The Bethe Ansatz Solution Of 1D Heisenberg Model And Quantum Metrology

Bethe ansatz is a method which was first presented by H. Bethe to diag-onalize the Hamiltonian of one-dimensional spin-1/2 Heisenberg model. It has been going through a rapidly development in last few decades and now becomes one of most efficient methods to obtain the exact solution of many-body system with interactions. This thesis consists of introduction, main text, conclusion and references. A brief introduction of the Heisenberg model and Bethe ansatz is presented in chapter one. In chapter two, we show the detailed calculation of the eigenvalues and eigenstates of one-dimensional spin-1/2 Heisenberg model. In the third chapter we present the procedure of solving the XXZ model by using Bethe ansatz and provide the analytical expressions. And in the forth chapter we intro-duce the concepts of classical Fisher information and quantum Fisher information at first, then discuss the relation between these two concepts and Cramer-Rao the-orem. The Cramer-Rao inequality implies that the quantum Fisher information describes the maximum value of the theoretical precision for the parameter under estimation. At the end of this chapter we consider the length of the chain as the parameter to be estimated and present the Fisher information of the eigenstates of the subspace corresponding to the reversal of only one spin. The expression of Fisher information implies that it decreases due to the increase of the length of the chain. The last chapter is the conclusion of this thesis.