Gaudin models.
Therefore, understanding quantum affine Gaudin models may allow understanding of the integrable structure of quantum integrable field theories.
Such classical field theories include the principal chiral model, coset sigma models and affine Toda field theory.