Hubbard model

The Hubbard model is a rough approximation method of solid state physics. It provides a description of the behavior of electrons in a assumed to be rigid lattice. The repulsive Coulomb forces are only for electrons residing on the same lattice site, taken into account. The kinetic electron energy fraction is modeled by an overlap integral, which comes from the tight-binding model. It is named after the British physicist John Hubbard.

The Hamiltonian for the Hubbard model is

It is

• The sum of the summation over all the grid squares,
• The sum of the sum over all pairs of adjacent grid locations,
• The sum of the summation of the two spin directions, and,
• And and for the fermionic creation and annihilation operators of an electron at lattice site with spin direction.

Determines the strength of the Coulomb repulsion, is calculated from the overlap of wave functions on adjacent lattice sites.

The sum of the Coulombterms determines the doubly occupied lattice sites. Therefore, can the value of the relevant place determined by the following integral:

In sum, for the hopping of electrons means that is summed only over neighboring lattice sites. Also, is automatically taken into account by the Operator constellation the Pauli principle.

The Hubbard model is the simplest model, in which one can study the interplay of kinetic energy, Coulomb repulsion, Pauli principle and band structure. Despite its simple structure, it is so far not succeeded, the exact solution of this model, except in the limiting cases of one and infinite dimensions to find.

There is, for example, in connection with

• Properties of electrons, which are relatively highly localized;
• Magnetism tape (Fe, Co, Ni, ...);
• Metal -insulator transition;
• High-temperature superconductivity

Discussed.

A variation approach to the Hubbard model is known as Gutzwiller approximation.

• Statistical Physics
• Solid State Physics