A question about Haldane's conjecture Haldane's conjecture states that the integer spin antiferromagnetic Heisenberg chains have a gap in the excitation spectrum. However, the dispersion relation of the antiferromagnetic spin wave is $\omega_k\sim k$ in the long wave length limit, meaning that the excitation energy could be zero. What is the matter?
 A: To clarify and correct some of the above points ...
There is no long range order in 1D systems at zero temperature, as explicitly proved by Pitaevskii and Stringari in 1991; see 'Uncertainty Principle, Quantum Fluctuations and Broken Symmetries', J. of Low Temp. Phys. 85, 377. But I agree with @NorbertSchuch in that I doubt if Mermin-Wagner says this. 
The correspondence between classical and quantum systems, which @EverettYou explains slightly mistakenly, holds mostly at zero-temperature criticality i.e. the effective $D+1$-th dimension in the classical model is infinite in extent only at $T=0$ of the quantum model.
To answer the original question, spin-wave theory always builds on a mean-field solution, which is almost always based on an educated guess. If the starting guess is incorrect, so will most of what results follow.
A: Spin wave theory simply does not apply for 1D spin system. The starting point of the spin wave theory is a magnetically ordered ground state. But Mermin-Wagner theorem states that 1D spin system can not order even at zero temperature, due to the strong quantum fluctuation. So 1D Heisenberg model does not lead to an antiferromagnetically ordered ground state, and hence the spin wave is not well defined, and the spin fluctuation does not follow the dispersion relation $\omega\sim k$. It is known[1] that 1D spin chain is gapped, as conjectured by Haldane.
[1] Z.-C. Gu and X.-G. Wen, Phys. Rev. B 80, 155131 (2009).
