Solid-liquid critical point

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Main Question or Discussion Point

In many phase diagrams of a single substance, there is a triple point, where the solid, liquid and gas phases coexist in equilibrium, and there is a liquid-gas critical point beyond which, the transition between liquid and gas becomes continuous, and the substance is known as a super-critical fluid.

Is there a simple reason why in most substances there doesn't appear to be a solid-liquid critical point, where, say pressures are so large that solids can flow (e.g., the mantle of the Earth)? Is there a special distinction between crystalline and amorphous solids as far as this point goes? Are there substances with known phase diagrams that have solid-liquid critical points?

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king vitamin
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The explanation of the (generic) non-existence of such a critical points lies in Landau theory. The main idea is to study the ways in which symmetry can be broken in the system, write down an analytic functional for a coarse-grained order parameter which can describe that symmetry breaking (the functional must satisfy all the symmetries of the microscopic system), and then to study how many parameters need to be tuned to obtain a critical point (that is, a non-analyticity in the free energy which is not a simple discontinuity in either the free energy itself or its first derivative).

In this approach, the issue with finding a critical point in the solid-liquid transition is that too many symmetries are broken at once in going between the two phases, since a solid simultaneously breaks the translational and rotational symmetries of the liquid. If you write down the Landau functional needed, the number of parameters which you need to tune in order to get a continuous transition is more than you have to play with in real life (where you usually just have, say, just temperature and pressure).

Is there a special distinction between crystalline and amorphous solids as far as this point goes? Are there substances with known phase diagrams that have solid-liquid critical points?
Yes, the existence of intermediate solid-like phases is the work around! If you have a system which can have an intermediate phase between the liquid and solid which breaks only one of the two symmetries, then you can imagine going from a liquid to a solid with critical transitions. For example, in two dimensions there can be an intermediate hexatic phase, and the famous "Kosterlitz-Thouless-Halperin-Nelson-Young" (or KTHNY) proposal outlined how transitions between the three phases can be continuous. I believe nematic systems can also undergo a multi-step series of transitions between liquid and solid phases which can be continuous in principle.

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If you have a system which can have an intermediate phase between the liquid and solid which breaks only one of the two symmetries, then you can imagine going from a liquid to a solid with critical transitions.
Do you know of any substances that have this property?

king vitamin
Here is a very recent paper outlining the melting of a skyrmion lattice via an intermediate hexatic phase in the material Cu$_{2}$OSeO$_{3}$. It touches directly on many of the things king vitamin detailed.