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The Mermin-Wagner theorem states that continuous symmetry cannot be spontaneously broken at any finite temperature in two dimensions or lower.

The Kosterlitz-Thouless (KT) transition is a phase transition on a symmetric system (no easy axis for mangetic moments to align) in two dimensions. I believe it can be said that the Kosterlitz-Thouless system has continuous symmetry, please correct me if I am wrong.

At zero temperature the KT system is a ferromagnetic state, which means all the magnetic moments are pointing in the same direction (some random direction since no easy axis). Since all of the magnetic moments are pointing in the same direction (chosen at random) at zero temperature then we can say that the continuous symmetry of the KT system has spontaneously broken. However this does not violate the Mermin-Wagner theorem since this occurs at zero temperature.

At small finite temperatures the KT system is no longer a ferromagnetic state. Instead vortex anti-vortex pairs form. Doesn't this mean that the continuous symmetry of the KT system is broken again at non-zero finite temperatures. Thus, violating the Mermin-Wagner theorem?

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At low temperature the spin-spin correlation function of the 2d O(2) model can be computed in perturbation theory, see, for example chapter 33 of Zinn-Justin, QFT and Critical Phenomena. The answer is $$ \langle e^{i\theta(0)}e^{-i\theta(r)}\rangle \sim r^{-t/(2\pi)} $$ where $t$ is the reduced temperature, and I have written the spin vector in terms of the polar coordinate $\theta$, $\vec{S}=(\cos\theta,\sin\theta)$. At any finite $t$ the correlation function decays algebraically (it decays as a fractional power), and there is no long range order.

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  • $\begingroup$ So your saying that at zero temperature the KT system has long range order, thus, the continuous symmetry of the KT system is spontaneously broken. And at finite temperatures the KT system does not have long range order, thus, the symmetry is not spontaneously broken. Am I correct? Also, what do you mean by spin correlation? Do you mean how neighbouring magnetic moments effect each other? $\endgroup$ – PiccolMan Oct 13 '16 at 20:36
  • $\begingroup$ Additionally, what do you mean when you say the correlation function decays algebraically? Are you saying that because it decays algebraically it is not spontaneous? If so, what is this correlation function? $\endgroup$ – PiccolMan Oct 13 '16 at 20:39
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    $\begingroup$ Added a few more comments. The spin-spin correlation function in a ferromagnet goes to a constant, but in the 2-d O(2) model it decays as a power (``algebraically''), so there is no long range order. Note that at $T_c$ the correlation function changes, and above $T_c$ there is screening (exponential decay). $\endgroup$ – Thomas Oct 13 '16 at 21:20
  • $\begingroup$ The correlation function measures the order of a system. Does it measure how well the magnetic moments will be aligned with one another at a certain temperature? For example, the ising model before $T_c$ is a ferromagnet, thus, it has a spin-spin correlation of some constant. After $T_c$ the ising model will be a paramagnet, thus, its correlation function will be 0? $\endgroup$ – PiccolMan Oct 13 '16 at 23:06
  • $\begingroup$ Ordered phase: Correlation function goes to a constant as $r\to\infty$. Disordered phase: Correlation function goes to zero as $r\to\infty$. Gapped system: Correlation function decays exponentially. $\endgroup$ – Thomas Oct 14 '16 at 15:24

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