Recently it has been shown theoretically and then experimentally that a liquid-liquid transition (LLT) takes place in pure supercooled water. That is to say, two distinct liquid phases have a direct transition between each other on the phase diagram. This was a very controversial result as I understand it (and is something people have argued about for a while apparently). It is not the existence of an LLT that is controversial, but rather the existence of an LLT in water.
I am rather confused by this whole idea, however. Ordinarily, when we talk about phase transitions, there are very clear dynamical and energetic reasons why the transition should take place. For instance, water has hydrogen bonds that get weaker and weaker as the temperature increases because the molecules deviate from the minimum energy structure more often which leads to repulsion. Thus, the existence of a phase change from liquid to vapor must exist because at some point the repulsions overcome the attraction. Similar kinds of arguments can be thought of when going from the liquid to solid, and even from solid to solid.
I cannot think of a similar simple competition which makes it clear to me why a liquid-liquid transition should ever happen. I have looked at ref.  which makes it clear that the two phases (of this mixture, not water) display different average hydrogen-bonding arrangements, one of which is bimodal. It isn't so surprising that there could be two different hydrogen-bonding arrangements in a phase diagram. What does surprise me is that the path between two hydrogen-bonding arrangements is not smooth.
So, to say it as simply as I can, what competing interactions are responsible for the existence of a LLT in water?
One important point is that all the qualitative features of this transition were described by the TIP4P 2005 water model in ref. . TIP4P has a potential energy function described by Lennard-Jones forces. This confounds me even more because the physics exists in a simple Lennard-Jones fluid with only pairwaise interactions.
: Singh, R. S., Biddle, J. W., Debenedetti, P. G., & Anisimov, M. A. (2016). Two-state thermodynamics and the possibility of a liquid-liquid phase transition in supercooled TIP4P/2005 water. The Journal of chemical physics, 144(14), 144504.
: Zhao, Z., & Angell, C. A. (2016). Apparent First‐Order Liquid–Liquid Transition with Pre‐transition Density Anomaly, in Water‐Rich Ideal Solutions. Angewandte Chemie, 128(7), 2520-2523.
: Bruijn, J. R., van der Loop, T. H., & Woutersen, S. (2016). Changing Hydrogen-Bond Structure during an Aqueous Liquid–Liquid Transition Investigated with Time-Resolved and Two-Dimensional Vibrational Spectroscopy. The journal of physical chemistry letters, 7(5), 795-799.