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The picture above shows Mermin-Wagner Theorem and its corollary. How can the corollary be derived from Mermin-Wagner Theorem?

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The corollary can be inferred from the Goldstone theorem: If the ground state breaks a continuous symmetry, there are gapless excitations. Thus, if the system is gapped, the symmetry cannot be spontaneously broken in the ground state.

This does, of course, not explain how the corollary follows from the Mermin-Wagner theorem (and it holds in any spatial dimension).

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