Phase separation in high-T$_c$ cuprates

We develop a minimal non-BCS model for the CuO$_2$ planes with the on-site Hilbert space reduced to only three effective valence centers CuO$_4$ with different charge, conventional spin, and orbital symmetry, combined in a charge triplet. Using the S=1 pseudospin algebra we introduce an effective spin-pseudospin Hamiltonian. To illustrate the possibilities of the molecular field approximation we start with the analysis of the atomic and the "large negative-$U$" limits of the model in comparison with the Bethe cluster approximation, classical and quantum Monte Carlo methods. Both limiting systems exhibit the phase separation effect typical of systems with competing order parameters. The $T$\,-\,$n$ phase diagrams of the complete spin-pseudospin model were reproduced by means of a site-dependent variational approach within effective field approximation typical for spin-magnetic systems. Limiting ourselves to two-sublattice approximation and $nn$-couplings we arrived at several N\'{e}el-like phases in CuO$_2$ planes for parent and doped systems with a single nonzero local order parameter: antiferromagnetic insulator, charge order, glueless $d$-wave Bose superfluid phase, and unusual metallic phase. However, the global minimum of free energy is realized for phase separated states which are bounded by the third-order phase transition line $T^{\star}(n)$, which is believed to be responsible for the onset of the pseudogap phenomenon. With a certain choice of the Hamiltonian parameters the model phase diagrams can quite reasonably reproduce the main features of experimental phase diagrams for T- and T$^{\prime}$-cuprates and novel nickelates. The superconducting phase of cuprates/nickelates is determined by the on-site composite boson transport, it is not a consequence of pairing of doped holes/electrons, but represents one of the possible phase states of parent systems.