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At the Berezinskii-Kosterlitz-Thouless (BKT) phase transition, the singular part of the free energy behaves as $\xi^{-2}$, where $\xi \propto e^{c/\sqrt{T-T_c}}$ (with $c>0$) is the correlation length. Hence $\xi$ has an essential singularity at $T_c$ so that the free energy $f$ is non-analytic at the phase transition. However, $f$ is still a smooth function, thus we classify the BKT transition as an infinite order phase transition.

Furthermore, the Mermin-Wagner theorem forbids spontaneous symmetry breaking of a continuous symmetry in two dimensions at finite $T$. Therefore the BKT transition does not break any symmetries (and, since it is a continuous phase transition, is thus not described within Landau theory).

I was wondering whether the BKT transition is special among classical phase transitions with regard to these two properties, therefore I have the following questions:

1) Are there examples of classical infinite order phase transitions, which are not in the universality class of the BKT transition?

2) Are there examples of classical continuous phase transitions without symmetry breaking, which are not in the universality class of the BKT transition?

3) On a slightly different note, can we say how intimately the properties of the BKT transition are linked to each other? In order to get a classical infinite order phase transition or a continuous phase transition without symmetry breaking, is it mandatory to have a logarithmic interaction between the relevant degrees of freedom (as with the vortices / antivortices in the XY-model) or a two-dimensional system? Does an infinite order phase transition imply that there is no symmetry breaking or vice versa?

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