How to Diagonalize Hamiltonian in Bloch Approximation for Spin Systems?

[Petar Mali]

How to diagonalise Hamiltonian in Bloch approximation?

$$\hat{H}=-\frac{1}{2}\sum_{\vec{n},\vec{m}}I_{\vec{n},\vec{m}}\hat{S}_{\vec{n}}^x\hat{S}_{\vec{n}}^x-\Gamma\sum_{\vec{n}}\hat{S}_{\vec{n}}^z$$

 

[kanato]

I would start by writing S^x in terms of raising and lowering operators. What is the Bloch approximation?

 

[Petar Mali]

Replacing spin with Bose operators I think!

$$S_{\vec{n}}^+=\sqrt{2S}B_{\vec{n}}$$


$$S_{\vec{n}}^-=\sqrt{2S}B_{\vec{n}}^+$$


$$S-S_{\vec{n}}^z=B_{\vec{n}}^{+}B_{\vec{n}}$$

 

[kanato]

So, maybe a little more background would be helpful. What exactly is your question?

If you want to calculate spin waves you need to choose a reference ground state which will affect the form of your Bose operators (right now it looks like you've chosen a ferromagnetic ground state). For spin waves then you need to Fourier transform your boson operators.

But I am a bit surprised by the appearance of the lone S_z operator, because usually the presence of a single spin operator is a problem for spin wave theory. But since S_z won't result in a single boson operator then it might be ok. Also, are both S_x operators acting on the same site, or is that a typo? If that's correct, you should carry out the sum over m first.