The Hubbard model is an approximation used in solid state physics to describe the transition between conducting and insulating systems. In particular, the Hubbard Model considers the hopping integral (the ability for electrons to jump between neighboring atoms), which is part of the tight-binding model from regular band theory, as the mode of conduction, but also considers electron-electron repulsion (i.e. the Coulomb potential between atoms) that is not usually considered in standard band theories. These two forces compete to determine whether a material is conducting or insulating. In particular, the Hubbard model can explain certain materials known as Mott insulators that are predicted to be conductors in standard band thoery models. Solid-state physics, the largest branch of condensed matter physics, is the study of rigid matter, or solids. ... Jump to: navigation, search Mott Insulators are a class of materials that are expected to conduct electricity under conventional band theories, but which in fact turn out to be insulators when measured. ...

Theory

The Hubbard model is based on the tight-binding approximation from solid state physics. In the tight-binding approximation, electrons are viewed as occupying the standard orbitals of their constituent atoms, and then 'hopping' between atoms during conduction. Mathematically, this is represented as a 'hopping integral' or 'transfer integral' between neighboring atoms, which can be viewed as the physical principle that creates electron bands in crystalline materials. However, the more general band theories do not consider interactions between electrons. By formulating conduction in terms of the hopping integral, however, the Hubbard model is able to include the so-called 'onsite repulsion', which is stems from the Coulomb-Coulomb repulsion between electrons. This sets up a competition between the hopping integral, which is a function of the distance between neighboring atoms, and the onsite repulsion, which is not. The Hubbard model can therefore explain the transition from conductor to insulator in certain transition metal oxides as they are are heated by the increase in nearest neighbor spacing, which reduces the 'hopping integral' to the point where the onsite potential is dominant. Similarly, this can explain the transition from conductor to insulator in such systems as rare-earth pyroxides as the atomic number of the rare-earth metal increases, because the lattice parameter increases as the rare-earth element atomic number increases, thus changing the relative importance of the hopping integral compared to the onsite repulsion.

Example: 1D chain of hydrogen atoms

The hydrogen atom has only one electron, in the so-called s orbital - it can have either a spin up () or down (). This orbital can be occupied by at most two electrons, one with spin up and one down. For a 1D chain of atoms, ignoring the existance of other orbitals, the Hubbard hamiltonian has the form: Jump to: navigation, search General Name, Symbol, Number hydrogen, H, 1 Chemical series nonmetals Group, Period, Block 1, 1, s Appearance colorless Atomic mass 1. ... Jump to: navigation, search Properties For alternative meanings see atom (disambiguation). ... The term orbital has several meanings: In physics and chemistry it is used to describe an atomic electron configuration, see also molecular orbital and atomic orbital. ...

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The first term, without contribution from the second, leads to conventional band spectrum in which each electron is delocalized throughout the crystal.

The second term, without contribution from the first, leads to localized magnetic moments.

See also

• Band theory
• Bloch wave
• Electronic band structure
• Energy band
• Solid state physics

In solid state physics, band theory is the theory of the behavior of the electrons in solids. ... A Bloch wave or Bloch state is the wavefunction of a particle (usually, an electron) placed in a periodic potential. ... In solid state physics, the electronic band structure (or simply band structure) of a solid is the series of forbidden and allowed energy bands that it contains. ... In solid-state physics, an energy band is a continuous range of values of energy that an electron may or may not have. ... Solid-state physics, the largest branch of condensed matter physics, is the study of rigid matter, or solids. ...

External links

• The Hubbard Model - A Numerical Introduction