Heisenberg model (quantum)

The Heisenberg model in the quantum mechanical formulation is a much used in theoretical physics mathematical model for describing ferromagnetism ( and antiferromagnetism and ferrimagnetism ) in solids. 1928 Werner Heisenberg and Paul Dirac realized that ferromagnetism can be described in a solid state by an effective Hamiltonian, which does not contain the quantum mechanical spatial functions because it is constructed only from interacting localized electron spins on a lattice ( the lattice ). The interaction is (initially) reduced to adjacent pins ( nearest neighbor interactions ):

The known and the quantum vector operators due spin quantum number S are ({1 /2, 1, 3/2, 2, ... }). The indices i and j refer to the lattice positions, with the grid is a chain ( 1-dimensional Heisenberg model ), a two-dimensional grid (such as a hexagonal lattice ) or a three-dimensional array (e.g., a cubic lattice ) can be meant. The spin, however, is always three-dimensional, which is why it is also referred to as a special case of the n-vector model with the Heisenberg model.

The aim of the analysis is to be modeled experimentally observed effects such as the spontaneous magnetization and the critical exponent of the phase transitions.

The exchange interaction between the localized spins is caused by the Coulomb repulsion and the Pauli principle and a restriction to nearest -neighbor interaction and isotropic (see below) with a single coupling constant, the so-called exchange energy, expressed. The model is suitable to ferromagnetism in insulators to describe qualitatively but fails for most metals ( the Hubbard model is more appropriate here). The model can be justified by a generalization of the Heitler-London approximation for the formation of diatomic molecules (see the relevant sub-chapter in magnetism ). For one-dimensional systems, it can be solved exactly; in two and three dimensions, there are, however, only approximate solutions, for example with quantum Monte Carlo methods. In contrast to the classical Heisenberg model, the spins are expressed by operators and obey the rules of quantum mechanics.

Notes to the model

Ferromagnetism of insulators is caused by localized magnetic moments that are due to an incompletely filled electron shell ( 3d, 4d, 4f and 5f). This localized magnetic moments is an angular momentum associated which can be expressed by the respective spinning:

The spin vector is given about the Spin-1/2-Operatoren is the Landé factor and the Bohr magneton. The exchange interaction between the magnetic moments can be expressed by the corresponding pins. Thus, the exchange interaction simulates the Coulomb repulsion and the Pauli principle. The coupling constants between the localized spins are therefore also called exchange integrals. It is believed that the exchange integrals are different for neighboring spins significantly from zero. Overall, therefore, one obtains an effective Hamiltonian which is designed merely to explain the ferromagnetism in insulators.

Generalizations of the model

The Heisenberg model can be generalized by making the coupling constant dependent on direction (ie by going from isotropic to anisotropic systems).

A special case of the generalized Heisenberg model is the XXZ model, which is generated by the reduction of and:

The Heisenberg model and its special cases are often considered in the context of an applied magnetic field in the z- direction. The Hamiltonian is then:

Another generalization involves the integration of linkages not only between nearest neighbors as well as from inhomogeneities:

The transitions to the XY model and the Ising model can be best represented in the n- vector model.

Model in the k-space

For the analysis of the model and to the consideration of the suggestions it makes sense to look at the model in k-space. Transformation ( discrete Fourier transform ) for the spin operators are:

The generalized Heisenberg model in a magnetic field with no directional dependence with and can be written as then:

Where the exchange integrals are wavenumber- dependent:

Ground state

This section of the ground state of the generalized Heisenberg model is considered in the magnetic field dependence without direction. The ground state is the eigenstate of the system with the lowest energy. This is highly dependent on the sign of the coupling constant.

Under a rotation of all the spin vectors of the Heisenberg model does not change, so it is invariant under a rotation. For it is energetically favorable for the spins to align in the same direction and one speaks of a ferromagnetic ground state. Due to the rotational invariance is no preferred direction, so the alignment is assumed in the z- direction. The direction in the solid state is determined by anisotropy or by a weak applied magnetic field. In the ferromagnetic ground state all spins are aligned in one direction. We specialize yet

Then the ground state energy can be expressed as:

Here, the eigenvalue of the operator was as used. For the Spin-1/2-Heisenberg-Modell.

For it is energetically more favorable when neighboring spins pointing in different directions. The ground state is therefore highly dependent on the underlying crystal lattice. This can then be antiferromagnetic or ferrimagnetic. For special crystal lattice, it can also lead to magnetic frustration.

Magnons and spin waves

In this section, the suggestions of the ferromagnetic ground state of the generalized Heisenberg model are considered in the magnetic field dependence without direction. The excited states are assigned to the quasiparticle Magnon. It involves collective excitations of the entire crystal lattice and these are therefore referred to as spin waves.

The eigenstates of the operator are also eigenstates of the Hamiltonian, there and commute:

The unique application of the operator on the ferromagnetic ground state is thus an excited eigenstate of the Heisenberg model and is called the ( normalized) one- Magnonenzustand:

The corresponding state of the energy is given as:

The excitation energy is attributed to the already mentioned magnon quasiparticles. If we consider the expectation value of the operator of this condition is obtained:

The left side of the equation is no longer the place i depends. This clearly means that the excitation from the ground state ( A - Magnonenzustand ) is not generated by simply flipping a spin on a lattice site, but that the A - Magnonenzustand is evenly distributed over the grid. Therefore, the state is viewed as a collective excitation and referred to as spin wave.

1D Heisenberg model

In the one-dimensional Heisenberg model, the spins are strung on a chain. For periodic boundary conditions, the chain is closed into a ring. The eigenstates and eigenenergies for the one-dimensional Heisenberg model has been accurately determined in 1931 by Hans Bethe with the Bethe approach.

Eigenvectors and eigenstates

Since the operator commutes with the Hamiltonian decomposes the whole Hilbert space into different rooms, which can be diagonalized separately.

The different subspaces can be described by their quantum numbers. This means that the eigenvectors are superpositions of basis states with the same quantum number. In the Bethe approach, these states are classified by the folded state from the ferromagnetic ground state. For example, the state with two folded spins is ( also) at the lattice sites and is expressed as:

The eigenvectors in a subspace with a quantum number are superpositions of all possible states

The coefficients are given by plane waves and the Bethe approach.

The parameters can be determined from the equations of the Bethe approach

The eigenvectors are given by all combinations of the Bethe quantum numbers that satisfy the equations of the Bethe approach. A classification of the eigenvectors is possible via the Bethe quantum numbers. However, the provision of all eigenvectors is not trivial. Diezugehörige energy of state is given as:

Jordan -Wigner transformation

The 1D Heisenberg model can be mapped onto spinless fermions with periodic boundary conditions by means of a Jordan -Wigner transformation to a chain with only nearest neighbor interaction. The Hamiltonian of the 1D Heisenberg model can therefore be written as:

They are creation and annihilation operators for spinless fermions.