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Hubbard model (Read 4214 times)
Gerrit-Jan Linker
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Hubbard model
30.05.10 at 14:46:33
 
Hubbard model
 
The Hubbard model is an approximate model used to describe the transition between conducting and insulating systems.
 
The Hubbard model is an approximation for particles in a periodic potential at sufficiently low temperatures that all the particles are in the lowest Bloch band, as long as any long-range interactions between the particles can be ignored.  
 
The Hubbard model is the simplest model of interacting particles in a lattice, with only two terms in the Hamiltonian: a kinetic term t allowing for tunneling ('hopping') of particles between sites of the lattice and a potential term U consisting of an on-site interaction. For electrons in a solid, the Hubbard model can be considered as an improvement on the tight-binding model, which includes only the hopping term.  
 
If interactions between particles on different sites of the lattice are included, the model is often referred to as the 'extended Hubbard model'.
 
Hubbard model Hamiltonian:
H = - ∑i≠j tija+a + ∑i Uini↑ni↓ + ∑εn
 
where:
a+ is the creation operator, creating an electron in orbital i with spin σ (either up ↑ or down ↓).
a is the annihilation operator, removing an electron in orbital i with spin σ (either up ↑ or down ↓).
n is the number operator gives the number of electrons in orbital i with spin σ
tij is the hopping integral, the energy associated with the transfer of an electron between neighboring lattice sites. (t is a Hamilton integral and not an overlap integral)
i≠j is the requirement that electrons are transferred between different orbitals. This is often restricted further to only allow transfers to neighbouring sites.
Ui is the electronic repulsion of two electrons with opposite spin in orbital i
ε is the energy of an electron with spin σ in orbital i. This third part of the Hubbard hamiltonian is sometimes referred to as the "free part"
 
 
u]source:[/u]
http://en.wikipedia.org/wiki/Hubbard_model
http://titus.phy.qub.ac.uk/group/Eunan/Project.pdf
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« Last Edit: 08.12.12 at 14:01:21 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
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Gerrit-Jan Linker
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Extended Hubbard model
Reply #1 - 06.06.10 at 21:36:20
 
Extended Hubbard model
 
In the extended Hubbard model another term is added to the Hamiltonian to describe the Coulomb interactions between neighbouring sites:  
 
i≠j Vijninj
 
Thus, the standard Hubbard model is when Vij=0
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Gerrit-Jan Linker
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Hubbard model: hopping integral t
Reply #2 - 03.12.13 at 18:49:29
 
Hubbard model: hopping integral t
 
The hopping integral t is equal to the Hamilton matrix element of determinants |aa| and |ab| where the underscore denotes beta spin.
 
t = < |aa| H |ab| >
 = < aa - aa| H |ab -ba >
 = < aa | H |ab > + <aa| H |ba >
 = <a|H|b> + <a|H|b> = <a|H|b>
 
This is roughly equivalent to the overlap β = <a|H|b>.
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« Last Edit: 03.12.13 at 19:37:47 by Gerrit-Jan Linker »  

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