We will study quantum coherence in systems with electron correlations by means of Green functions, renormalized many-body theory and numerical simulations. We will extend an approximation earlier developed by us with a two-particle self-consistency from the reduced parquet equations qualitatively correctly describing the Kondo strong-coupling limit of the metallic dot. The general theory will be applied to a model of quantum dot attached to superconducting leads with the aim to explain and understand its behavior at the transition from the spin singlet to the spin doublet state (zero-pi transition). The dot will be studied in an applied weak magnetic field in order to understand this transition and the properties of the spin doublet state with a degenerate ground state. The magnetic solution in a consistent theory must continuously match the non-magnetic one in the limit of the vanishing field. We further extend the static approximation from the reduced parquet equations to a dynamical one to make it applicable to low-dimensional lattice systems with long-range quantum coherence.

**Aims of the project:**
The objective is to build an approximation with a two-particle self-consistency suppressing spurious mean-field transitions and reliably describing quantum coherence in the strong- coupling limit applied on the zero-pi transition in the superconducting quantum dot in an external magnetic field.