Bethe-Ansatz
The Bethe approach is an analytical method for the exact calculation of one-dimensional quantum mechanical Vielteilchenproblemen. 1931 Hans Bethe presented this method to calculate the exact eigenvalues (self- energies) and eigenvectors of the one-dimensional Heisenberg model. The actual Bethe approach it describes the parameterization of the eigenvectors as an approach to the solution of the eigenvalue problem ( Schrödinger equation).
Variants of the Bethe approach lead to the exact solution of the Kondo model, which became independent in 1980 found by Paul Wiegmann and Natan Andrei, and the Anderson model (PB Wiegmann and N. Kawakami, A. Okiji, 1981).
Bethe approach for the 1D Heisenberg model
The Bethe approach was originally developed for the one-dimensional Heisenberg model with nearest-neighbor interaction and periodic boundary conditions:
Depending on the sign of the coupling constant J of the ground state is ferromagnetic or anti- ferromagnetic:
The ferromagnetic ground state is the starting point for the Bethe approach. In the ferromagnetic ground state all spins are aligned in one direction. This is assumed in the z- direction o.B.d.A. So that the ground state can be described as:
In the Bethe approach the states by means of the folded states are classified by the ferromagnetic ground state. For example, the state with two folded spins indicated at the lattice sites and as:
The eigenstates of the Hamiltonian of the Heisenberg model are given as superpositions of these states. In this case, only linear combinations of states having the same number r of folded Pins are permitted. This is due to the fact that the operator commutes with the Hamiltonian and hence the eigenvectors are linear combinations of spins with the same quantum number must exist. To calculate these states Bethe went on iterative and initially looked at states with only one folded spin. This is then extended to superpositions of states with r folded spins.
R = 1
The eigenvectors consisting of superpositions of states with only one folded Spin N at lattice site:
The eigenvectors are solutions of the stationary Schrödinger equation. By means of comparing coefficients one finds N linearly independent equations for the coefficients:
Solutions of these equations, which also fulfill the condition for periodic boundary conditions, are plane waves:
Thus, the eigenvectors are given consisting of superpositions of states with only one folded spin. The energy of these states follows from the Schrödinger equation:
The next step is to look at superpositions of states with two folded spins.
R = 2
The approach for the eigenvectors is:
Bethe's approach for the coefficients are again plane waves with unknown amplitudes and:
The parameters and are determined by the insertion into the Schrödinger equation. This yields the following amplitude ratio:
Where you put in the approach for the coefficients:
With the periodic boundary conditions can be found overall that the wave numbers and the angle must satisfy the following equations:
Where the integers Bethe quantum numbers are called. Thus all eigenvectors for determined by all possible pairs that satisfy the equations. The energy is then give by:
The final step is the generalization of eigenvectors, which consist of superpositions of states with r folded spins.
R any
For eigenvectors, which consist of superpositions of states with r folded spins, is the approach:
With the coefficients:
The sum runs over all possible permutations of the numbers. This choice of the coefficients of the plane waves is called the Bethe approach. Insertion into the Schrödinger equation and the periodic boundary conditions lead to the Bethe- equations approach:
The eigenvectors are given with all combinations of the Bethe quantum numbers that satisfy the Bethe equations approach. A classification of the eigenvectors is possible via the Bethe quantum numbers. However, the provision of all eigenvectors is not trivial. However, the energy of the corresponding state can then easily by
Be specified.