Electronic and magnetic structures of cuprates with spin-orbit interaction

The effective Hamiltonian for the ${\mathrm{CuO}}_2$ plane of high-${\mathit{T}}_{\mathit{c}}$ cuprates is derived by taking into account the spin-orbit interaction. A finite-size exact-diagonalization technique is applied to the Hamiltonian, and the magnetic and electronic structures of the ${\mathrm{CuO}}_2$ plane in the low-temperature orthorhombic (LTO) and low-temperature tetragonal (LTT) phases of the ${\mathrm{La}}_2$ ${\mathrm{CuO}}_4$-type crystals are examined. It is shown that the contribution from the in-plane oxygen 2${\mathit{p}}_{\mathit{z}}$ orbital perpendicular to the plane is essential for the emergence of the weak ferromagnetism in the LTO-phase ${\mathrm{La}}_2$ ${\mathrm{CuO}}_4$. In the LTT phase, either the uniaxial antiferromagnetism or the weak ferromagnetism is induced, depending on how the 2${\mathit{p}}_{\mathit{z}}$ orbital contributes to the anisotropic superexchange interaction of the single Cu–O–Cu bond. We study the dynamics of a hole in an extended t-J model, and show that the lattice distortion works to change the symmetry of the electronic ground state via the spin-orbit coupling. The effect is, however, small and seems irrelevant to the anomalous properties of ${\mathrm{La}}_{1.88}$ ${\mathrm{Ba}}_{0.12}$ ${\mathrm{CuO}}_4$.