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Hubbard Model

The greatest success of band theory of solids is that it can classify a large number of crystalline solids as metals or insulators. However, band theory fails in some cases because it ignores correlation effects (electron-electron repulsions). Since a band with orbital degeneracy nn can hold 2n2n electrons per unit cell, it is necessary to have an even number of electrons per unit cell to fully occupy all the bands. Therefore, there are an odd number of electrons per unit cell, the material cannot be an insulator.

The Hubbard model is the simplest many-body Hamiltonian which describes two opposing tendencies:

  • The kinetic energy (electron hopping) acts to delocalize the electrons into itinerant (Bloch) states, leading to metallic behavior.
  • The electron-electron interaction (approximated here by the onsite Coulomb interaction) wants to localize the electrons onto sites.

Metal Insulator Transitions​

Metal insulator transition (MIT), first proposed by Mott, is a prime example of strongly correlated electron system. The most successful model for describing MIT is the Hubbard model. If a material has vanishingly small electrical conductivity when a weak electrostatic field is applied at zero temperature, we classify it as insulator. Electrons are responsible of the flow of charge in a material, which are subjected to the Coulomb interaction with ions and other electrons. The first class of insulators are the band insulators which can be understood in terms of single electrons that interact with the electrostatic field of the ions.

On the other hand, there is another class of insulators where the insulating behavior is understood as a many-electron cooperative phenomena. These type of insulators are Mott insulators.

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