Renormalisation group analysis of the two-dimensional Hubbard model: Spin correlations and d-wave superconductivity

The repulsive two-dimensional Hubbard model is analysed within a functional renormalisation group (RG) approach. The flow equation for the effective interactions is evaluated on 1-loop level. The effective interactions diverge at a finite energy scale which is exponentially small for small bare interactions. To analyse the nature of the instabilities signalled by the diverging interactions, we compute the flow of the superconducting susceptibilities for various pairing symmetries and also charge and spin density susceptibilities. Depending on the choice of the model parameters (hopping amplitudes, interaction strength and band-filling) we find spin density wave instabilities or d-wave superconductivity as leading instability close to half-filling. This calculation establishes the existence of d-wave superconductivity in the Hubbard model beyond doubt.