One-dimensional orbital fluctuations and the exotic magnetic properties of $\mathrm{Y}\mathrm{V}{\mathrm{O}}_3$

Starting from the Mott insulator picture for cubic vanadates, we derive and investigate the model of superexchange interactions between ${\mathrm{V}}^{3+}$ ions, with nearly degenerate $t_{2g}$ orbitals occupied by two electrons each. The superexchange interactions are strongly frustrated and demonstrate a strong interrelation between possible types of magnetic and orbital orders. We elucidate the prominent role played by fluctuations of $yz$ and $xz$ orbitals which generate ferromagnetic superexchange interactions even in the absence of Hund’s exchange. In this limit, we find orbital valence bond state which is replaced either by $C$-type antiferromagnetic order with weak $G$-type orbital order at increasing Hund’s exchange, or instead by $G$-type antiferromagnetic order when the lattice distortions stabilize $C$-type orbital order. Both phases are observed in $\mathrm{Y}\mathrm{V}{\mathrm{O}}_3$, and we argue that a dimerized $C$-type antiferromagnetic phase with stronger and weaker ferromagnetic bonds alternating along the $c$ axis may be stabilized by large spin-orbital entropy at finite temperature. This suggests a scenario which explains the origin of the exotic $C$-type antiferromagnetic order observed in $\mathrm{Y}\mathrm{V}{\mathrm{O}}_3$ in the regime of intermediate temperatures and allows one to specify the necessary ingredients of a more complete future theory.

Figure 1

An artist view of the energy splittings between $t_{2g}$ orbitals in $\mathrm{Y}\mathrm{V}{\mathrm{O}}_3$ in different temperature regimes. The orbital splitting ▵, which occurs below the structural transition at $T_s$ and persists in the $C\text{−}\mathrm{AF}$ phase, favors the occupied $xy$ orbitals, but allows also for weak orbital fluctuations. Such fluctuations are quenched in the $G\text{−}\mathrm{AF}$ phase at $T<T_{N2}$.

Figure 2

Virtual charge excitations $d_i^2{d}_j^2\to d_i^3{d}_j^1\to d_i^2{d}_j^2$ within a bond $⟨ij⟩$ along the $c$ axis, which contribute to the superexchange in cubic vanadates. Orbital fluctuations which support the FM superexchange occur when different active orbitals $a$ and $b$ are occupied at both sites, as in cases (a) and (b). If the same orbitals are occupied at both sites, e.g., the orbital $a$ as in case (c), the superexchange is AF; then, a double occupancy of the occupied (and active) orbital is generated in the excited state, which next dissociates to a configuration with either (c) the same orbital occupancies or (d) with interchanged occupied orbitals at sites $i$ and $j$.

Figure 3

(Color online) Schematic picture of the classical phases with magnetic and orbital orders for $n_{ic}=1$ in the $ac$ plane: (a) OVB phase, with alternating strong FM bonds stabilized by orbital singlets represented by double lines, and weak AF bonds (with disordered $a∕b$ orbitals); (b) $C\text{−}\mathrm{AF}$ spin order accompanied by $G\text{−}\mathrm{AO}$ order; and (c) $G\text{−}\mathrm{AF}$ spin order accompanied by $C\text{−}\mathrm{AO}$ order with repeated either $a$ or $b$ orbitals along the $c$ axis. In cases (b) and (c), spins and orbitals alternate along the $b$ direction (not shown). These latter states follow the Goodenough-Kanamori rules (Refs. 5, 54) and are analyzed below for $\mathrm{Y}\mathrm{V}{\mathrm{O}}_3$.

Figure 4

(Color online) Exchange interactions $J_c$ and $J_{ab}$ as functions of Hund’s exchange $\eta$ as obtained for (a) OVB phase with alternating strong FM $J_{c1}$ (solid line) and weak AF $J_{c2}$ (dashed-dotted line) exchange interaction, (b) $C\text{−}\mathrm{AF}$ phase with (weak) $G$-type OO, and (c) $G\text{−}\mathrm{AF}$ phase stabilized by the orbital interactions $\{V_a,V_c\}$ which induce the $C\text{−}\mathrm{AO}$ order. FM (AF) exchange interactions along the $c$ axis in $C\text{−}\mathrm{AF}$ $\left(G\text{−}\mathrm{AF}\right)$ phase are shown by solid lines, while AF interactions in the $ab$ planes are shown by dashed lines.

Figure 5

(Color online) Spin-wave dispersion ${\omega }_{\mathbf{k}}$ (full lines) and orbital triplet excitation energy ${\Omega }_{\mathbf{k}}$ (dashed line), as obtained for the OVB phase along high-symmetry directions in the Brillouin zone at $\eta =0$ and $V_c=0$. For the present parameters, one finds the following values of spin-exchange constants: $J_{ab}=0.3125J$, $J_{c1}=-0.250J$, and $J_{c2}=0.125J$. The high-symmetry points are $\Gamma =(0,0,0)$, $M=(\pi ,\pi ,0)$, $R=(\pi ,\pi ,\pi )$, and $Z=(0,0,\pi )$.

Figure 6

(Color online) Spin-wave dispersions ${\omega }_{\mathbf{k}}$ (full lines) as obtained in the LSW theory for the parameters motivated by experiment (Ref. 36) (for $J=40\mathrm{meV}$): (a) $G\text{−}\mathrm{AF}$ phase with $J_{ab}=0.1425J$ and $J_c=-0.1425J$ and (b) $C\text{−}\mathrm{AF}$ phase with $J_{ab}=0.0650J$ and $J_c=0.0775J$. Orbital excitations ${\Omega }_{\mathbf{k}}$ (dashed lines) were obtained (a) within the LOW theory for the $G\text{−}\mathrm{AF}$ phase and (b) for a disordered orbital state in $C\text{−}\mathrm{AF}$ phase. Other parameters are $\eta =0.13$, $V_a=0.3J$, and $V_c=0.84J$. High-symmetry points as in Fig. 5.

Figure 7

(Color online) Energies of different phases for increasing Hund’s exchange $\eta$: $C\text{−}\mathrm{AF}$ phase (solid line), OVB phase (dashed-dotted line), and $G\text{−}\mathrm{AF}$ phase (long-dashed line). The energy of the $G\text{−}\mathrm{AF}$ phase is shown for $2V_a=V_c=0.45J$ (the other energies do not depend on $V_c$). A constant energy term $-2J$ was neglected in all phases. The circles show, for a comparison, the energy obtained for decoupled FM chains along the $c$ axis, with orbital correlations described by the 1D pseudospin Heisenberg model.

Figure 8

Mean-field phase diagram of the spin-orbital model [Eq. (2.18)] as obtained for cubic vanadates in the $(\eta ,V)$ plane at $T=0$ for $V_c=2V_a=2V$. At the spectroscopic value of $\eta \simeq 0.13$, two phases are possible: $C\text{−}\mathrm{AF}$ phase (for $V<V_0$) and $G\text{−}\mathrm{AF}$ phase (for $V>V_0$), with $V_0\simeq 0.43J$. These two AF states are observed at low temperature in $\mathrm{La}\mathrm{V}{\mathrm{O}}_3$ $\left(C\text{−}\mathrm{AF}\right)$ and in $\mathrm{Y}\mathrm{V}{\mathrm{O}}_3$ $\left(G\text{−}\mathrm{AF}\right)$, respectively.

Figure 9

(Color online) Orbital correlation functions $⟨{\vec{\tau }}_i\bullet {\vec{\tau }}_{i+1}⟩$ at even and odd bonds along a dimerized 1D chain, as obtained within the $XY$ model for increasing anisotropy ${\delta }_o$ in the exchange constants, see Eq. (5.2). The energy $E\left({\delta }_o\right)$ (dashed line), obtained using spinless fermions [Eq. (5.5)], decreases with increasing ${\delta }_o$, while the average energy in an orbital chain with the same exchange interaction $J_o$ at each bond would increase (dotted line).

Figure 10

(Color online) Exchange constants $J_{ab}$ (dashed line) and $J_{c1}$ and $J_{c2}$ (solid lines), all in meV, as obtained for the idealized $C\text{−}\mathrm{AF}$ phase with condensed $xy$ orbitals $(n_c=1)$ [Eqs. (2.8)] and orbital disordered state along the $c$ axis with dimerized orbital correlations. Vertical line indicates the value of $\eta =0.13$ estimated from the atomic data (Ref. 47) and from the optical data (Ref. 10) for $\mathrm{La}\mathrm{V}{\mathrm{O}}_3$. The parameters are $J=30\mathrm{meV}$, $⟨{\vec{\tau }}_i\bullet {\vec{\tau }}_{i+1}⟩=-0.4431$, and ${\delta }_{\tau }=0.12$ [Eq. (5.6)].

Figure 11

(Color online) Reduction of exchange constants ${\mathcal{J}}_{ab}$ (dashed line) and ${\mathcal{J}}_{c1}$ and ${\mathcal{J}}_{c2}$ (solid lines), all in meV, in the dimerized $C\text{−}\mathrm{AF}$ phase due to orbital bond fluctuations between FM and AF bonds, as given by Eqs. (5.7, 5.8, 5.9). Orbital disordered state along the $c$ axis is assumed at $p=0$. Experimental values of exchange constants found for the $C\text{−}\mathrm{AF}$ phase of $\mathrm{Y}\mathrm{V}{\mathrm{O}}_3$ (Ref. 36), shown by circle $(J_{ab}\simeq 2.6\mathrm{meV})$ and by diamonds ($J_{c1}\simeq 4.2\mathrm{meV}$ and $J_{c2}\simeq 2.0\mathrm{meV}$), are nearly reproduced for moderate fluctuations with $p=0.30$ (vertical dashed line) The parameters are $J=30\mathrm{meV}$, $\eta =0.13$, and ${\delta }_s=0.35$.

Figure 12

(Color online) Spin-wave dispersions ${\omega }_{\mathbf{k}}$ (full lines) as obtained in the LSW theory along the representative directions in the Brillouin zone for the dimerized $C\text{−}\mathrm{AF}$ phase with experimental exchange constants (Ref. 36) $J_{ab}=2.6\mathrm{meV}$, $J_c=3.1(1\pm {\delta }_s)\mathrm{meV}$, and ${\delta }_s=0.35$. These interactions are obtained by considering plaquette fluctuations of spin-exchange interactions as described in the text (see also Fig. 11). The parameters are $J=30\mathrm{meV}$, $\eta =0.13$, ${\delta }_s=0.35$, and $p=0.30$. The experimental points of Ref. 36 measured by neutron scattering at $T=85\mathrm{K}$ are reproduced by circles (the effective linewidths are not shown). The high-symmetry points are $\Gamma =(0,0,0)$, $M=(\pi ,\pi ,0)$, $R=(\pi ,\pi ,\pi )$, and $Z=(0,0,\pi )$.

Figure 13

(Color online) Result of the self-consistent calculation of intersite correlations in the dimerized $C\text{−}\mathrm{AF}$ phase along the $c$ axis for increasing temperature: (a) orbital $⟨{\vec{\tau }}_i\bullet {\vec{\tau }}_{i+1}⟩$ and (b) spin $⟨{\vec{S}}_i\bullet {\vec{S}}_{i+1}⟩$. The correlations on stronger (weaker) FM bonds are shown by solid (dashed) lines. Dashed-dotted line in (b) shows the order parameter $⟨S^z⟩$ in the $C\text{−}\mathrm{AF}$ phase as obtained from the LSW theory. The parameters are $\eta =0.13$, $J=30\mathrm{meV}$, and $p=0.30$, see Fig. 11.

Figure 14

(Color online) Entropy of the $C\text{−}\mathrm{AF}$ $\left(S_C\right)$ and $G\text{−}\mathrm{AF}$ $\left(S_G\right)$ phases as obtained for the spin-orbital model [Eq. (2.18)] using the experimental values of magnetic exchange constants in both phases. The dominating contributions result from spin excitations (dashed-dotted lines), while the orbital contributions (dashed lines) are much smaller but also give a higher entropy in the $C\text{−}\mathrm{AF}$ phase. The parameters are $J=40\mathrm{meV}$, $\eta =0.13$, $V_a=0.30J$, and $V_c=0.84J$.

Figure 15

(Color online) Free energies of the $C\text{−}\mathrm{AF}$ (${\mathcal{F}}_C$, solid line) and $G\text{−}\mathrm{AF}$ (${\mathcal{F}}_G$, dashed line) phase as obtained for the spin-orbital model [Eq. (2.18)] using the experimental values of magnetic exchange constants (Ref. 36) in both phases. The experimental magnetic transition temperatures $T_{N2}\simeq 77\mathrm{K}$ and $T_{N1}\simeq 116\mathrm{K}$ are indicated by arrows. The parameters are the same as in Fig. 14.