Gaudin model

In physics, the Gaudin model, sometimes known as the quantum Gaudin model, is a model, or a large class of models, in statistical mechanics first described in its simplest case by Michel Gaudin. They are exactly solvable models, and are also examples of quantum spin chains. The simplest case was first described by Michel Gaudin in 1976, with the associated Lie algebra taken to be , the two-dimensional special linear group. Let be a semi-simple Lie algebra of finite dimension . Let be a positive integer. On the complex plane , choose different points, . Denote by the finite-dimensional irreducible representation of corresponding to the dominant integral element . Let be a set of dominant integral weights of . Define the tensor product . The model is then specified by a set of operators acting on , known as the Gaudin Hamiltonians. They are described as follows. Denote by the invariant scalar product on (this is often taken to be the Killing form). Let be a basis of and be the dual basis given through the scalar product. For an element , denote by the operator which acts as on the th factor of and as identity on the other factors. Then These operators are mutually commuting. 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 , sometimes referred to as the Gaudin Hamiltonian. It depends on a complex parameter , and also on the quadratic Casimir, which is an element of the universal enveloping algebra , defined as This acts on representations by multiplying by a number dependent on the representation, denoted . This is sometimes referred to as the index of the representation. The Gaudin Hamiltonian is then defined Commutativity of for different values of follows from the commutativity of the . When has rank greater than 1, the commuting algebra spanned by the Gaudin Hamiltonians and the identity can be expanded to a larger commuting algebra, known as the Gaudin algebra.