Interacting Spins , Heisenberg Hamiltonian , Tensor product

[Phileas.Fogg]

Hello,
I'm studying the Heisenberg Model. Given the Hamiltonian

$$H = - 2 \frac{J}{\hbar^2} \vec{S}_1 \vec{S}_2$$

with

$$\begin{equation} \vec{S} = \frac{\hbar}{2} \; \left( \begin{array}{ccc} \sigma_x \\ \sigma_y \\ \sigma_z \end{array} \right) \end{equation}$$

$$\sigma_{x,y,z} \quad {\text are \; the \; Pauli \; Matrices}$$

Supposed, there are two electrons interacting antiferromagnetically.
So I have a spin-spin interaction. The first is up and the second spin is down.

The texts says that this results in the following Hamiltonian:

$$\begin{equation} H = - J \; \left(\begin{array}{cccc} 0,5 & 0 & 0 & 0 \\ 0 & -0,5 & -1 & 0 \\ 0 & -1 & -0,5 & 0 \\ 0 & 0 & 0 & 0,5 \end{array} \right) \end{equation}$$

I know it has something to do with the tensor product of both spin vector operators.
I tried to calculate by myself, but didn't get the correct result.

Could anyone explain, how this works?!

Thanks.

Regards,
Phileas

 

[peteratcam]

You haven't made it very clear what goes wrong when you try to derive this.

You need to expand the dot product first (in your orginal hamiltonian you haven't included a dot product):
$$H = J(\sigma_x\otimes\sigma_x + \sigma_y\otimes\sigma_y + \sigma_z\otimes\sigma_z)$$
If you have Matlab, the kron function does the direct products exactly as required.
Did you get that far?

Perhaps the issue is appreciating that $$S_1^x$$ (a 4x4 matrix) is not the same as $$\sigma_x$$ (a 2x2 matrix) ?

Maybe question 3 of 'Some questions on spin' and solutions here might help:
http://www-thphys.physics.ox.ac.uk/people/PeterConlon/teaching.shtml
And if you have Matlab/Octave, here also:
http://www-thphys.physics.ox.ac.uk/people/PeterConlon/notes/spin.m

 

[Phileas.Fogg]

Thanks peteratcam,
I used the wrong expansion before, so I've the correct result now.

How can J look like in this case? I think, it must be a 4x4 Matrix, right?
Is the product of matrix J and the spin-spin-coupling matrix (I call it A) a tensor product, or just a normal matrix multiplication?
$$\begin{equation}A = \left(\begin{array}{cccc} 0,5 & 0 & 0 & 0 \\ 0 & -0,5 & -1 & 0 \\ 0 & -1 & -0,5 & 0 \\ 0 & 0 & 0 & 0,5 \end{array} \right) \end{equation}$$

Finally, I have to diagonalize the resulting Hamiltonian matrix, right? I do this, because then I find the eigenvalues on the diagonal of the matrix? Or, does it have to do something with numerics?

Regards,
Phileas

 

[peteratcam]

J is not an operator/matrix, it is just a number, the strength of the coupling. (If you really want, you could think of it as proportional to the identity operator, but this doesn't really get you anything.)

In an anisotropic model, you might have separate J_x, J_y, J_z, but then H would look like:
$$H = J_x \sigma_x\otimes\sigma_x + J_y \sigma_y\otimes\sigma_y +\ldots$$
The J are still just parameters which are numbers.

The products of operators which appear in a hamiltonian are always matrix products. The direct product comes into construct the appropriate operators which act on composite systems.

For example, the operator for the x-component of the first spin when the composite system is two spin-1/2 systems is:
$$S^{(1)}_x = \sigma_x\otimes \mathbf 1$$
and for the second spin
$$S^{(2)}_x = \mathbf 1\otimes \sigma_x$$
You can then show that the matrix product $$S^{(1)}_xS^{(2)}_x$$ is the same as the direct product $$\sigma_x\otimes\sigma_x$$.

This all assumes that the basis for the composite system is constructed in the obvious way from the basis you use for the single particle systems.

Do you want to diagonalise the Hamiltonian? I don't know, depends what you want to do. If you do, you will notice that the eigenvectors are the singlet and triplet states.

 

[Phileas.Fogg]

Okay, thanks.

I found an article, where the author says:

$$J_{i,j}$$ is a symmetric matrix containing the exchange parameters between spins at sites i and j.

How does the spin vector operator look like, for Ions with spin e.g. 3/2 or 5/2 ?

Regards,
Phileas

 

[lemma]

Hi there,

I need to solve the problem for 3 spins (find the Hamiltonian), and I am really not certain about how to construct it.
It is meant to be a 8x8 matrix, and is defined as H=〖J(S〗_1 S_2+S_2 S_3+S_3 S_1).

Any help or hint would be most appreciated.

Thanks in advance,

- lemma