So it seems that while absolute zero Kelvin isn't a possible temperature/thermodynamic state, a sub-absolute zero or negative temperature is, which still abides by thermodynamics but acts differently than classical heat does -- sort of like a negative charge in electricity, but with heat/thermodynamic energy.
"Using lasers and magnetic fields, [Ulrich Schneider, a physicist on the research team at the Ludwig Maximilian University in Munich, Germany, and colleagues] kept the individual atoms in a lattice arrangement. At positive temperatures, the [potassium] atoms repel, making the configuration stable. The team then quickly adjusted the magnetic fields, causing the atoms to attract rather than repel each other. [...] At positive temperatures, such a reversal would be unstable and the atoms would collapse inwards. But the team also adjusted the trapping laser field to make it more energetically favourable for the atoms to stick in their positions. This result, described today in [the journal] Science, marks the gas’s transition from just above absolute zero to a few billionths of a Kelvin below absolute zero."
Negative temperature gas states appear to be able to mimic dark energy. Writes Zeeya Merali for the journal Nature, "Another peculiarity of the sub-absolute-zero gas is that it mimics 'dark
energy', the mysterious force that pushes the Universe to expand at an
ever-faster rate against the inward pull of gravity. Schneider notes
that the attractive atoms in the gas produced by the team also want to
collapse inwards, but do not because the negative absolute temperature
stabilises them."
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