A second order phase transition happens gradually when we lower or raise the temperature. Freezing of saline water in a puddle happens gradually as the temperature drops.
Specific heat measurements suggest that the transition to superconductivity or superfluidity happens gradually when the temperature is lowered. The heat from the phase transition is released over an interval of from one kelvin to several kelvins.
Melting of a time crystal versus an ordinary crystal
An ordinary crystal is an almost static lattice where atoms or molecules sit. There is vibration of the atoms around their mean position, as well as collective vibrations of the lattice, but in the big picture the system is simple and static. There is just one, or a few, compact (not fiber-like) building blocks from which we build the lattice in 3D space.
Melting of an ordinary crystal is a simple monotonic process and can happen completely at a sharp temperature.
On the other hand, a time crystal may be extremely complex because it is built from atoms and their movements. Movements can take complicated forms. An atom and its movement in spacetime constitutes a thread. Building a time crystal requires weaving these threads together in a 4D space.
A time crystal might be analogous to a 3D crystal built from polymers. Such a crystal presumably does not have a sharp melting point.
Conjecture. Most phase transitions of time crystals are second order. There is no sharp temperature which destroys all the order in a time crystal.
An example: the conducting electron gas in a metal crystal undergoing zero-point vibration
Let us have a crystal of metal. Our initial configuration is a uniform gas of conducting electrons.
1. + + + + + +
2. + + + + + +
crystal
The lattice of the crystal is undergoing vibrations which move positive charge back and forth between the center and the ends of the crystal. The stages of the process are marked 1 and 2 in the diagram.
We can presumably lower the energy of the system by letting an electron adjust its path so that it masks some of the extra positive charge at the stage 1.
We can mask all of the extra positive charge by adjusting the path of several electrons. We created a time crystal in the electron gas and the lattice.
How does the time crystal "melt" if we raise the temperature?
If we remove one electron from the time crystal and release to it to the random gas of electrons, then the rest of the electrons are more tighly bound because there is now more excess positive charge at the stage 1.
The time crystal melts gradually as we raise the temperature. It is a second order phase transition.
In this example it was not essential that the system is a time crystal. The essential point was that the excess positive charge at the stage 1 is large and that it requires many electrons to mask the excess charge.
Our example is similar to the ionization of a many electron atom. We have to raise the temperature to get more electrons out.
Conclusions
Melting of an ordinary crystal at a precise temperature is a special case where the simplicity of the process allows it to happen at a sharp temperature.
Since a time crystal in a lattice of a metal is a more complicated system, it probably cannot melt at a sharp temperature.
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