Understanding Ising Cell Hamiltonian

[LagrangeEuler]

I don't understand this idea. For example we have cubic crystal which has a lot of unit cells. We define spin variable of center of cell like S_c. And spin variable of nearest neighbour cells with S_{c+r}. So the cell hamiltonian is

\hat{H}=\frac{1}{2}J\sum_{c}\sum_{r}(S_c-S_{c+r})^2+\sum_cU(S_c^2)
This model is simulation of uniaxial feromagnet.

I have three question:
1. What's the difference between Ising model and 1d Heisenberg model?

2. Why this model is better than Ising model with no cells? Where we have just spins which interract.
\hat{H}=-J\sum_iS_{i}S_{i+1}

3. What \sum_cU(S_c^2) means physically?

Tnx.

 

[cgk]

I have three question:
1. What's the difference between Ising model and 1d Heisenberg model?

If I am not mistaken, in the Heisenberg model the spins can have an arbitrary polarization, not just up or down (and there are Heisenberg models with different J paramters in different directions, like the XXZ Heisenberg model). The "1d" in the model then applies to the /lattice/ dimension. That is, for the 1d Heisenberg model, you might think of a one-dimensional lattice in 3d space, where the spins are not actually one-dimensional.

2. Why this model is better than Ising model with no cells? Where we have just spins which interract.
\hat{H}=-J\sum_iS_{i}S_{i+1}
3. What \sum_cU(S_c^2) means physically?

I have little knowledge of spin models, but the model you wrote down looks to me like it includes some variant of a "Hubbard U" term. In electron models, such terms represent a strong local interaction which gives a penalty for two electrons occupying the same lattice site (a kind of screened Coulomb interaction, if you wish). In such models the U is used to tune a system between weakly correlated limits (simple metals) to strongly correlated limits (Mott-insulating anti-ferromagnet), and maybe other phases depending on the lattice type.

In the Hubbard case, the U term is normally written as U\cdot n_{c\uparrow}\cdot n_{c\downarrow} where the n are the up/down spin occupation number operators of electrons (thus giving only a contribution if there are both up and down electrons on the same site). But this form can be re-formulated into a similar form involving the total electron number operator and squared spins: \langle n_{c\uparrow}\cdot n_{c\downarrow}\rangle=\langle n\rangle - 2/3 \langle S_c^2\rangle (or something to that degree..don't nail me on the prefactors). In spin models the site occupation is of course normally fixed at one spin per site, but maybe the squared spin term still can fulfill a similar role.

 

[LagrangeEuler]

Tnx for the answer. So I can say in 1 - dimensional Heisenberg spin can pointed in any direction.

I'm not quite sure about Hubbard but I will look at it.

Do you know maybe the answer of second question?