Revision 2

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Title · changed: Revision 1: Do superfluid-stiffness bounds on $T_c$ extend to three dimensions; Revision 2: Do superfluid-stiffness bounds on $T_c$ extend to three dimensions?
Plain: In two-dimensional superconductors the transition temperature is provably limited by how stiff the superconducting phase is, which in turn is limited by how easily electrons move; whether a similar proof works in three dimensions is unknown.
Precise: For two-dimensional superconductors the Berezinskii-Kosterlitz-Thouless relation and the optical sum rule give rigorous upper bounds on $T_c$ in terms of band dispersion and density (2019). Determine whether a comparable rigorous bound on $T_c$ exists in three dimensions, and in flat-band systems where the superfluid stiffness has a quantum-geometric contribution. An answer is a proof with stated assumptions or a counterexample model.
Settled by: A proof bounding $T_{c}$ by stiffness or kinetic-energy sum rules in 3D, or a solvable 3D model violating any such bound.
Kind: well-posed
Answer type: proof
References: (empty)
Literature status: open
Status note: (empty)
Posed since: 2019

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