Take carbon tetrahedral diamonds who’s negative thermal coefficient means that it grows as it becomes colder means that you can add in heavier (or lighter) substances between the containing carbon and once it is locked into place will begin to cool as well until it forces a chain reaction and brings the carbon down to it’s least possible temperature.
Simple experiment: take an ice pop mold and fill it with one warmer material with a large negative coefficient. Fill the core with a second material with a larger negative coefficient, and both will swell until they are at their atomic limits but the center being stronger will force their still “warm” center to pull the cooling through the system cooling the carbon further, or at least the center atom(s), meaning you can get lower than near proper kelvin zero.
You don’t need the carbon to change form into a liquid when you can use a superfluid or a least atoms attempting to become a superfluid between the remaining spaces between the carbon atoms unable to escape that is weighed down by the carbon so that it either remains in a floating position or falls as wanted, or chained together or weighted with carbine which also expands due to the negative thermal coefficient and is attached to the bottom of the cooling canisters walls. They would need proper pressurization to insure that the canister didn’t pop along a seam.