Breakdown of the Landau-Fermi liquid in two dimensions due to umklapp scattering

We study the renormalization-group (RG) flow of interactions in the two-dimensional ${t-t}^{\prime }$ Hubbard model near half-filling in an N-patch representation of the whole Fermi surface. Starting from weak to intermediate couplings the flows are to strong coupling, with different characters depending on the choice of parameters. In a large parameter region elastic umklapp scatterings drive an instability which on parts of the Fermi surface exhibits the key signatures of an insulating spin liquid (ISL), as proposed by Furukawa, Rice, and Salmhofer [Phys. Rev. Lett. 81, 3195 (1998)] rather than a conventional symmetry-broken state. The ISL is characterized by both strong d-wave pairing and antiferromagnetic correlations; however, it is insulating due to the vanishing local charge compressibility and a spin liquid because of the spin gap arising from the pairing correlations. We find that the unusual RG flow, which we interpret in terms of an ISL, is a consequence of a Fermi surface close to the saddle points at the Brillouin-zone boundaries which provides an intrinsic and mutually reinforcing coupling between pairing and umklapp channels.