Violation of the Fermi-liquid picture in two-dimensional systems owing to Van Hove singularities

We consider the two-dimensional t-t' Hubbard model with the Fermi level being close to Van Hove singularities. The phase diagram of the model is discussed. Far from the quantum phase transition (QPT) into ferro- (antiferro-) magnetic states. the self-energy at the singularity points has a nearly linear energy dependence, the density of states being proportional to ln(3)\ epsilon \. in the quantum-disordered and quantum-critical regimes near the QPT a scaling approach is used. At small energies the self-energy demonstrates a powerlike behavior with the exponent changing from unity to zero in a narrow energy interval. Provided that the system is close to ferromagnetic instability which occurs at t'/t --> -1/2, the quasiparticle spectral weight has, instead of a narrow peak. a broad maximum and is nonmonotonous at small energies. The application of the results to cuprates is discussed.