A toy model for two-dimensional spin-fluctuation-induced unconventional superconductivity

Superconductivity had been one of the most enigmatic phenomena in condensed matter physics, puzzling the best theorists for 45 years, since the original discovery by Kamerlingh-Onnes in 1911 till the final solution by Bardeen, Cooper, and Schrieffer (BCS) in 1957. The original BCS proposal assumed the highest-symmetry form for the superconducting order parameter Δ, namely, a constant, and a uniform pairing interaction due to attractive mediation of ionic vibration. While it was rather soon realized that generalizations onto k-dependent order parameters and anisotropic pairing interaction was straightforward, only thirty years later, upon the discovery of high-temperature superconductivity in cuprates, high-order angular dependence of Δ and repulsive interaction, mediated by spin fluctuations or Coulomb repulsion brought such “unconventional” into the spotlight. In 2008 yet another such system was discovered, and eventually the idea of repulsion-mediated unconventional superconductivity was generally accepted. Apart from the two specific systems mentioned above, a large number of various specific implementations of this idea have been proposed, and it is becoming increasingly clear that it is worth studying mathematically how unconventional superconductivity emerges, and with what properties, for a simple, but sufficiently general theoretical model. In our project, we study systematically unconventional superconductivity in an isotropic two-dimensional model system of electrons, subjected to repulsive interactions of a simple, but physically motivated form: a delta function peaked at a particular momentum (from 0 to twice the Fermi momentum), or Gaussian of varying widths.