Bethe approach

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The Bethe approach is an analytical method for the exact calculation of one-dimensional quantum mechanical many-body problems. In 1931 Hans Bethe presented this method for calculating the exact eigenvalues (eigenenergies) and eigenvectors of the one-dimensional Heisenberg model . The actual Bethe approach describes the parameterization of the eigenvectors as an approach for the solution of the eigenvalue problem ( Schrödinger equation ).

Variants of the Bethe approach lead to the exact solution of the Kondo model , which was independently found in 1980 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 the closest neighbor interaction and periodic boundary conditions:

Depending on the sign of the coupling constant , 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 to be in the direction. The basic state can thus be described as:

In the Bethe approach, the states are classified from the ferromagnetic ground state by means of the flipped states. For example, the state with two flipped spins at the lattice sites and is given as:

The eigen-states of the Hamilton operator of the Heisenberg model are given as superpositions of these states. Only linear combinations of states with the same number r of flipped spins are permitted. This is due to the fact that the -operator commutes with the Hamilton operator and therefore the eigenvectors must consist of linear combinations of spins with the same -quantum number. To calculate these states, Bethe proceeded iteratively and initially considered states with only one flipped spin. This is then extended to superpositions of states with flipped spins.

r = 1

The eigenvectors consisting of superpositions of states with only one flipped spin at the lattice site :

The eigenvectors are solutions of the stationary Schrödinger equation . By comparing coefficients one finds linearly independent equations for the coefficients :

Solutions of these equations, which also fulfill the condition for periodic boundary conditions, are plane waves:

This gives the eigenvectors consisting of superpositions of states with only one flipped spin. The energy of these states follows from the Schrödinger equation:

The next step is to look at superpositions from states with two flipped spins.

r = 2

The approach for the eigenvectors is:

Bethes approach for the coefficients are again plane waves with as yet unknown amplitudes and :

The parameters and are determined by inserting them into the Schrödinger equation. This gives the following amplitude ratio:

which one adds in the approach for the coefficients:

With the periodic boundary conditions, we find that the wave numbers and the angle must satisfy the following equations:

where the integers are called Bethe quantum numbers . Thus all eigenvectors for are determined by all possible pairs that satisfy the equations. The energy is then given by:

The last step is the generalization for eigenvectors, which consist of superpositions of states with flipped spins.

r any

For eigenvectors consisting of superpositions of states with flipped spins, the approach is:

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 approach equations :

The eigenvectors are given with all combinations of the Bethe quantum numbers that satisfy the Bethe approach equations. A classification of the eigenvectors is therefore possible via the Bethe quantum numbers. The determination of all eigenvectors is however not trivial. The energy of the associated state can then easily be obtained using

can be specified.

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  1. H Bethe: On the theory of metals . In: Journal of Physics A Hadrons and Nuclei . Volume 71, No. 3-4, 1931, pp. 205-226. doi : 10.1007 / BF01341708 .
  2. ^ PB Wiegmann, Soviet Phys. JETP Lett. , 31 , 392 (1980).
  3. ^ N. Andrei: Diagonalization of the Kondo Hamiltonian . In: Phys. Rev. Lett. . 45, No. 5, August 1980, pp. 379-382. doi : 10.1103 / PhysRevLett.45.379 .
  4. ^ PB Wiegmann: Towards an exact solution of the Anderson model . In: Physics Letters A . 80, No. 2-3, September 1980, pp. 163-167. doi : 10.1016 / 0375-9601 (80) 90212-1 .
  5. Kawakami, Okiji: Exact expression of the ground-state energy for the symmetric anderson model . In: Physics Letters A . 86, No. 9, 1981, pp. 483-486. doi : 10.1016 / 0375-9601 (81) 90663-0 .