Local intra-unit-cell order parameters in cuprates

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Starting with the on-site Hilbert space reduced to only three effective valence centers, nominally Cu¹⁺,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, nₕ=0,1,2, conventional spin s=1/2 for Cu²⁺ and s=0 for Cu¹⁺,³⁺ centers, and different orbital symmetry:B₁g for the ground states of the Cu²⁺ center and A₁g for the Cu¹⁺,³⁺ centers, respectively. The latter does result in a spontaneous orbital symmetry breaking with emergence of the IUC orbital nematic order parameter of the B₁g∝ dₓ₂₋y₂ symmetry. To describe the quantum local charge order we develop an S=1 pseudospin model. Conventional spin density ρₛ for mixed valence superpositions can vary inbetween 0 and 1 in accordance with the weight of the Cu²⁺ 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 ¹A₁g the ground state of the hole Cu³⁺ center in cuprates should be described by a complex ¹A₁g-¹,³B₂g-¹,³Eᵤ multiplet, formed by a competition of conventional hybrid Cu 3d-O 2p b₁g(σ)∝ dₓ₂ ₋y₂ state and purely oxygen nonbonding O 2pπ states with a₂g(π) and eux,y(π) 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.