Magnetism and orbital ordering in an interacting three-band model: A dynamical mean-field study

Single-site dynamical mean-field theory is used to determine the magnetic and orbital-ordering phase diagram for a model of electrons moving on a lattice with three orbital states per site and with the fully rotationally invariant Slater-Kanamori on-site interactions. The model captures important aspects of the physics of transition-metal oxides with partially filled $t_{2g}$ shells and of electron-doped ${\text{C}}_{60}$. We introduce an unbiased, computationally simple, and inexpensive method for estimating the presence of two-sublattice order, determine the regimes in which spatially uniform and two-sublattice spin and orbital orderings are present and give physical arguments for the origins of the different phases. Guidelines are determined for optimizing the presence of ferromagnetism, which may be desirable in applications.

Figure 1

(Color online) Variation in the on-site orbital occupancy $\left(n\right)$ and magnetization $\left(m\right)$ with number of iterations of the dynamical mean-field self-consistency condition, computed for a semicircular density of states at density $n=1$ with $\beta t=50$, $U/t=16$, and $J=U/6$. Panel A: iteration number dependence of the magnetization (summed over all orbitals); saturation indicates a ferromagnetic state with moment $m\approx 0.65n$. Panel B: iteration number dependence of occupancies of the three orbitals, obtained from a translation-symmetry-unbroken [Eq. (7) with $G_B\to G_A$] dynamical mean-field equation; oscillations of the occupancy indicate two sublattice orbital ordering. Panels C and D: iteration number dependence of occupancies of the three orbitals for sublattice A (panel C) and sublattice B (panel D), obtained from the general self-consistent condition Eq. (7); saturation of occupancies to a two-sublattice orbital state is evident. Error bars have the same size as the data points.

Figure 2

(Color online) Iteration number dependence of the orbital occupancies for a simulation with parameters as in Fig. 1 but started in a broken-symmetry state with different orbital occupancies than the stable state. The simulation relaxes to the same final state as in the case of an orbitally symmetric initial state (panel B of Fig. 1).

Figure 3

(Color online) Main panel: phase diagram of the three-band model with semicircular density of states at $\beta t=50$ and $J=U/6$ in the plane of particle density $n$ and interaction strength $U$. The vertical lines indicate the Mott insulating phases at integral values of $n$. The magnetic state is labeled by P, F, and A (two sublattice antiferromagnetic) while the labels $\text{O}\left(n\right)$ denote the three classes of orbital ordering discussed in the text. The heavy dashed line (orange online) gives the boundary of the non-Fermi-liquid frozen-moment phase discovered in Ref. 23. Inset: Hartree-Fock phase diagram for magnetic phases of the same model. Magnetic phase boundaries are indicated by solid lines and orbital-ordering boundaries by dashed lines. OO and OS stand for the orbitally ordered and orbitally symmetric phases, respectively. All transitions are second order except the FM-AFM transition and the orbital-ordering transitions at $U\gtrsim 12t$ and small $n$.

Figure 4

(Color online) Internal energy $E$ computed as a function of density per site $n$ at $\beta t=50$, $U/t=16$. Vertical lines give the positions of phase boundaries presented in Fig. 3.

Figure 5

(Color online) Magnetization, $m$, plotted against interaction strength $U$ for different inverse temperatures $\beta$ at density $n=1$ indicating a very rapid onset of magnetism as the material enters the Mott insulating phase and a weakly interaction-strength dependent Curie temperature at large $U$.

Figure 6

(Color online) Square of the magnetization (normalized to density) for various magnetic phases at $\beta t=50$ and $U/t=16$. the curve shown for density $n=1$ corresponds to an insulating phase, while all other phases are metallic. The PO phases, $n\sim 1.2,1.5,1.75$ have higher Curie temperature $\left(k_{B}T/t\sim 0.05–0.07\right)$ than the orbitally ordering phases.

Figure 7

(Color online) Relative orbital-ordering strength for various phases at $\beta t=50$ and $U/t=16$. “Relative orbital-ordering strength” is the deviation of the occupancy of one orbital from the paraorbital value $\left(n/3\right)$ normalized to the value at our lowest measurement temperature $k_{B}T=0.02t$; mathematically $\left[n_{\alpha }^A\left(k_{B}T>0.02t\right)-\frac{n}{3}\right]/\left[n_{\alpha }^A\left(k_{B}T=0.02t\right)-\frac{n}{3}\right]$. The transitions from the orbital selective to the paraorbital phase are apparently second order. The transition temperatures are not sensitive to the metallic or insulating (M/I), paramagnetic of ferromagnetic (P/F) or orbital ordering natur (O1,02,O3; see Table ) of the low-$T$ phase.