I have learned that the critical exponents for phase transitions is independent of the microscopic structure of the substance and is dependent on the symmetry. For instance the phase transition for a ferromagnetic at the Curie temperature and the phase transition for water at the critical point have the same critical exponents. The ferromagnet has $\mathbb{Z}_2$ symmetry is related to spin up ground state and the spin down ground state. What will be the $\mathbb{Z}_2$ symmetry in water?
1 Answer
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There is no $\mathbb{Z}_2$ symmetry in water. This is why you need to tune both the temperature $T\to T_c$ and the pressure $P\to P_c$ to get to the critical point. The $\mathbb{Z}_2$ symmetry is emergent at the critical point, and only at long distances.
By contrast, in what you call a ferromagnet the symmetry is always present and you only need to tune the temperature $T$. If you turn on an external magnetic field $H$, this will break the symmetry. Correspondingly, you will now need to tune both the temperature $T\to T_c$ and the magnetic field $H\to 0$ to get to the critical point.
So, in some sense, water at a generic point $(P,T)$ in the phase diagram is closer to a ferromagnet in a magnetic field. The analogy is not perfect, because the phase diagram of a ferromagnet has a line of enhanced symmetry at $H=0$, while there's no such line for water.
The idea of universality is more about there being much fewer types of critical behavior than critical systems. The symmetry is important, but it does not determine everything.
By the way, most ferromagnets have critical exponents from a different universality class. For the one you have in mind, you need to look for uniaxial ferromagnets.
