Why do phase transitions even exist? Why not smooth density change curves? Why do phase transitions even exist? Why not smooth density change curves? What properties of matter, quantum or otherwise, predicts that matter will undergo phases at different pressures and temperatures?
Some materials have all phases, others are missing some.
If this were to be researched from very little existing knowledge, a great place to start would be by examining the differences between materials that are missing some phases, and comparing them with ones that aren't.
 A: Phase transitions are a many-body effects. You can not generate sharp transition with a finite number of degrees of freedom (or particles). However as you add particles the features of the system may become sharper. In the limit of infinitely many particles (thermodynamic limit) you get a truly discontinuous transition.
In practice nothing is infinite. The typical number of atoms in normal matter is however 10^(23) which is indistinguishable from infinity in the sense that phase transitions appear perfectly sharp.
The mechanism from which the transition occurs depends on the particular system and transition that you consider. For example, in the case of water freezing, you have a competition in between disorder (temperature) and atom-atom interaction. The atoms want to stick together in ordered pattern (crystal) but the temperature wants them to have random position. There is a critical temperature above which the disorder wins and below which it is the potential energy that is stronger.
A: It's not the details of the interaction that provide the clue, it's the cooperativity of the effect. For example, in a ferromagnet, there's no point to any one spin lining up unless its neighbors are aligned. That leads to the abrupt change from no long-range magnetic order to the onset of some long-range magnetic order starting at a well-defined temperature in an infinite system. Likewise, there's no point to a water molecule lining up in a particular way if its neighbors are tumbling around. Again, that cooperative effect leads to the abruptness of the transition from liquid to solid. So it takes not only the above-mentioned infinite number of degrees of freedom but also some cooperativity among them to generate a phase transition.
