Role of rotational symmetry in the magnetism of a multiorbital model

Effect of rotationally invariant Hund's rule coupling on a magnetism of multiorbital Hubbard models is studied within a dynamical mean-field theory framework. Comparison of static magnetic susceptibilities and local densities of states of two- and three-orbital models of a complete rotationally invariant Coulomb interaction and a “density-density” Hartree type interaction shows the different role of spin-flip interactions for different band fillings. In the particle-hole symmetric case, the Mott-Hubbard physics dominates due to the strong effective Coulomb interaction, while for the multiple electronic configurations away from half-filling (two electrons in the three-band model) the formation of local magnetic moments due to Hund's exchange interaction becomes the most significant effect for itinerant magnetic systems. A shift of the temperature of magnetic ordering due to the rotationally invariant Hund's rule coupling is found to be the largest in a three-orbital model with a two-electron occupancy where the single particle spectrum is metallic and is not sensitive to different forms of the Coulomb vertex. A larger enhancement of the effective mass in a model with a rotationally invariant interaction is discussed. In the half-filled case we find a drastic change in the density of states close to the Mott transition which is related to the spin-flip Kondo fluctuations in a degenerate orbital case, while the corresponding shift of the magnetic transition temperature is relatively small. It is shown that a change in the ground-state degeneracy due to a different symmetry of the Coulomb interaction in the density-density model leads to a breakdown of the quasiparticle peak at the Fermi level in the proximity of a Mott transition on the metallic side. We discuss the relevance of rotationally invariant Hund's interaction in the transition metal magnetism.

Figure 1

The temperature dependence of the inverse magnetic susceptibility of a single unhybridized atomic orbital, defined by full rotationally invariant Hamiltonian (3) (solid lines) and (4) (dashed lines) in the particle-hole symmetric regime at $U=3.6,J=1$ for orbital dimensionalities $1,2,3$.

Figure 2

Curie-Weiss dependence of the inverse magnetic susceptibility ${\chi }^{-1}\left(T\right)$ over temperature of the half-filled two- and three-orbital Hubbard model at $U/t=4,J=1.2t$ and $U=8t,J=1.2t$ (for the two-orbital case only). The curves for the rotationally invariant interaction are named “full” and plotted with straight lines and filled points, while the curves which correspond to the density-density type of interaction are named “nn” and plotted with dashed lines and empty points. The dotted lines are linear fits of the inverse susceptibilities in the paramagnetic regime. The Neel temperature is defined as the crossing point of interpolation lines with the ${\chi }^{-1}\left(T\right)=0$ axis. The temperatures obtained are $T_{N=2}^{\mathrm{full}}=0.42$ in the rotationally invariant model and $T_{N=2}^{\mathrm{nn}}=0.49$ in the density-density variant of the two-orbital model. In the three-orbital case the corresponding values are $T_{N=3}^{\mathrm{full}}=0.7$ and $T_{N=3}^{\mathrm{nn}}=0.83$.

Figure 3

Curie-Weiss dependence of the inverse magnetic susceptibility ${\chi }^{-1}\left(T\right)$ over temperature of the two-orbital model at the single-electron filling and the three-orbital model at two-electron filling. The curves are obtained at $U=8,J=1.2t$. The notations are the same as in Fig. 2. The obtained magnetic transition temperatures in the three-orbital model at one-third filling differ by a factor of 2 and have values $T_{N=3;n=2}^{\mathrm{full}}=0.14$ in the rotationally invariant model and $T_{N=3;n=2}^{\mathrm{nn}}=0.27$ in the “density-density” model.

Figure 4

(a) Family of densities of states of the two-orbital Hubbard model for various values of $U$ in the vicinity of the metal-insulator transition, $J=U/4,T=0.04t$. All curves are produced by using the analytic continuation by the maximum entropy method.[47] The upper part of the graph contains results for the rotationally invariant Coulomb interaction (3), and the lower part of the graph contains results for its density-density part (4). (b) Density of states at the Fermi level $A\left(0\right)$ for rotationally invariant and density-density models as a function of $U$, $J=U/4$ at temperatures $0.04t,0.05t,0.07t$. The points were obtained from the extrapolation of Matsubara Green's function to the zero value and showed no difference from the value ${\beta G(\tau =\beta /2)=A(\omega =0)|}_{T\to 0}$.

Figure 5

Family of densities of states of the two-orbital Hubbard model for various values of temperature for the models with the (a) rotationally invariant Coulomb interaction at $U=3.14t,J=U/4$ and (b) density-density interaction at $U=2.9t,J=U/4$.

Figure 6

Family of the density of states of the three-orbital Hubbard model at half-filling for various values of $U$ in the vicinity of the metal-insulator transition, $J=U/4,T=0.05t$. The curves are plotted in the same fashion as in Fig. 4; the upper part corresponds to the rotationally invariant interaction and the lower to its “density-density” part.

Figure 7

Densities of states of the one-third filled (two electrons) three-orbital model at $U=2.6t,J=U/4$ for $T=0.04$ and $0.05t$ obtained by maximum entropy analytic continuation.[47]

Figure 8

Imaginary part of the self-energy in the Matsubara domain plotted as a function of the number of Matsubara frequency in the three-orbital model (2) with one-half and one-third fillings at $U=2.6t,J=U/4,T=0.05t$. The half-filled curve shows an insulating diverging behavior, whereas the curve for a one-third filling reaches a finite value with a proximity to zero. The inset shows the curves for one-third filling zoomed at first Matsubara frequencies.

Figure 9

Effective mass as a function of temperature in the one-third-filled three-orbital model (2) at $U=2.6t,J=U/4$ for the density-density and rotationally invariant Coulomb interactions. (Inset) Imaginary part of the self-energy at $\omega =0$ as a function of temperature for the same parameters.