Orbitally induced string formation in the spin-orbital polarons

We study the spectral function of a single hole doped into the $ab$ plane of the Mott insulator ${\text{LaVO}}_3$, with antiferromagnetic (AF) spin order of $S=1$ spins accompanied by alternating orbital (AO) order of active $\left\{d_{yz},d_{zx}\right\}$ orbitals. Starting from the respective $t\text{−}J$ model, with spin-orbital superexchange and effective three-site hopping terms, we derive the polaron Hamiltonian and show that a hole couples simultaneously to the collective excitations of the AF/AO phase, magnons, and orbitons. Next, we solve this polaron problem using the self-consistent Born approximation and find a stable quasiparticle solution—a spin-orbital polaron. We show that the spin-orbital polaron resembles the orbital polaron found in $e_g$ systems, as e.g., in ${\text{K}}_2{\text{CuF}}_4$ or (to some extent) in ${\text{LaMnO}}_3$, and that the hole may be seen as confined in a stringlike potential. However, the spins also play a crucial role in the formation of this polaron—we explain how the orbital degrees of freedom: (i) confine the spin dynamics acting on the hole as the classical Ising spins and (ii) generate the string potential which is of the joint spin-orbital character. Finally, we discuss the impact of the results presented here on the understanding of the phase diagrams of the lightly doped cubic vanadates.

Figure 1

(Color online) Artist’s view of a single hole introduced into the spin and orbitally ordered $ab$ plane of ${\text{LaVO}}_3$, which leads to the spin-orbital polaron considered in the paper. The electrons occupy the $d_{yz}$ and $d_{zx}$ degenerate orbitals forming the classical AO state (their projections along the $a$ and $b$ axis are shown) whereas the electron spins alternate on the neighboring sites forming the classical AF state. Only the spins 1/2 of the electrons in the $d_{yz}$ and $d_{zx}$ orbitals are shown. At each site an additional electron occupies the $d_{xy}$ orbital (not shown). Due to Hund’s coupling the two electron spins form a total spin $S=1$, while at the site occupied by the hole $S=1/2$.

Figure 2

(Color online) The three-site hopping term $\tau$ [Eq. (2.42)] (solid line), the effective spin exchange $J_S$ [Eq. (2.35)] and orbital exchange $J_O$ [Eq. (2.34)] interaction (dashed and dotted line) for increasing Hund’s exchange coupling $\eta$ [Eq. (2.9)]. The realistic value of $\eta =0.15$ for ${\text{LaVO}}_3$ (cf. Ref. 14) is indicated by the light vertical dotted line.

Figure 3

(Color online) Diagrammatic representation of the SCBA equations: top—the Dyson’s equation for the $G_{AA}\left(\mathbf{k},\omega \right)$ and $G_{BB}\left(\mathbf{k},\omega \right)$ Green’s functions; bottom — equations for the respective self-energies. Note the appearance of the two wiggly lines in the calculation of the self-energies coming from simultaneous orbiton and the magnon excitations.

Figure 4

The spectral functions derived within the SCBA for the hole doped at half-filling to spin-orbital model (2.1) into: (a) $a$ orbital, and (b) $b$ orbital. Parameters: $J=0.2t$ and $\eta =0.15$ (i.e., $J_S=0.03t$, $J_O=0.08t$, and $\tau =0.06t$). Broadening $\delta =0.01t$ and cluster size $16\times 16$.

Figure 5

The spectral functions derived within the SCBA for the hole doped at half-filling to spin-orbital model (2.1) into: (a) $a$ orbital, and (b) $b$ orbital. Parameters: $J=0.6t$ and $\eta =0.15$ (i.e., $J_S=0.09t$, $J_O=0.23t$, and $\tau =0.19t$). Broadening $\delta =0.01t$ and cluster size $16\times 16$.

Figure 6

(Color online) Quasiparticle properties for increasing $J$ for the spin-orbital model discussed in the present paper with the Hund’s coupling $\eta =0.15$ (solid line), compared with the spin model of Ref. 32 (dashed line), and with the orbital model of Ref. 12 (dotted line). Different panels show: (a) the quasiparticle bandwidth $W$, (b) the quasiparticle spectral weight $a_{\text{QP}}$ averaged over the 2D Brillouin zone, and (c) the pseudogap △ (energy distance between the quasiparticle state and the first-excited state) at $\mathbf{k}=\left(\pi /2,\pi /2\right)$. Note that △ is not shown for the spin model as it cannot be defined there. The two light solid lines on panel (c) show the ${\left(J/t\right)}^{2/3}$ curves fitted to the data at low values of $J$ for the orbital and the spin-orbital model.

Figure 7

(Color online) Spectral function along the $\Gamma -M$ direction of the Brillouin zone for: the spin-orbital model (top panels), toy-spin model (middle panels), and toy-orbital model (bottom panels). Parameters: $J=0.2t$ and $\eta =0$ (i.e., $J_S=0.05t$ and $J_O=0$) on the left panels whereas $J=0.2t$ and $\eta =0.15$ (i.e., $J_S=0.03t$ and $J_O=0.08t$) on the right panels; in both cases $\tau \equiv 0$ (see text). Note that $\tau \equiv 0$ implies (Ref. 13) $A_a\left(\mathbf{k},\omega \right)=A_b\left(\mathbf{k},\omega \right)\equiv A\left(\mathbf{k},\omega \right)$. Broadening $\delta =0.01t$ [$\delta =0.02t$ on panel (f)] and cluster size $16\times 16$.

Figure 8

(Color online) Spectral function along the $\Gamma -M$ direction of the Brillouin zone for: the spin-orbital model (top panels), toy-spin model (middle panels), and toy-orbital model (bottom panels). Parameters: $J=0.6t$ and $\eta =0$ (i.e., $J_S=0.15t$ and $J_O=0$,) on the left panels, whereas $J=0.6t$ and $\eta =0.15$ (i.e., $J_S=0.09t$ and $J_O=0.23t$) on the right panels; in both cases $\tau \equiv 0$ (see text). Note that $\tau \equiv 0$ implies (Ref. 13) $A_a\left(\mathbf{k},\omega \right)=A_b\left(\mathbf{k},\omega \right)\equiv A\left(\mathbf{k},\omega \right)$. Broadening $\delta =0.01t$ [$\delta =0.02t$ on panel (f)] and cluster size $16\times 16$.

Figure 9

(Color online) Comparison between the spectral functions of the classical limit (dashed lines) and full quantum version (solid lines) of the spin-orbital model at a single point in the Brillouin zone (the spectra are $\mathbf{k}$ independent) for: (a) $J=0.2t$ and $\eta =0$ (i.e., $J_S=0.05t$ and $J_O=0$), and (b) $J=0.6t$ and $\eta =0$ (i.e., $J_S=0.15t$ and $J_O=0$); in both cases $\tau \equiv 0$ (see text). Note that $\tau \equiv 0$ implies (Ref. 13) $A_a\left(\omega \right)=A_b\left(\omega \right)\equiv A\left(\omega \right)$. Broadening $\delta =0.01t$ $\left(\delta =0.015t\right)$ in panel (a) [(b)] and cluster size $16\times 16$.

Figure 10

(Color online) Three neighboring sites along the $a$ direction with such an AO and AF order that the electron on site $\mathbf{i}+\hat{\mathbf{a}}$ can move over the intermediate site $\mathbf{i}$ via the superexchange process $\propto \tau =J/4$ to site $\mathbf{i}-\hat{\mathbf{a}}$ without disturbing the AO and AF order.