Liquid sodium can exist at room temperature at EXTREMELY high pressures. Why?
Any liquid can exist at any temperature provided the pressure is high enough (except of course at temperatures where the substance in question does not have a liquid state such a dry ice).
That must be because there is a liquid phase with higher density.
Normally, sodium crystallizes in the body-centered cubic (bcc) structure, which is not close packed. At high pressures there is probably a transition to cubic or hexagonally close packed. This structure may have a melting point below room temperature.
This is exactly right, but it appears that there's more to it than just that. Most metals that lack a close-packed structure exhibit some degree of negative melting. In sodium, however, this is unusually strong.
There's a very recent paper on this that I've yet to read : http://scitation.aip.org/getabs/ser...00094000018185502000001&idtype=cvips&gifs=yes
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