Is there any way for a planet to host, even a tiny amount, of matter in a superfluid form, without outside intervention, given that Helium-4 becomes a superfluid below 2.17K, and that according to this answer, the lowest possible stable temperature is 2.7k (background radiation temperature)?
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I googled your question and fell down a rabbit hole of heady dreamlike concepts I have never before encountered. I am way out of my depth. But that has never stopped me from posting here before. Strap in...
I learned of particles called excitons, where an electron and the absence of an electron from a space act together like a particle.
An exciton is a bound state of an electron and an electron hole which are attracted to each other by the electrostatic Coulomb force… The wavefunction of the bound state is said to be hydrogenic, an exotic atom state akin to that of a hydrogen atom. However, the binding energy is much smaller and the particle's size much larger than a hydrogen atom.
Trippy enough: a thing and the space where it used to be acting together as a new thing.
Then I learned of the crossbreeding of excitons with waves of light (photons), to form exciton-polaritons.
Exciton-polaritons are a type of polaritons, hybrid light and matter quasiparticles arising from the strong coupling of the electromagnetic dipolar oscillations of excitons (either in bulk or quantum wells) and photons.
Here is a video that describes a way in which these things can be made. https://www.youtube.com/watch?v=sWmvZ0IGrsU
I know about photons but have never heard of light being trapped into a stable relationship with matter.
I encountered a fair number of publications describing superfluid behavior of this bizarre state of matter. These authors in Nature Physics describe superfluid behavior to occur at room temperature. Room-temperature superfluidity in a polariton condensate. The abstract states
Even in the case of ultralight quasiparticles such as exciton-polaritons, superfluidity has been demonstrated only at liquid helium temperatures2. In this case, the limit is not imposed by the mass, but instead by the small binding energy of Wannier–Mott excitons, which sets the upper temperature limit. Here we demonstrate a transition from supersonic to superfluid flow in a polariton condensate under ambient conditions. This is achieved by using an organic microcavity supporting stable Frenkel exciton-polaritons at room temperature. This result paves the way not only for tabletop studies of quantum hydrodynamics, but also for room-temperature polariton devices that can be robustly protected from scattering.
A throwaway line in one of the articles I encountered comments that this state of matter is confined to two dimensions. But I also found it described as gaslike. Wherever it exists it needs captive light to sustain it. So probably you could not have a pitcher full.
Is this helpful? For a work of fiction, maybe. Good scifi jumps off something real and makes something unreal that advances the storyline. Initiates and devotees could call bullshit (for example, on wormholes where this call is made routinely), or chuckle and appreciate the points the authors actually got right.
For your story you could read about as much as I did on this exotic state of matter, wrap your head around it, and press it into service. My own sketchy concept involves this stuff crawling over crystals, which are translucent and allow transmission of the light waves you need to be the father, with some property of the overlying substance providing the excitons.
So far we have observed super fluidity in He-4, He-3 and other exotic liquids (fermionic gases). However, He-4 is the hottest superfluid so far (fermionic gases have been shown to be super fluid at 50 nK).
Based on the answer you link, the answer to your question is NO.
If per absurdum such a place existed, it would be colder than either the rest of the planet or the background radiation. In both cases, according to thermodynamic, it would end up absorbing energy and thus warming up until the equilibrium is reached.