Two-site fluctuations and multipolar intersite exchange interactions in strongly correlated systems

An approach is proposed for evaluating dipolar and multipolar intersite interactions in strongly correlated materials. This approach is based on the single-site dynamical mean-field theory (DMFT) in conjunction with the atomic approximation for the local self-energy. Starting from the local-moment paramagnetic state described by DMFT, we derive intersite interactions by considering the response of the DMFT grand potential to small fluctuations of atomic configurations on two neighboring sites. The present method is validated by applying it to one-band and two-band $e_g$ Hubbard models on the simple-cubic $3d$ lattice. It is also applied to study the spin-orbital order in the parent cubic structure of ternary chromium fluoride ${\mathrm{KCrF}}_3$. We obtain the onset of a G-type antiferro-orbital order at a significantly lower temperature compared to that in real distorted ${\mathrm{KCrF}}_3$. In contrast, its layered A-type antiferromagnetic order and Néel temperature are rather well reproduced. The calculated full Kugel-Khomskii Hamiltonian contains spin-orbital coupling terms inducing a misalignment in the antiferro-orbital order upon the onset of antiferromagnetism.

Figure 1

Calculated intersite interactions $J_n$ for the first three coordinational shells. The values of $J_2$ and $J_3$ are multiplied by 10. The dash-dotted line is the $-4t^2/U$ asymptote.

Figure 2

Calculated values for the mean-field Néel temperature $T_N$ compared with those obtained within QMC from (a) Ref. [10], (b) Ref. [43], and (c) Ref. [44] as well as with the large-$U$ asymptote $T_N=\frac{6t^2}{U}$.

Figure 3

Calculated superexchange intersite interactions for the $e_g$ Hubbard model with (a) the Hund's rule coupling $J_H=0$ and (b) $J_H=0.5W$. The black dashed lines are the values obtained from analytical formulas (33).

Figure 4

The total and projected spectral functions of ${\mathrm{KCrF}}_3$ calculated by the DFT+DMFT method within the Hubbard-I approximation using $U=3.75$ eV.

Figure 5

(a) Specific heat (per formula unit) as a function of temperature obtained by solving the Hamiltionian (34) with the values of superexchange interactions calculated at $U=3.75$ eV. (b) The G-type antiferro-orbital order obtained below $T_{\mathrm{OO}}=340$ K (plotted by xcrysden [65], the real-space representation of the orbitals is generated with the help of the wplot [66] program). (c) The A-type antiferromagnetic phase, stable below $T_N=102$ K, obtained with the interactions calculated with $U=3.75$ eV. (d) The C-type antiferromagnetic phase obtained using the interactions calculated with $U=5$ eV.

Figure 6

The spin magnetic moment and orbital misalignment angle $\mathrm{\Delta }\theta$ as a function of temperature in the A-type structure obtained for $U=3.75$ eV.