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We present a new method to treat the two-dimensional (2D) Hubbard model forparameter regimes which are relevant for the physics of the high-$T_c$superconducting cuprates. Unlike previous attempts to attack this problem, ournew approach takes into account all fluctuations in different channels on equalfooting and is able to treat reasonable large lattice sizes up to 32x32. Thisis achieved by the following three-step procedure: (i) We transform theoriginal problem to a new representation (dual fermions) in which all purelylocal correlation effects from the dynamical mean field theory are alreadyconsidered in the bare propagator and bare interaction of the new problem. (ii)The strong $1/(i\nu)^2$ decay of the bare propagator allows us to integrate outall higher Matsubara frequencies besides the lowest using low order diagrams.The new effective action depends only on the two lowest Matsubara frequencieswhich allows us to, (iii) apply the two-particle self-consistent parquetformalism, which takes into account the competition between differentlow-energy bosonic modes in an unbiased way, on much finer momentum grids thanusual. In this way, we were able to map out the phase diagram of the 2D Hubbardmodel as a function of temperature and doping. Consistently with theexperimental evidence for hole-doped cuprates and previous dynamical clusterapproximation calculations, we find an antiferromagnetic region at low-dopingand a superconducting dome at higher doping. Our results also support the roleof the van Hove singularity as an important ingredient for the high value of$T_c$ at optimal doping. At small doping, the destruction of antiferromagnetismis accompanied by an increase of charge fluctuations supporting the scenario ofa phase separated state driven by quantum critical fluctuations.

Dual parquet scheme for the two-dimensional Hubbard model: modelling low-energy physics of high-$T_c$ cuprates with high momentum resolution