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We consider the competition of magnetic and charge ordering in high-Tc cuprates within the framework of the simplified static 2D spin-pseudospin model. This model is equivalent to the 2D dilute antiferromagnetic (AFM) Ising model with charged impurities. We present the mean-field results for the system under study and make a brief comparison with classical Monte Carlo (MC) calculations. Numerical simulations show that the cases of strong exchange and strong charge correlation differ qualitatively. For a strong exchange, the AFM phase is unstable with respect to the phase separation (PS) into the charge and spin subsystems, which behave like immiscible quantum liquids. An analytical expression was obtained for the PS temperature.The Hamiltonian of the static spin-pseudospin model is:where S zi is a z-projection of the on-site pseudospin S = 1 and σ zi = P 0i s zi /s is a normalized z-projection of conventional spin s = 1/2 operator, multiplied by the projection operator P 0i = 1 − S 2 iz . The ∆ = U/2 is the on-site correlation and V > 0 is the inter-site density-density interaction, J =J/s 2 > 0 is the Cu 2+ −Cu 2+ Ising spin exchange coupling, h =h/s is the external magnetic field, µ is the chemical potential, so we assume the total charge constraint, nN = S zi = const, where n is the density of doped charge. The sums run over the sites of a 2D square lattice, ij means the nearest neighbors. This spin-pseudospin model generalizes the 2D dilute AFM Ising model with charged impurities. In the limit ∆ → −∞ it reduces to the S = 1 2 Ising model with fixed magnetization. At ∆ > 0 the results can be compared with the Blume-Capel model [27,28,29] or with the Blume-Emery-Griffiths model [30].An analysis of the ground state (GS) phase diagrams was done within the mean field approach [24,25]. It was shown, that the five GS phases are realized in two limits. In a weak exchange limit, atJ < V , all the GS phases (COI, COII, COIII, FIM) correspond to the charge ordering (CO) of a checker-board type at mean charge density n. While the COI phase is the charge-ordered one without spin centers, the COII and COIII phases are diluted by the non-interacting spins distributed in one sublattice only. This ferrimagnetic spin ordering is a result of the mean-field approach, so the classical MC calculations show a paramagnetic response at low temperatures. The FIM phase is also formally ferrimagnetic. Here the AFM spin ordering is diluted by the non-interacting charges distributed in one sublattice. In a strong exchange limit, atJ > V , there are only COI phase and AFM phase with the charges distributed in both sublattices. The absence of charge transfer in the Hamiltonian (1) is the most important limitation of our present model for comparison with the actual phase diagram of cuprates. But, as shown in [31], the accounting for two-particle transport enriches the GS phase diagram of the spin-pseudospin model with superfluid and supersolid phases competing with CO phases.The paper is organized as follows. We present the mean-field result...

We addressed a simplified static 2D spin-pseudospin model which takes into account both conventional Heisenberg spin exchange coupling and the on-site and inter-site charge correlations. Classical Monte-Carlo calculations for large square lattices show that homogeneous ground state solutions found in a mean-field approximation are unstable with respect to phase separation with the charge and spin subsystems behaving like immiscible quantum liquids. For instance, with lowering the temperature one can observe two sequential phase transitions: first, antiferromagnetic ordering in the spin subsystem diluted by randomly distributed charges, then, the charge condensation in the charge droplets. Thermodynamic properties and phase diagram of the 2D spin-pseudospin system are studied by Monte-Carlo simulation with a special attention given to the role played by the on-site correlation.

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