Understanding Bose-Hubbard Model: Senior Guide

[victorphy]

Would anyone help explain Bose-Hubbard Model for me?More detailed,more helpful.
Besides,I am trying to read a paper named Boson localization and the superfluid-insulator transition,but I find I can hardly understand it.What can I do if I want to access to as more knowledge as possible? I am a senior and just have elementary knowledge of QM.
Thanks in advance.

 

[xepma]

The Bose-Hubbard is a lattice model. It's quite a rich model, and can only be solved exactly in certain limits. Apart from being a relatively simple model to write down, it turns out to contain a wealth of funky physics, and people apply all sorts of techniques to understand the behavior of the model in certain limiting cases.

The model has a number of applications: it turns out to approximate the behavior of electrons in a lattice, and it is also related to superconductivity.

The basic rules of the game are as follows:

(1) consider a lattice -- it can be defined on any lattice with an arbitrary dimensions, but for simplicity just imagaine the two-dimensional square lattice.

(2) Each lattice point is either occupied by a boson or not. (The Hubbard model is the case where you would talk about fermions instead of bosons). The total number of bosons present is not fixed, and can take on any value. But only for particular values (i.e. half filling: half as many bosons as there are lattice points) are we able to perfroms some of the calculations (the number of bosons can be fixed by inserting a chemical potential term -- this is a standard technique in condensed matter physics)

(3) The Hamiltonian of the system has two "terms" (each term is a summation). A hopping term and an interaction term.

(4) The hopping term is like a kinetic energy term. A boson can hop from one lattice point to the next, and this term measures the energy associated to that process. The difference with a normal kinetic term is that it actually favors hopping of the bosons: the more the bosons hop around, the lower the energy of the system is. The strength of the hopping is given by $$t$$.

(5) The interaction term, on the other hand, is an on-site interaction. It only contributes to the Hamiltonian if there are two bosons on the same site. In the interesting case this interaction is taken to be repulsive: if two bosons occupy the same site, this gives a positive energy contribution. To lower the energy you would want a maximum of one boson per site. The strength of the repulsion is given by $$U>0$$

This is basically the model. Now you can start to tweak the parameters. As you can see, the hopping and interaction term are competing against each other. If the hopping term is dominating (t>U) the bosons roam around freely. If the interaction term is dominating, the behavior of the bosons highly depends on the number of bosons. If there are an equal number of bosons as there are lattice points, then the ground state of the system equals the case where there is exactly one boson per site. If you are at half-filling things start to get a little more interesting.

 

[victorphy]

Thanks for your explanation.

If you are at half-filling things start to get a little more interesting.

I am interested,what will happen in this situation?

 

[victorphy]

About the article I mentioned above,I know I have to learn more to get it right,like advanced QM,statistic physics,field theory and so on.What I want to know is if there are some articles on the same topic but more easier to understand.Or what I have to learn first if I have to finish that article?
Thanks

 

[xepma]

Well, I know that for the fermionic Hubbard model (i.e. instead of bosons you have fermions which carry spin -- the model is a little more involved, due to the fact that you now have two types of particles (spin up and spin down) and also have to deal with the antisymmetric nature of the wavefunction) at half filling the degrees of freedom of the system are equally well described by the anti-ferromagnet of spin-1/2 particles. And that system can be solved using either a Bethe ansatz or bosonization techniques -- really advanced stuff (there's a book by Fradkin on this, but I doubt that it gives any clarification).

I don't think I can help you with the problem of a good source. You really need a good understanding of statistical physics to tackle that article. I know that the book by Marder on Condensed Matter Physics treats the Hubbard model at low-level, but I doubt that it would be of any use.

 

[victorphy]

Well,thanks a lot.