Magnetism of iron and nickel from rotationally invariant Hirsch-Fye quantum Monte Carlo calculations

We present a rotationally invariant Hirsch-Fye quantum Monte Carlo algorithm in which the spin rotational invariance of Hund's exchange is approximated by averaging over all possible directions of the spin quantization axis. We employ this technique to perform benchmark calculations for the two- and three-band Hubbard models on the infinite-dimensional Bethe lattice. Our results agree quantitatively well with those obtained using the continuous-time quantum Monte Carlo method with rotationally invariant Coulomb interaction. The proposed approach is employed to compute the electronic and magnetic properties of paramagnetic $\alpha$ iron and nickel. The obtained Curie temperatures agree well with experiment. Our results indicate that the magnetic transition temperature is significantly overestimated by using the density-density type of Coulomb interaction.

Figure 1

Temperature dependence of the inverse uniform magnetic susceptibility as obtained by DMFT for the two-band (upper panel) and three-band (lower panel) Hubbard models on the infinite-dimensional Bethe lattice. The straight lines depict the least-squares fit to the Curie-Weiss law. The extracted Curie temperatures and effective local magnetic moments are presented in Table . The CT-QMC results were taken from Ref. 10.

Figure 2

Partial $t_{2g}$ (black) and $e_g$ (red) densities of states for paramagnetic $\alpha$ iron as obtained by the rotationally invariant HF-QMC within LDA+DMFT in comparison with the nonmagnetic LDA results. The Fermi level is indicated by the vertical (gray) line at zero energy. Inset: imaginary parts of the obtained self-energies.

Figure 3

Temperature dependence of the inverse uniform magnetic susceptibility for $\alpha$ iron as obtained by the LDA+DMFT. The straight lines depict the least-squares fit to the Curie-Weiss law. The experimental value $T_{\mathrm{C}}^{\exp }=1043$ K is denoted by the (black) arrow. The experimental value of the effective local magnetic moment is ${\mu }_{\mathrm{eff}}^{\exp }=3.13{\mu }_B$ (Ref. 33).

Figure 4

Partial densities of states for paramagnetic Ni obtained by rotationally invariant HF-QMC within LDA+DMFT in comparison with the LDA ones. The Fermi level is indicated by the vertical (gray) line at zero energy. Inset: imaginary parts of the obtained self-energies.

Figure 5

Spin-spin correlation functions on the real and imaginary (inset) energy axes for $\alpha$ iron and nickel calculated by LDA+DMFT at $T=2.5T_{\mathrm{C}}$.

Figure 6

Temperature dependence of the inverse uniform magnetic susceptibility for Ni as obtained by LDA+DMFT. The straight lines depict the least-squares fit to the Curie-Weiss law. The experimental value $T_{\mathrm{C}}^{\exp }=631$ K is denoted by the (black) arrow. The experimental value of the effective local magnetic moment is ${\mu }_{\mathrm{eff}}^{\exp }=1.62{\mu }_{\text{B}}$ (Ref. 33).