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I've been reading about the Mermin-Wagner theorem recently. I think I understand pretty much every computation need to derive its result from the Bogoliub inequality, but there is one thing I don't understand. In the expression for the Heisenberg hamiltonian in the original proof, $$H = - \sum_{i,j} J_{ij} \vec{S}_i \vec{S}_j - b \sum_i S^z_i e^{- i \vec{K} \vec{R}_i},$$ I don't understand where the exponential and the $\vec{K}$ are coming from. I read in The absence of finite-temperature phase transitions in low-dimensional many-body models: a survey and new results by Axel Gelfert, Wolfgang Nolting, that

"where the factor $e^{- i \vec{K} \vec{R}_i}$ already accounts for antiferromagnetic order by changing the sign of the spins in one sublattice (given that $\vec{K}$ has been properly chosen to achieve just that)."

In the original paper, pretty much the same detailless argument is given, ie

"To rule out ferromagnetism we will take $\vec{K}$ to be 0, and to exclude antiferromagnetism, we chose it such that $e^{- i \vec{K} \vec{R}_i}=1$ when $\vec{R}$ connects sites in the same sublattice, and -1 when it connects sites in different sublattices; [...]"

What I understand from these statements is that $\vec{K}$ is some parameter that we don't define now, because it will be different if we're in the case of ferromagnetism or antiferromagnetism. And at the end of the proof, when we have the expression of the inequality for the magnetization, we could replace $\vec{K}$ by the value we want to have the inequality valid for either ferromagnetism or antiferromagnetism. (but we don't because it doesn't really matter in the limit of the eternal field $B_0 \rightarrow 0$.

So my question is, did I understand this alright (or am I making some terrible, terrible mistake somewhere) ?

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  • $\begingroup$ Isn't it a reciprocal basis vector? $\endgroup$ – Mauricio Jan 7 at 15:12
  • $\begingroup$ well that's what i thought at first but then I don't understand why we need to go to the reciprocal lattice ? $\endgroup$ – Reflets de Lune Jan 7 at 15:47
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You understand it right. What the Mermin-Wagner theorem asserts is that there is no kind of magnetic ordering -- this includes ferromagnetic order ($\vec K =(0,0)$), antiferromagnetic order ($\vec K=(\pi,\pi)$), and any other order (such as mixed ferro-/antiferromagnetic order $\vec K=(0,\pi)$, and so forth).

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It's the (magnetic) ordering wave vector / momentum. It's very natural to have it in the order parameter, since it defines the right order parameter. E.g. for a one-dimensional chain, $k=0$ for a ferromagnet, or $k=\pi$ for an antiferromagnet, gives you a ferromagnetic or staggered magnetization, respectively. By putting it in the Hamiltonian, Gelfert and Nolting are compactly describing a magnetic field directed on each site as to cause the ordering vector of interest - assuming the system would order. Again, for the one-dimensional chain, $k=0$ would correspond to a uniform field, and $k=\pi$ a staggered one.

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