Calculating Magnetization for 2x2 Ising Model Lattice

[LagrangeEuler]

Homework Statement

Calculate magnetisation for partition function
$Z=12+4\cosh (8\beta J)$ for Ising model 2x2 lattice.

Homework Equations

$$F=-k_BTln Z$$
$$M(H,T)=-\frac{\partial}{\partial H}(\frac{F}{k_BT})$$

The Attempt at a Solution

For me it looks that magnetisation is zero. By just doing the derivatives. But solution of problem is $\frac{16+8e^{\beta J}}{Z}$. If I combine relation from relevant equation it looks like for me that solution is zero. Where I making the mistake.

 

[M Quack]

Where does the partition function come from?

There is no H in the partition function, so the system does not react to any external field, just as you calculated.

What are the units of J?

 

[LagrangeEuler]

$J$ is constant. Well you have 2x2 lattice. You calculate
$$Z=\sum e^{-\beta H}$$

 

[kreil]

You are confusing notation here. The derivative to find the magnetization should be with respect to h, the external magnetic field. H is the Hamiltonian used to find the partition function. If there is no term dependent on h in your partition function expression, then you have used an incorrect Hamiltonian. Assuming a constant interaction J and constant external field h, the 2D Ising Model has a Hamiltonian of the form

$$H=-J \sum_{i,j} \sigma_i \sigma_j - h\sum_j \sigma_j$$

The first sum is over all nearest neighbors.As written your partition function trivially has no magnetization.

 

[Oxvillian]

I think there are a few sources of confusion here...

It seems that the problem presents a certain partition function as a given. Presumably it's some kind of approximation for the 2-d Ising model Z.

Also it looks like J is some parameter that's proportional to the applied magnetic field. So dZ/dH won't be zero.

 

[Mute]

I think there are a few sources of confusion here...

It seems that the problem presents a certain partition function as a given. Presumably it's some kind of approximation for the 2-d Ising model Z.

Also it looks like J is some parameter that's proportional to the applied magnetic field. So dZ/dH won't be zero.

J is typically the spin-spin coupling in Ising models, and I suspect that is the case here. The partition function given simply doesn't depend on the magnetic field.

To the OP: why should one expect the magnetization to be non-zero in the absence of a magnetic field? You have a 2x2 Ising model at finite temperature. Furthermore, I assume that the spin-spin interactions are nearest neighbor only, is that correct?

If so, consider this question: what is the difference between your 2x2 Ising model and a 4-spin 1d Ising model with periodic boundary conditions (and no external field)?