Normally magnetic fields destroy superconductivity not help create it! However, in 2001 Uji and collaborators discovered that while the organic charge transfer salt (BETS)2FeCl4 had an insulating ground state applying a magnetic field parallel to the layers could create a metal, and for sufficiently high magnetic fields, superconductivity! This can be explained in terms of the exchange interaction between the magnetic Fe3+ ions and the pi electrons in the BETS molecules. When this exchange interaction is cancelled by the applied field (the Jaccarino-Peter effect, first proposed in the 1960's) the electron spins effectively see zero magnetic field.
I was very happy that in our paper Olivier Cepas, Jaime Merino, and I were able to predict the magnetic field range in which one should observe magnetic-field induced superconductivity for a specific material and this was subsequently observed.
One can tune between Mott insulating, metallic, and superconducting states by varying the magnetic field, temperature, or the relative concentration of magnetic Fe3+ ions and non-magnetic Ga3+ ions. This leads to a rich phase diagram such as that shown below. For a review see here.
Because the relevant field and temperature scales are so small these means that the superconducting and Mott insulating phase are very close in energy. i.e., within a few meV, which is orders of magnitude smaller than the energy scales in the underlying Hubbard Hamiltonian. Such a competition occurs naturally in RVB theory.
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