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In the wiki article phase transition, the ferromagnetic phase transition is listed both in Ehrenfest second order phase transition and continuous phase transition in modern classification. Is there a contradiction?

Since according to the article:

Though useful, Ehrenfest's classification has been found to be an incomplete method of classifying phase transitions, for it does not take into account the case where a derivative of free energy diverges (which is only possible in the thermodynamic limit).

From this, I conclude that ferromagnetic phase transition should not have divergent susceptibility since it is in the second order phase transition in Ehrenfest classification.

But below this the article also writes:

...continuous phase transitions. They are characterized by a divergent susceptibility, an infinite correlation length, and a power-law decay of correlations near criticality. Examples of second-order phase transitions are the ferromagnetic transition...

From this, I will conclude that ferromagnetic phase transition has divergent susceptibility.

How to reconcile these two statement?

If we accept that a divergent derivative is discontinuous, then Ehrenfest classification is perfectly OK, all the phase transition can be classified, why not adopt this convention but invent a new classification method?

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If we accept that a divergent derivative is discontinuous, then Ehrenfest classification is perfectly OK, all the phase transition can be classified, why not adopt this convention but invent a new classification method?

I agree with you, and I don't agree with the author of the Wiki article. A divergent function is discontinuous, so there is no contradiction here: in the ferromagnetic transition, the isothermal susceptibility

$$\chi_T = - \left(\frac{\partial^2 F}{\partial H^2}\right)_T$$

diverges at the Curie temperature $T_c$. A divergent quantity is by definition discontinuous, therefore:

  • It is a second-order phase transition according to the Ehrenfest criterion.
  • It is a continuous (or "second-order") phase transition according to the modern criterion.

The modern point of view is that the most meaningful difference is that between first order phase transitions and all the rest, and that the Ehrenfest criterion is somewhat unnecessarily detailed.

First order phase transitions have some features which separates them from all the higher order (in the Ehrenfest classification) transitions, such as the presence of latent heat and a discontinuity in the order parameter.

Actually, a somewhat better definition is that given by Binder (Theory of first-order phase transitions, 1987): a first-order transition is one in which the order parameter disappears discontinuously at the transition, and a second-order transition is one in which the order parameter disappears continuously at the transition.

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The ferromagnetic transition is indeed a second-order transition in the sense of Ehrenfest, as well as a continuous one in the modern sense.

The Ehrenfest classification has proven to be useful for the classification of the first observed phase transitions such as the magnetic transitions, and the solid/liquid/gas ones. However, this classification is not powerful enough to account for more complicated transitions such as the superconducting, BKT, or topological phase transitions which show more subtle features (non-trivial behaviour of correlation length, phase stiffness, robust ground state degeneracy...), hence the need for a new classification method.

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