Quantum magnetic phase transitions in a Kugel-Khomskii model including spin-orbit coupling

Abstract

Using the formalism of pseudospin and isospin operators the Hamiltonian of an effective Kugel-Khomskii model with spin-orbit coupling is derived with an exact account of the $t_{2g}$ multiplet splitting by the crystal field. An analytical solution is obtained for an arbitrary relation between the Hubbard repulsion and crystal field splitting, i.e., interpolating the cases of Mott-Hubbard and charge-transfer insulators. A description of orbital orders is given in terms of octupole moments. The ground-state phase diagram is constructed in the parameter space spanned by spin-orbit coupling, Hund's exchange, and Hubbard interaction. We investigate a quantum phase transition between a state exhibiting hidden magnetic and orbital long-range order and a ferromagnetic state with a reduced magnetic moment accompanied by antiferroorbital order. It is shown that the cooperative effect of Hund's and spin-orbit interactions gives rise to an easy-plane-type anisotropy.

Figures (3)

• Figure 1: $J$-dependences for the model \ref{['eq:H_eff']}: (a) of orbital filling $N_{im} = \langle X_{i}^{mm}\rangle$ of sublattices, $N_{\mathrm{A}xz} = N_{\mathrm{B}yz}$, $N_{\mathrm{A}yz} = N_{\mathrm{B}xz}$; (b) of absolute value of site magnetization $|\bf{M}_{s}|$. The parameter values $\tilde{\lambda} = \lambda U/t^2 = 0.35, 1.0, 2.0, 3.5$ are presented. Solid lines correspond to $t \ll \Delta_{\rm CF}\ll U$ ($\beta =2$), and dashed lines to $U \ll \Delta_{\rm CF}$ ($\beta =1$).
• Figure 2: The phase diagram in $J-U-\lambda$ variables for the model \ref{['eq:H_eff']} with $t \ll \Delta_{\rm CF}\ll U$. AFOct is a phase with hidden antiferrooctupole ordering Jackeli2009a, FM-AFOct is a ferromagnetic phase.
• Figure 3: The dependence of the pseudoorbital-space polar angle $\theta$ in $J-\lambda$ variables for the FM-AFOct ferromagnetic phase with $t \ll \Delta_{\rm CF}\ll U$, $U = 20t$.