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Magnetic Interaction in Doped 2D Perovskite Cuprates with Nanoscale Inhomogeneity: Lattice Nonlocal Effects vs Superexchange

We have studied the superexchange interaction $J_{ij}$ in doped $2D$ cuprates. The AFM interaction strongly depends on the state of the lattice of a CuO$_2$ layer surrounded by two LaO rock salt layers. In a static $U$ and $D$ stripe nanostructure, the homogeneous AFM interaction is impossible due to the $U/D/U...$ periodic stripe sequence and $T_N=0$. In a dynamic stripe nanostructure, the ideal CuO$_2$ layer with nonlocal effects and the homogeneous AFM interaction are restored. However the interaction $J_{ij}$ decreases by the exponential factor due to partial dynamic quenching. The meaning of the transition from the dynamic to the static cases lies into the spontaneous $θ$-symmetry breaking with respect to the rotation of all the tilted CuO$_6$ octahedra by an orientation angle $δθ= n \cdot \left( {45^\circ } \right)$ (where $n=1\div4$) in the $U$ and $D$ stripe nanostructure of the CuO$_2$ layer. Moreover, the structural features help to study various experimental data on the charge inhomogeneity, Fermi level pinning in the p type cuprates only and a time reversal symmetry breaking from a unified point of view.