Local intra-unit-cell order parameters in cuprates

Starting with the on-site Hilbert space reduced to only three effective valence centers, nominally Cu$^{1+,2+,3+}$, we present an unified approach to the description of the local intra-unit-cell (IUC) order parameters in cuprates. Central point of the model implies the occurrence of unconventional on-site quantum superpositions of the three valent states characterized by different hole occupation, $n_h$=0,1,2, conventional spin s=1/2 for Cu$^{2+}$ and s=0 for Cu$^{1+,3+}$ centers, and different orbital symmetry:$B_{1g}$ for the ground states of the Cu$^{2+}$ center and $A_{1g}$ for the Cu$^{1+,3+}$ centers, respectively. The latter does result in a spontaneous orbital symmetry breaking with emergence of the IUC orbital nematic order parameter of the $B_{1g}\propto d_{x2-y2}$ symmetry. To describe the quantum local charge order we develop an S=1 pseudospin model. Conventional spin density $\rho_s$ for mixed valence superpositions can vary inbetween 0 and 1 in accordance with the weight of the Cu$^{2+}$ center in the superposition. We show that the superconductivity and spin magnetism are nonsymbiotic phenomena with competing order parameters. Furthermore we argue that instead of a well-isolated Zhang-Rice (ZR) singlet $^1A_{1g}$ the ground state of the hole Cu$^{3+}$ center in cuprates should be described by a complex $^1A_{1g}$-$^{1,3}B_{2g}$-$^{1,3}E_u$ multiplet, formed by a competition of conventional hybrid Cu 3d-O 2p $b_{1g}(\sigma)\propto d_{x^2 -y^2}$ state and {\it purely oxygen nonbonding} O 2p$\pi$ states with $a_{2g}(\pi)$ and $e_{ux,y}(\pi)$ symmetry. In contrast with inactive ZR singlet we arrive at several novel competing IUC orbital and spin-orbital order parameters, e.g., electric dipole and quadrupole moments, Ising-like net orbital magnetic moment, orbital toroidal moment, intra-plaquette's staggered order of Ising-like oxygen orbital magnetic moments.