Mott insulators and Hubbard Model
Shuhua Liang人教版七年级下册语文
UTK physics
sliang@utk.edu
Hubbard Model
Band theory is very successful at describing 'small interaction', 'low temperature' and 'ordered' systems. However, it can not explain why some metal oxides that were predicted to be conductors by band theory were in fact insulators. Mott found out the answer. It is the effect of 'Column interaction'. When all sites are half filled and the electronelectron interaction is much larger than the hoping term ( the overlap of two neighbor electrons wave function), two electron cannot be in one site, so they can not move anymore. In other words, we can say: becau band theory assume electrons are delocalized and solve the total system in a "one electron" wave function. This is the difference.
In order to show the repulsion of two electron, Hubbard introduced a
'U*n1*n2' term into the Hamiltonian.
So, the total energy will be very high if there are two electrons on one site causing the system to prefer to choo a lower energy state.
(this picture shows the column repulsion
U term opens a gap with a width U
猛洞河between the full occupied site and non
full occupied site)
After introducing the Hubbard Model, the strong interaction will open a gap between the previous 'continuous' band in 'band theory when U is sufficiently large. This rembles a competition between the kinetic energy and the potential energy. If 't/U' is larger enough, the two parated bands are going to merge together!!
矮冬瓜
The time when the two
bands merge together is
the point this material
changes from Mott
insulator into conductor.
It is called 'insulator
metal' transition.
Hubbard model successfully explained why some oxide is insulator why some is not. It is all dependent upon the fraction of (t/U). So , to some material where "t" is nsitive to pressure, we can change it from insulator to metal just by adding more pressure. Then, the atoms distance will decrea, and there will be more overlap "t" of the neighbor electrons' wave function. Since the hopping constant "t" increas, it is possible for electrons to move. This is only one way of "MIT"
Hubbard Model's disadvantage and tJ Model
How ever, it is too difficult to solve the Hubbard Model completely, becau the degree of freedom is proportional to 4^N (N is the number of sites). Today even computers cannot solve system with more than 100 electron using Hubbard model.
动漫桌面So what we can do is find out the best approximation of the Hubbard model in each particular restriction.
When U>>t, two electrons do not want to share one site any more. So, people introduced the tJ model for average electron number per site smaller than 1 situation. This means that when we choo tJ model, the maxim dimension is 3^N instead of 4^N. This greatly simplified the problem.
The bas, using in the tJ model, is a sub
space of Hubbard model bas. Each site only
contains:(empty, up, down) three
20年后的家乡configurations.
In simplest two sites model, they are:
合同补充协议书范本
Totally 9=3^2 in the low band
However, the they cannot be constructed into the t of “2 electron” eigenstates of Hubbard model Hamiltonian which containing full occupied site state. We neglect the up band 7 configurations.
Project from whole space into subband space
The official way to derive the tJ model by using a projector to parate the total Hamiltonian into 3 parts. The first (1) and the final (3) term reprent the previous two sub band parated by strong Column repulsion.
(1)
(2)
(3)
烽火四起In (1), no site is full occupied.
In (3) no site is empty and at least one site is fully occupied.
In (3) the hopping term reprent the electron moving from one full occupied site to a half occupied site, making it into a fully occupied site. The cond term (2) reprents the interaction of the two su
b band. In the “large U” approximation(e picture in page 1). The up and low bands are not connected, so they should not have any hopping between them, as that electron jump from full occupied site to empty site. However, it is a very important part with an complex value and many orders added together which will infect the final result. It is similar to valcum energy in high energy
physics.
Actually, the mixed term is equivalent to the spin interaction term. Under low temperature and high U/t, the up band will be empty, so the (3) term in the Hamiltonian is neglectable.
Finally, we get the effective Hamiltonian:
in the Kinetic energy
火的偏旁有哪些字
term. Only half filled
site can donate a non
zero term to total energy. It means, the electron can only jump from a half filled site to an empty site.
2 sites Hubbard model and TJ model
Since the TJ model is an approximation of Hubbard model. I tried to u the half filled sites as the ground state to calculate the total energy with the perturbation theory.
here are two sites two electrons Hamiltonian(total spin zero):
of the Hubbard model
0 0 t t
0 0 t t
t t U 0
t t 0 U
and form TJ model (2X2) matrix. (I enlarge the matrix to compare with Hubbard model)
J/2 J/2 0 0
J/2 J/2 0 0
0 0 0 0
0 0 0 0
