Tight Binding - Second Quantization

Second Quantization

Modern explanations of electronic structure like t-J model and Hubbard model are based on tight binding model. If we introduce second quantization formalism, it is clear to understand the concept of tight binding model.

Using the atomic orbital as a basis state, we can establish the second quantization Hamiltonian operator in tight binding model.

,

- creation and annihilation operators

- spin polarization

- hopping integral

-nearest neighbor index

Here, hopping integral corresponds to the transfer integral in tight binding model. Considering extreme cases of, it is impossible for electron to hop into neighboring sites. This case is the isolated atomic system. If the hopping term is turned on electrons can stay in both sites lowering their kinetic energy.

In the strongly correlated electron system, it is necessary to consider the electron-electron interaction. This term can be written in

This interaction Hamiltonian includes direct Coulomb interaction energy and exchange interaction energy between electrons. There are several novel physics induced from this electron-electron interaction energy, such as metal-insulator transitions (MIT), high-temperature superconductivity, and several quantum phase transitions.