The Bethe-Ansatz for Gaudin Spin Chains

Abstract

We investigate a special case of the quantum integrable Heisenberg spin chain known as Gaudin model. The Gaudin model is an important example of quantum integrable systems. We study the Gaudin model for the Lie algebra s[z(<C). The key problem is to find the spectrum and the corresponding eigenvectors of the commuting Hamiltonians. The standard method to solve this type of classical problem was introduced by H. Bethe and is known as the Bethe-Ansatz. Bethe's technique has proven to be very powerful in various areas of modem many-body theory and statistical mechanics. [19], [14], [4] Following Sklyanin's ideas in [19], we derive the Bethe-Ansatz equations for sl2(<C). Solving the Bethe-Ansatz equations is equivalent to finding polynomial solutions of the Lame differential equation, which has a meaning in electrostatics. We derive this equation for sl2(<C), and investigate its special cases. We discuss classical and more recent results on the Gaudin spin chain for sl2(<C) and provide numerical evidence for new observations in the real case of the Lame equation. Using roots of classical polynomials known as Jacobi polynomials, which are solutions to a special case of the Lame equation, we numerically approximate solutions to the Lame equation in more complicated settings. We discuss the Gaudin model associated to the Lie algebra sl3(C). Using the Bethe-Ansatz equations for sl3(C), we provide solutions in special cases.