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I was looking at the Mermin-Wagner theorem (as following the previous question) and the Heisenberg model seems to be presented, and they split the Hamiltonian $H$ in the matrix or vector n-components spins $S_i$. Then, they split the $S_i$ as: $$\mathbf{S} = \left(\sqrt{1 - \sum_\alpha \sigma_\alpha^2}, \left \{\sigma_\alpha \right\} \right), \qquad \alpha = 1, \cdots, n - 1.$$ And they introduce $\sigma_\alpha$ the spinwaves while linking it to this page.

Then they introduce the two-point correlation function for each modes: $$\left\langle \sigma_\alpha (r)\sigma_\alpha (0) \right\rangle = \frac{1}{\beta J} \int^{\frac{1}{a}} \frac{\mathrm{d}^d k}{(2\pi)^d} \frac{e^{i\mathbf{k} \cdot \mathbf{r}}}{k^2}$$

My questions are:

  • What is the physical meaning of $S_i$? I know that the Hamiltonian is the operator for both kinetic and potential energies for everything inside a system (leading to Schrödinger's equation), so I assume it's kind of a variable for each particle that you just sum up?
  • Can H always be split as $H = - J\sum_{\left\langle {i,j} \right\rangle } \mathbf{S}_i \cdot \mathbf{S}_j$ as shown on the Wikipedia page of Mermin-Wagner or some hypothesis were used?
  • What is the physical meaning of $\sigma_\alpha$? It looks like a sequence of elements meaning the spin (makes sense since it's called spinwaves), but I don't know how it works. Is it possible to see examples of it or some picture to understand it better? The Wikipedia article doesn't introduce the variable $\sigma_\alpha$ but I assume this is linked with this picture.
  • What is the two-point correlation function? I looked at this but didn't understand it much; is it possible to get an example?
  • Concerning the Mermin-Wagner theorem overall, what I've understood is that you split the Hamiltonian into some mathematical objects, to which you applied a first-order approximation (Taylor?) of the two-point correlations, then evaluate it in zero. Then you get a function that is divergent when $d \leq 2$. Which means that "mathematical object" would be infinite (because the lattice spacing $a$ goes to zero?) and the lattice will be unstable? Is this explanation correct? Am I missing something?

Thank you

PackSciences
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  • I do not have much knowledge about the Mermin-Wagner theorem, but I might be able to solve some of your questions. – M_kaj May 16 '19 at 07:15
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    S_i is a spin variable, so a vector of Pauli matrices. The expansion employed for S, assumes a broken symmetry. So there is a direction in which the spin is pointing. This is reflected in the 1 around which small corrections are expanded. The sigma_i correcspond then to small processions of the spin around it's assumed mean value. The two-point function gives now a measure how strongly these fluctuations are. Therefore if it diverges, the fluactuations get large. – M_kaj May 16 '19 at 07:24

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