Effective magnetic interactions in spin-orbit coupled $d^4$ Mott insulators

Transition metal compounds with the ${\left(t_{2g}\right)}^4$ electronic configuration are expected to be nonmagnetic atomic singlets both in the weakly interacting regime due to spin-orbit coupling, as well as in the Coulomb dominated regime with oppositely aligned $L=1$ and $S=1$ angular momenta. However, starting with the full multiorbital electronic Hamiltonian, we show the low-energy effective magnetic Hamiltonian contains isotropic superexchange spin interactions but anisotropic orbital interactions. By tuning the ratio of superexchange to spin-orbit coupling $J_{\mathrm{SE}}/\lambda$, we obtain a phase transition from nonmagnetic atomic singlets to novel magnetic phases depending on the strength of Hund's coupling, the crystal structure and the number of active orbitals. Spin-orbit coupling plays a non-trivial role in generating a triplon condensate of weakly interacting excitations at antiferromagnetic ordering vector $\vec{k}=\vec{\pi }$, regardless of whether the local spin interactions are ferromagnetic or antiferromagnetic. In the large $J_{\mathrm{SE}}/\lambda$ regime, the localized spin and orbital moments produce anisotropic orbital interactions that are frustrated or constrained even in the absence of geometric frustration. Orbital frustration leads to frustration in the spin channel opening up the possibility of spin-orbital liquids with both spin and orbital entanglement.

Figure 1

(a) The single site total angular momentum is zero in both the $jj$ and $LS$ coupling schemes. (b) Schematic phase diagram of the spin-orbital model appearing in (5) pitting spin-orbit coupling $\lambda$ against superexchange $J_{\mathrm{SE}}$ where $\lambda$ is the spin-orbit coupling energy scale and $J_{\mathrm{SE}}$ is the superexchange energy scale with $z$ being the coordination number. Starting with a van Vleck phase with no atomic moments at large $\lambda$ we find a triplon condensate at $\mathit{k}=\vec{\pi }$ for all values of the Hund's coupling $J_H/U$. The intermediate regime where $\lambda \approx z{J}_{\mathrm{SE}}$ has not been explored. At large $J_{\mathrm{SE}}$, we obtain effective magnetic Hamiltonians that have isotropic Heisenberg spin interactions (antiferromagnetic for small $J_H/U$ and ferromagnetic for large $J_H/U$) but the orbital interactions are more complex and anisotropic. We expect novel magnetic phases arising from orbital frustration in the intermediate and large $J_{\mathrm{SE}}/\lambda$ regimes.

Figure 2

(a) The $N_{\mathrm{orb}}=2$ model is an approximation of oxygen mediated electron hopping between $t_{2g}$ orbitals in a simple cubic lattice. Both $d_{xy}$ and $d_{yz}$ orbitals participate in hopping along the $y$ direction. (b) The $N_{\mathrm{orb}}=1$ model is an approximation of direct hopping between $t_{2g}$ orbitals on the face of a face-centered cubic lattice. The $d_{xy}$ orbitals are most relevant for hopping in the $xy$ plane.

Figure 3

The Hamiltonian in equation (1) is solved for a two-site system. The local total angular momentum squared on one site, $\langle J_i^2\rangle$, is plotted for small and large values of Hund's coupling, $J_H/U=0.1$ and $J_H/U=0.2$, for the three types of hopping matrices used in the text. (a) and (b) Hopping using $N_{\mathrm{orb}}=3$ produces sizable local moments. For small Hund's coupling, the local moment gradually forms as $t$ is turned on. For large Hund's coupling, there is an abrupt formation of large local moments due to an energy level crossing. (c) and (d) Hopping using $N_{\mathrm{orb}}=2$ produces qualitatively similar behavior to the $N_{\mathrm{orb}}=3$ case. (e) and (f) Hopping using $N_{\mathrm{orb}}=1$ produces negligible moments.

Figure 4

(a) The virtual process leaves the first site in a low spin, $S=\frac{1}{2}$, configuration and results in antiferromagnetic superexchange. (b) The virtual process leaves the first site in a high spin, $S=\frac{3}{2}$, configuration and results in ferromagnetic superexchange.

Figure 5

Energy eigenvalues of the two-site superexchange Hamiltonian (7) are plotted for (a) $N_{\mathrm{orb}}=3$ using (8), (b) $N_{\mathrm{orb}}=2$ using (10), and (c) $N_{\mathrm{orb}}=1$ using (12). In addition to a spin-AF ground state, a spin-F ground state can be favored when Hund's coupling is large in both the $N_{\mathrm{orb}}=3$ and $N_{\mathrm{orb}}=2$ models.

Figure 6

Expectation values of different angular momentum correlators are plotted for the two-site effective Hamiltonian in (5) using the three different $N_{\mathrm{orb}}$ models with the parametrization $\lambda =cos\theta , t^2/U=sin\theta , J_H/U=0.1$, and $\lambda /J_H=1$. The $N_{\mathrm{orb}}=3$ model features full rotational symmetry while the $N_{\mathrm{orb}}=2$ and 1 models only have one axis of rotational symmetry to make the $z$ correlators different than the $x$ and $y$ correlators. The effect of increasing $J_H/U$ is to push the crossing point from spin-AF to spin-F behavior further left in these plots.

Figure 7

Orbital frustration is graphically illustrated for the $N_{\mathrm{orb}}=2$ model. The orbitals shown on the vertices of the plaquette are the doubly occupied orbital on each site in a square lattice. Once the first bond, labeled as 1, is chosen to be of a particular type, either (a) AF or (b) F, the next bonds, labeled as 2, are immediately fixed by this choice. The result is that the last bond on the plaquette, labeled as 3, then takes a configuration which is neither the most energetically favorable AF bond nor the most energetically favorable F bond.

Figure 8

The triplon condensation mechanism is graphically illustrated. When there exists a triplon excitation on a site, superexchange can move the excitation to neighboring sites. This effective hopping causes the triplon's energy to disperse in $k$ space. When superexchange becomes large enough, condensation of triplon excitations occurs as the bottom of the triplon band becomes lower in energy than the original $J_i=0$ level.

Figure 9

Energy levels of the effective Hamiltonian $H_{\mathrm{SE}}+H_{\mathrm{SOC}}$ appearing in (20) and (21) with the parametrization $\lambda =cos\theta$ and $J_{\mathrm{SE}}=sin\theta$. The levels are labeled by their good quantum numbers. In the $\theta =0$ limit, the eigenstates of spin-orbit coupling are used, and, in the $\theta =\pi /2$ limit, the eigenstates of the spin-orbital superexchange Hamiltonian are used. The interpretations of the states are discussed in the main text.

Figure 10

The flow of angular momentum is graphically shown where ingoing arrows are incoming angular momentum and outgoing arrows are outgoing angular momenta. Wigner-3j symbols and Clebsch-Gordan coefficients are vertices with three legs while the scalar contraction of four Wigner-3j symbols (right) is a Wigner-6j symbol [42, 43]. (a) Equation (C6) is shown in graphical form for the $T_{\zeta }^{†}$ part of the equation. A $J=0$, state is decomposed into its $L=1$ and $S=1$ components which are acted on by the ${\left(L_i\right)}_m^l$ and ${\left(S_i\right)}_{\sigma }^s$ operators. The resulting $L=1$ and $S=1$ are combined together to give a $J=1$ state with quantum number $\zeta =m+\sigma$. (b) The projection of equation (C5) to (C7) conserves angular momentum. Equation (C8) will appear similarly except that $1m^{\prime }$ and $1m$ add to yield $LM$ instead.