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The Mermin-Wagner Theorem - Condensed Matter Theory Group

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How symmetry breaking occurs in principle<br />

Actors<br />

Proof of the <strong>Mermin</strong>-<strong>Wagner</strong> <strong>The</strong>orem<br />

Discussion<br />

<strong>The</strong> Bogoliubov inequality<br />

<strong>The</strong> <strong>Mermin</strong>-<strong>Wagner</strong> <strong>The</strong>orem<br />

Substituting what we have found in the Bogoliubov inequality and<br />

summing over all the wavevectors of the first Brillouin zone we get:<br />

βS(S + 1) ≥ M2<br />

N2g 2<br />

j µ2 <br />

1<br />

B |B0M| + k<br />

k<br />

22NQS(S + 1)<br />

We are finally ready to prove the <strong>Mermin</strong>-<strong>Wagner</strong> <strong>The</strong>orem. In the<br />

thermodynamic limit we find:<br />

S(S + 1) ≥ m2vdΩd β(2π) dg 2<br />

j µ2 k0<br />

B 0<br />

k d−1 dk<br />

|B0M| + k 2 2 QS(S + 1)<br />

Andreas Werner <strong>The</strong> <strong>Mermin</strong>-<strong>Wagner</strong> <strong>The</strong>orem

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