Dependence of exchange interaction on system size in Ising model

[cosmicraga]

In the well known Ising model, without any external field (H=0), the energy (E), spins (s) and exchange interaction (J) are related as in the following equation
$$
E = -\sum_{<ij>}J_{ij}s_{i}s_{j}
$$

J_ij is site dependent and consists of three components J_AA, J_BB and J_ABwhere A is say up spin and B is down spin on a lattice (say SL).

For a material of type A_xB_1-x with system size N = 100 (N is number of spins), I know the values of J_AA, J_BB and J_AB. Here J_AA=E(x=1)/n_AA and J_BB=E(x=0)/n_BB are constant for any x. J_AB changes with x. n_AA is number of AA bonds.

E(x=1) means Energy of the system when x=1 for A_xB_1-x material, i.e. Energy of the system when the system consists of only up spins (all A). Similarly E(x=0) means energy of the system when there is only down spins (all B).

x is composition, x=0.25 means 25% of N is A spins and rest are B spins.

**Queston 1**: If I increase my system size N to 200, then shall the values of J_AA, J_BB and J_AB change?

**Question 2**: If the values of J_AA, J_BB and J_AB change with N, then with what factor shall I increase it?

 

[daveyrocket]

Um this is confusing.
You are describing a system where J is spin dependent. This might be written as
E = -\sum_{&lt;ij&gt;}J_{ij}(s_i,s_j)s_is_j
This is NOT the Ising model. The Ising model has a constant J which is independent of the spin. But you are also describing a system which lacks some fundamental symmetries, so it's very confusing.

It sounds like what you want to describe is an inhomogeneous system, where you have two types of atoms, A and B. The exchange interaction is different between different types of atoms, which is why you have three values for J. This is an Ising model for a binary system.

Usually the answer to your question 1 is no. J is generally regarded as a microscopic interaction parameter which is rather localized and it is not affected by the extent of the system. But you have not specified your boundary conditions. If you have some sort of surface, the value of J might be different at the surface. If you are using periodic boundary conditions then J should not change.

 

[cosmicraga]

>> $$ E = -\sum_{<ij>}J_{ij}(s_i,s_j)s_is_j $$

Yes, this is better representation.

>> This is an Ising model for a binary system.

Yes, you are right.

>> But you have not specified your boundary conditions.

Yes, I am using periodic boundary condition in all directions.

Thanks for your answer. :)