Quantum fluctuations in the effective pseudospin-$\frac{1}{2}$ model for magnetic pyrochlore oxides

The effective quantum pseudospin-$1/2$ model for interacting rare-earth magnetic moments, which are locally described with atomic doublets, is studied theoretically for magnetic pyrochlore oxides. It is derived microscopically for localized Pr${}^{3+}$ $4f$ moments in Pr${}_2{\mathit{TM}}_2$O${}_7$ ($\mathit{TM}=$Zr, Sn, Hf, and Ir) by starting from the atomic non-Kramers magnetic doublets and performing a strong-coupling perturbation expansion of the virtual electron transfer between the Pr $4f$ and the O $2p$ electrons. The most generic form of the nearest-neighbor anisotropic superexchange pseudospin-$1/2$ Hamiltonian is also constructed from the symmetry properties, which is applicable to Kramers ions Nd${}^{3+}$, Sm${}^{3+}$, and Yb${}^{3+}$, potentially showing large quantum effects. The effective model is then studied by means of a classical mean-field theory and the exact diagonalization on a single tetrahedron and on a 16-site cluster. These calculations reveal appreciable quantum fluctuations leading to quantum phase transitions to a quadrupolar state as a melting of spin ice for the Pr${}^{3+}$ case. The model also shows the formation of a cooperative quadrupole moment and pseudospin chirality on tetrahedrons. A sign of a singlet quantum spin ice is also found in a finite region in the space of coupling constants. The relevance to experiments is discussed.

Figure 1

(a) Pr${}^{3+}$ ions form tetrahedra (dashed lines) centered on ${\mathrm{O}}^{2-}$ ions (O1) and are surrounded by ${\mathrm{O}}^{2-}$ ions (O2) in the $D_{3d}$ symmetry as well as by transition-metal ($\mathit{TM}$) ions. Each Pr magnetic moment (thick arrow) points to either of the two neighboring O1 sites. $({\mathit{x}}_{\mathit{r}},{\mathit{y}}_{\mathit{r}},{\mathit{z}}_{\mathit{r}})$ denotes the local coordinate frame. (b) Local coordinate frame from the top. Upward and downward triangles of ${\mathrm{O}}^{2-}$ ions (O2) are located above and below the hexagon of $\mathit{TM}$ ions, respectively.

Figure 2

Local level scheme for $f$ and $p$ electrons, and local quantization axes ${\mathit{z}}_{\mathit{r}}$ and ${\mathit{z}}_{{\mathit{r}}^{\prime }}$.

Figure 3

(a) Two $f$-$p$ transfer integrals: $V_{\mathit{pf}\sigma }$, between $p_z$ and $f_{(5z^2-3r^2)z}$ orbitals, and $V_{\mathit{pf}\pi }$, between $p_x$/$p_y$ and $f_{x(5z^2-r^2)}$/$f_{y(5z^2-r^2)}$. (b) $f$-$p$ virtual electron hopping processes. $n$ ($n^{\prime }$) and $\ell$ in state $f^n{p}^{\ell }{f}^{n^{\prime }}$ represent the number of $f$ electrons at the Pr site $\mathit{r}$ (${\mathit{r}}^{\prime }$) and that of $p$ electrons at the O1 site.

Figure 4

Pyrochlore lattice structure. The phase ${\varphi }_{\mathit{r},{\mathit{r}}^{\prime }}$ appearing in Eq. (18) takes $-2\pi /3$, $2\pi /3$, and $0$ on the blue, red, and green bonds, respectively, in our choice of the local coordinate frames [Eqs. (12)].

Figure 5

A phase diagram for the sign of the dimensionless Ising coupling $\mathrm{J̃}$, defined through Eq. (19), as functions of $\gamma$ and $V_{\mathit{pf}\pi }/V_{\mathit{pf}\sigma }$ for $\beta =\gamma /3$, $\gamma /6$, $0$, $-\gamma /6$, and $-\gamma /3$. In each case, $\mathrm{J̃}$ is positive in the shaded region and negative otherwise.

Figure 6

Coupling constants (a) $\delta$ and (b) $q$ as functions of $\gamma$ for several choices of $\beta /\gamma =1/3$, $1/6$, $0$, $-1/6$, and $-1/3$. We have adopted $V_{\mathit{pf}\pi }/V_{\mathit{pf}\sigma }=-0.3$.

Figure 7

Classical mean-field phase diagram of the model given by Eq. (18). The $\mathit{Q}=0$ planar ferropseudospin (PF) phase physically represents the antiferroquadrupole (AFQ) phase. The planar antiferropseudospin (PAF) phase is characterized by Bragg rods $\mathit{Q}\parallel [111]$ and physically represents a 2D AFQ phase for $q>0$ or a 2D noncoplanar ferroquadrupole (FQ) phase for $q<0$, where quadrupole moments are aligned only within the plane perpendicular to Brag rod vectors $\mathit{Q}\parallel [111]$ in the mean-field level. In the region around the N.N. Heisenberg antiferromagnet, the present mean-field theory becomes less accurate. The point $X=(\delta ,q)=(0.51,0.89)$ for Pr${}^{3+}$ is also shown.

Figure 8

Symmetry of the ground states of the effective Hamiltonian ${\mathrm{Ĥ}}_{\mathrm{eff}}$ in the space of $\delta$ and $q$ in a single-tetrahedron analysis.

Figure 9

(a) Outward normal vectors [(green) arrows] of the surfaces of the tetrahedron, used to define the chirality ${\mathrm{\kappa ̂}}_T$. (b) Solid angle subtended by four pseudospins ${\mathit{\sigma }}_{{\mathit{r}}_i}$. (c) Distribution of the tetrahedral magnetic moment $\langle {\stackrel{\wedge }{\mathcal{M}}}_T\rangle$ in a cooperative ferroquadrupolar state with $\langle {\stackrel{\wedge }{\mathcal{Q}}}_T^{\mathit{zz}}\rangle >0$. Arrows represent the lattice deformation linearly coupled to $\langle {\stackrel{\wedge }{\mathcal{Q}}}_T^{\mathit{zz}}\rangle$.

Figure 10

Symmetry properties of the ground states obtained with the exact diagonalization of the cube with the periodic boundary conditions. The dashed curve is the boundary between the “2-in, 2-out” singlet and the “2-in, 2-out”+“4-in”/“4-out” doublet for the ground state of the model on a single tetrahedron, as shown in Fig. 8. The point $X=(\delta ,q)=(0.51,0.89)$ for Pr${}^{3+}$ is also shown.

Figure 11

Calculated neutron scattering profile (a) $S(\mathit{q})/M_0^2$ and (b) $S(\mathit{q})F_{{\mathrm{Pr}}^{3+}}(\mathit{q}){}^2/M_0^2$ for $\mathit{q}=\frac{2\pi }{a}(\mathit{hhl})$, with the form factor $F_{{\mathrm{Pr}}^{3+}}(\mathit{q})$.

Figure 12

Powder neutron-scattering intensity. Theoretical curves with(without) the form factor [blue(magneta) curve] and experimental results with Pr${}_2$Sn${}_2$O${}_7$ from Ref. 19.

Figure 13

Magnetization (left) and energy (right) per site calculated for the $I$-odd(–) ground state and the $I$-even(+) state by adding the Zeeman term $-\mathit{H}·\mathit{M}$ to ${\mathrm{Ĥ}}_{\mathrm{eff}}$, i.e., Eq. (18), at the applied field $\mathit{H}\parallel [111]$. The symmetry of the ground-state manifold does not change until a level cross occurs at ${\mu }_{B}H/J_{\mathrm{n}.\mathrm{n}.}~4.6$ to the almost fully polarized state. Experimental data on Pr${}_2$Ir${}_2$O${}_7$ at $T=0.5$ and 0.06 K from Ref. 21 are also shown for comparison, although they include additional contributions from Ir conduction electrons.

Figure 14

Upper and lower panels: dominant and subdominant forms, respectively, of quadrupole-quadrupole correlations $\langle {\stackrel{\wedge }{\mathcal{Q}}}_T^{\mathit{ii}}{\stackrel{\wedge }{\mathcal{Q}}}_{T^{\prime }}^{\mathit{jj}}\rangle$ between tetrahedrons $T$ and $T^{\prime }$ displaced by (a) $\mathit{R}={\mathit{R}}_1$, (b) ${\mathit{R}}_2$, and (c) ${\mathit{R}}_3$ in the cooperative ferroquadrupolar state with the wave vector ${\mathit{k}}_Z$ and $\langle {\stackrel{\wedge }{\mathcal{Q}}}_T^{\mathit{ZZ}}\rangle \ne 0$. In particular, ${\mathit{\lambda }}_{{\mathit{R}}_1,1}=(-0.803,0.274,0.529)$, ${\mathit{\lambda }}_{{\mathit{R}}_2,1}=(0.274,-0.893,0.529)$, and ${\mathit{\lambda }}_{{\mathit{R}}_3,1}=(1,-1,0)/\sqrt{2}$. Red and blue regions represent positive and negative values of ${\mathcal{Q}}_{\mathit{R},1}$, respectively.