Quantum Melting of Spin Ice: Emergent Cooperative Quadrupole and Chirality

Quantum melting of spin ice is proposed for pyrochlore-lattice magnets ${\mathrm{Pr}}_2{\mathrm{TM}}_2{\mathrm{O}}_7$ ($\mathrm{TM}=\mathrm{Ir}$, Zr, and Sn). The quantum superexchange Hamiltonian having a nontrivial magnetic anisotropy is derived on the basis of atomic non-Kramers magnetic doublets. The ground states exhibit a cooperative ferroquadrupole and pseudospin chirality, forming a magnetic analog of smectic liquid crystals. Our theory accounts for dynamic spin-ice behaviors experimentally observed in ${\mathrm{Pr}}_2{\mathrm{TM}}_2{\mathrm{O}}_7$.

Figure 1

(a) ${\mathrm{Pr}}^{3+}$ ions (red) form tetrahedrons (dashed lines) centered at ${\mathrm{O}}^{2-}$ ions (O1) (blue), and are surrounded by ${\mathrm{O}}^{2-}$ ions (O2) (blue) in the $D_{3d}$ symmetry as well as by TM ions (green). Each Pr magnetic moment (bold arrow) points to either of the two neighboring O1 sites. (b) The local coordinate frame (${\overrightarrow{x}}_{\mathit{r}}$, ${\overrightarrow{y}}_{\mathit{r}}$, ${\overrightarrow{z}}_{\mathit{r}}$) from the top. Upward and downward triangles of the ${\mathrm{O}}^{2-}$ ions (O2) are located above and below the hexagon of the TM ions. (c) The Pr pyrochlore lattice. The phase ${\phi }_{\mathit{r},{\mathit{r}}^{\prime }}$ in Eq. (2) depends on the color of the bonds. The global coordinate frame ($X$, $Y$, $Z$) is also shown.

Figure 2

(a) Local level scheme for $f$ and $p$ electrons, and the local quantization axes ${\overrightarrow{z}}_{\mathit{r}}$ and ${\overrightarrow{z}}_{{\mathit{r}}^{\prime }}$. (b) $f\mathrm{\text{−}}p$ transfer integrals. (c) Virtual hopping processes. $n$ ($n^{\prime }$) and $\ell$ in the 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 3

(a) Coupling constants $\delta$ and $q$ versus $\beta (=\gamma /3)$. The arrow points to the experimentally estimated value of $\beta$ [17]. (b) Outward normal vectors (green arrows) of the surfaces of the tetrahedron, used to define the chirality ${\kappa }_T$. (c) Solid angle subtended by four pseudospins ${\overrightarrow{\sigma }}_{{\mathit{r}}_i}$. (d) Distribution of the tetrahedral magnetic moment ${\overrightarrow{M}}_T$ in the cooperative ferroquadrupolar state ($⟨Q_T^{ZZ}⟩>0$). The arrows represent the lattice deformation linearly coupled to $Q_T^{ZZ}$.

Figure 4

(a) $S(\mathit{q})/M_0^2$ constructed from the local, nearest-neighbor, and second-neighbor correlations. (b) The theoretical curve (red) $I(q)F(q{)}^2$ with the form factor $F(q)$ and the powder neutron-scattering data (green) on ${\mathrm{Pr}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ at 1.4 K [15]. $I(q)$ (blue curve) is the angle average of $S(\mathit{q})/M_0^2$. (c) The magnetizations (left) and energies (right) per site for the $I\mathrm{\text{−}}\mathrm{\text{odd}}(-)$ ground state and the $I\mathrm{\text{−}}\mathrm{\text{even}}(+)$ state under the magnetic field $\overrightarrow{H}\parallel ⟨111⟩$. (d–f) Quadrupole correlations $⟨Q_T^{ii}{Q}_{T^{\prime }}^{jj}⟩$ between the tetrahedrons $T$ and $T^{\prime }$ displaced by $\mathit{r}={\mathit{r}}_X$ (d), ${\mathit{r}}_Y$ (e), and ${\mathit{r}}_Z$ (f) in the cooperative ferroquadrupolar state with the wave vector ${\mathit{k}}_Z$ and $⟨Q_T^{ZZ}⟩\ne 0$. The matrix $⟨Q_T^{ii}{Q}_{T^{\prime }}^{jj}⟩$ in $i$, $j$ has been diagonalized to yield two orthogonal forms of quadrupoles, ${\mathcal{Q}}_{\mathit{r}\mu }=\sum _i{\lambda }_{\mathit{r}\mu }^i{Q}_T^{ii}$ ($\mu =1$, 2). The shape of ${\mathcal{Q}}_{\mathit{r}1}$ showing the dominant correlation amplitude is shown. Red (blue) regions represent positive (negative) values of ${\mathcal{Q}}_{\mathit{r}1}$.