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It is known, that there's no local order parameter in Kosterlitz-Thouless transition.

Is the order parameter in Kosterlitz-Thouless transition non-local?

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Correlation function have different behaviour: $$ \langle\psi(x)\psi^\dagger(y)\rangle \sim \frac{e^{-r/\xi}}{\sqrt{r}} \;\;\;\;\;\;\;\;\;\; T \gg T_c $$ $$ \langle\psi(x)\psi^\dagger(y)\rangle \sim \frac{1}{r^\eta} \;\;\;\;\;\;\;\;\;\; T\ll T_c $$

This suggests that there may be a phase transitions between them. The fact that the order parameter for this phase transition is non-local – it involves the position of fields at two distinct points rather than one is our first hint that this phase transition has a slightly different smell from others.

Non-local order parametr is correlation function.

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