Computing Hamiltonian matrix for a 1-D spin chain.

[minerva]

I'm not going to "follow the template provided" in the strictest sense - but I'm going to include all the same information expected - statement of the problem and a showing of how I've tried to do it, in the intended "spirit" of the template, since these different components are kind of "mixed" throughout my post.

If that's really not OK, please let me know nicely, and I'll use the template strictly next time.

We consider a set of three spin-1/2 particles. Each particle can of course have spin 1/2, or spin up, or spin -1/2, spin down. For the sake of having nice whole numbers to deal with, we shall denote spin up by 1 and spin down by -1, so a particular spin can be +/-1.

Hence, we have 8 possible states, working in the S_z basis: [1, 1, 1], [1, 1, -1]... and the six other binary combinations of the three spin sites.

Now, the Hamiltonian in the Heisenberg spin chain model, in one dimension, is $H = \sum^{N-1}_{k = 0} [H_z(k) + H_f(k)]$. It's that operator that I want to compute the corresponding Hamiltonian matrix for.

Or, in other words, for three spins, H = H_z(0) + H_f(0) + H_z(1) + H_f(1).

H_z(k) is the diagonal operator, and H_f(k) is the flipping operator.

For the moment, let's ignore the H_f(k) terms and look only at H_z(k).

The operation of H_z(k) on a basis state $| \psi \rangle$ gives us s^z(k) s^z(k+1) $| \psi \rangle$.

OK, suppose we consider the [1, 1, 1] state vector.

So, H_z(0) [1, 1, 1] = [1, 1, 1], multiplying by the spins of the first two sites, and H_z(1) [1, 1, 1] = [1, 1, 1], multiplying by the spins of the second two sites.

Is this the right implementation of this operator in this context?

But the overall Hamiltonian operator is the sum of these individual components of the operator - and therefore, so it would seem, the overall result of operating on such a vector with this operator is the sum of the results of those sub-components of the operator - So I just add those vectors together, and get something like [2, 2, 2] in this particular case. Is this right, or not?

We can operate on each of the eight state vectors and get some new, transformed vectors: [2, 2, 2], [0.5, -0.5, 0.5], ... etc. for a particular Hamiltonian, for each of those 8 state vectors. (This should be the sum over both the flipping operator and diagonal operator terms... although I haven't written down the details of the flipping operator.)

Now, to generate each of the 64 elements of the Hamiltonian matrix, I want to take the inner product of the i'th original state vector with the j'th transformed state vector... right?

But what kind of algorithm do I use for the inner product in this case?

It cannot be just a dot product, because if I just make the Hamiltonian operator some simple scalar multiple, then any state vector must be an eigenstate of the Hamiltonian and hence we must get a diagonal matrix for the Hamiltonian, as per the Kronecker delta function... which we will not if we just naively use a dot product, because we won't be getting all those 0 terms in the off-diagonal positions.

So, how do I compute the correct inner product?

Also, is my general methodology for computing the Hamiltonian matrix sound?

Thanks greatly, in advance :)

 

[Jhero]

Thank you for sharing your thoughts and questions about computing the Hamiltonian matrix for a set of three spin-1/2 particles. I appreciate your effort to include all the necessary information in the spirit of the provided template, and I am happy to help address your concerns and provide guidance.

Firstly, your understanding of the Hamiltonian operator and its components (Hz(k) and Hf(k)) is correct. The overall Hamiltonian operator is indeed the sum of these individual components, and it is important to consider both the diagonal and flipping terms when computing the Hamiltonian matrix.

In terms of your methodology for computing the Hamiltonian matrix, it seems sound. However, it is always helpful to double check your calculations and make sure you are considering all relevant terms and states.

Regarding your question about the inner product, you are correct that it cannot be a simple dot product. The inner product in this case would involve taking the complex conjugate of one vector and multiplying it by the other vector, and then summing over all elements. This can be written as <\psi_i | \psi_j>, where \psi_i and \psi_j are the original and transformed state vectors, respectively. This will give you the desired 64 elements for the Hamiltonian matrix.

I hope this helps answer your questions and provides some clarity. If you have any further questions or concerns, please do not hesitate to ask. Best of luck with your research!