Rigorous Fermi liquids and superconductivity

4 problems

• Off-diagonal long-range order in a many-body model of attractive fermions

  Open

  BCS theory explains superconductivity by electron pairing, but it is an approximation. No one has proven that a full many-electron model with attraction actually forms a pair condensate in an infinite system.

• Fermi-liquid behavior of weakly interacting fermions above the pairing scale

  Open

  Show that electrons with weak repulsion in a 3D continuum act like slightly modified free particles down to very low temperatures. At the lowest temperatures they are expected to pair, so the statement must stop above that scale.

• Kohn-Luttinger superconductivity from purely repulsive interactions

  Open

  Kohn and Luttinger argued that even purely repelling electrons pair up and superconduct at very low temperature, because the surrounding electrons screen the repulsion unevenly. This has never been proven.

• Itinerant ferromagnetism in the Hubbard model at finite hole density

  Open

  With infinitely strong repulsion and exactly one missing electron, the Hubbard model is proven to be a ferromagnet. Whether this survives for a small but finite fraction of missing electrons is unknown.