Magnetic exchange in $\alpha$-iron from ab initio calculations in the paramagnetic phase

Applying the local density approximation (LDA) and dynamical mean field theory to paramagnetic $\alpha$-iron, we reinvestigate the origin of its ferromagnetism. The analysis of local magnetic susceptibility shows that at sufficiently low temperatures $T<1500$ K, both $e_g$ and $t_{2g}$ states equally contribute to the formation of the effective magnetic moment with spin $S=1$. The self-energy of $t_{2g}$ states shows sizable deviations from Fermi-liquid form, which accompany earlier found nonquasiparticle form of $e_g$ states. By considering the nonuniform magnetic susceptibility we find that the nonquasiparticle form of $e_g$ states is crucial for ferromagnetic instability in $\alpha$-iron. The main contribution to the exchange interaction, renormalized by the effects of electron interaction, comes from the hybridization between $t_{2g}$ and $e_g$ states. We furthermore suggest the effective spin-fermion model for $\alpha$-iron, which allows us to estimate the exchange interaction from paramagnetic phase, which is in agreement with previous calculations in the ordered state within the LDA approaches.

Figure 1

Temperature dependence of inverse local magnetic susceptibility, and the corresponding $e_g$ and $t_{2g}$ orbital contributions. Dashed lines show linear behavior in different temperature intervals.

Figure 2

Temperature dependence of the effective magnetic moment and instantaneous average $\langle {\left(s^z\right)}^2\rangle$ and ${\mu }_{\mathrm{eff}}^2$ in $\alpha$-iron, extracted from the temperature dependence of local susceptibility, together with the contribution of the $e_g$ and $t_{2g}$ orbitals.

Figure 3

Frequency dependence of $\mathrm{Im}{\mathrm{\Sigma }}_{t_{2\mathrm{g}}}\left(\mathrm{i}\nu \right)$ (black solid lines), calculated at Matsubara frequencies ${\nu }_n=\pi T(2n+1)$ (marked by dots) at temperatures (from top to bottom) $T=1/5,1/10,1/20,1/30,1/40$ eV. The green dot-dashed line presents fits to the Fermi-liquid dependence in the range ${\nu }_n<1$ eV, while blue dashed lines present fits to the non-Fermi-liquid dependence (see text).

Figure 4

Orbital-resolved momentum dependence of ${\chi }_{\mathbf{q}}^{0,mn}$ at $T=290$ K calculated in high symmetry directions of the Brillouin zone. The contributions ${\chi }_{\mathbf{q}}^{0,d},{\chi }_{\mathbf{q}}^{0,t_{2g}\text{−}{t}_{2g}},{\chi }_{\mathbf{q}}^{0,e_g\text{−}{e}_g}$, and the hybridization part ${\chi }_{\mathbf{q}}^{0,e_g\text{−}{t}_{2g}}$ are shown by black, red, green, and blue lines, respectively. Solid (dashed) lines correspond to LDA+DMFT (LDA) results. The LDA+DMFT estimate for $J_q^{\left(1\right)}{({\mu }_B/I)}^2$ is shown by magenta short-dashed line.

Figure 5

The same as in Fig. 1 of the main text for $U=4.0$ and $I=1.0$ eV.

Figure 6

The same as in Fig. 4 of the main text for $U=4.0$ and $I=1.0$ eV.