Hubbard model

The Hubbard model (after the British physicist John Hubbard ) is a roughly approximate model of a solid and is therefore of great importance in solid-state physics . It describes the behavior of electrons in a lattice assumed to be rigid . The repulsive Coulomb forces are only taken into account for those electrons that are at the same lattice location. The share of the kinetic energy of the electrons is modeled by an overlap integral that comes from the tight binding model. ${\displaystyle t}$

The Hubbard model is the simplest model that can be used to study the interplay of kinetic energy, Coulomb repulsion , the Pauli principle and band structure . Despite its simple structure, it has not yet been possible to find the exact solution of this model, except in the limiting cases of one and infinitely many dimensions.

It is discussed e.g. B. in connection with

• Properties of electrons that are relatively strongly localized ;
• Tape magnetism (Fe, Co, Ni ...);
• Metal-insulator junction ;
• High temperature superconductivity .

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

formulation

The Hamilton operator for the Hubbard model is

${\displaystyle H=U\sum _{i}c_{i\uparrow }^{\dagger }c_{i\uparrow }c_{i\downarrow }^{\dagger }c_{i\downarrow }-t\sum _{\langle ij\rangle ,\sigma }\left(c_{i\sigma }^{\dagger }c_{j\sigma }+c_{j\sigma }^{\dagger }c_{i\sigma }\right)}$

It stands

• the sum over for the summation over all grid locations, ${\displaystyle i}$
• the sum over for the sum over all pairs of neighboring grid locations, ${\displaystyle \langle ij\rangle }$
• the sum over for the summation over both spin directions and , ${\displaystyle \sigma }$ ${\displaystyle \uparrow }$ ${\displaystyle \downarrow }$
• ${\displaystyle c_{i,\sigma }^{\dagger }}$and for the fermionic creation and annihilation operators of an electron at the lattice site with spin direction . ${\displaystyle c_{i,\sigma }}$ ${\displaystyle i}$ ${\displaystyle \sigma }$
• ${\displaystyle U}$ defines the strength of the Coulomb repulsion
• ${\displaystyle t}$is calculated from the overlap of wave functions at neighboring grid locations.

The sum of the coulomb term determines the double occupied grid positions. Therefore, the value of can be determined at the respective location using the following integral: ${\displaystyle U}$ ${\displaystyle \mathbf {x_{i}} }$

${\displaystyle U(\mathbf {x_{i}} )=\int d^{3}\mathbf {r_{1}} \int d^{3}\mathbf {r_{2}} \,\,\left|\Psi (\mathbf {r_{1}-x_{i}} )\right|^{2}{\frac {e^{2}}{\left|\mathbf {r_{1}-r_{2}} \right|}}\left|\Psi (\mathbf {r_{2}-x_{i}} )\right|^{2}}$

The sum for the hopping of the electrons means that it is only added over neighboring grid positions. In addition, the Pauli principle is automatically observed through the operator constellation. ${\displaystyle \langle ij\rangle }$

Individual evidence

1. ↑ HUBBARD, John. Electron correlations in narrow energy bands. In: Proceedings of the royal society of london a: mathematical, physical and engineering sciences. The Royal Society, 1963. pp. 238-257.