Quantum Spin Ice Dynamics in the Dipole-Octupole Pyrochlore Magnet ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$

Neutron scattering measurements on the pyrochlore magnet ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ reveal an unusual crystal field splitting of its lowest $J=5/2$ multiplet, such that its ground-state doublet is composed of $m_J=\pm 3/2$, giving these doublets a dipole-octupole (DO) character with local Ising anisotropy. Its magnetic susceptibility shows weak antiferromagnetic correlations with ${\theta }_{\mathrm{CW}}=-0.4(2)\mathrm{K}$, leading to a naive expectation of an all-in, all-out ordered state at low temperatures. Instead, our low-energy inelastic neutron scattering measurements show a dynamic quantum spin ice state, with suppressed scattering near $|\mathbf{Q}|=0$, and no long-range order at low temperatures. This is consistent with recent theory predicting symmetry-enriched U(1) quantum spin liquids for such DO doublets decorating the pyrochlore lattice. Finally, we show that disorder, especially oxidation of powder samples, is important in ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ and could play an important role in the low-temperature behavior of this material.

Figure 1

(a) The inverse magnetic susceptibility of a powder sample of ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$. The red curve is the Van Vleck susceptibility calculated with the CEF Hamiltonian of ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$. (Top left inset) The AIAO and 2I2O magnetic ground-state spin configurations on a pair of tetrahedra. (Bottom right inset) The low-temperature magnetic susceptibility that yields ${\theta }_{\mathrm{CW}}=-0.4(2)\mathrm{K}$ and a paramagnetic moment of $1.3(1){\mu }_B$ and shows no signature of magnetic order or spin freezing down to 0.5 K. (b) Inelastic neutron scattering spectra of ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ at $T=5\mathrm{K}$ with incident neutron energy $E_i=150\mathrm{meV}$. Two strong excitations can be identified as magnetic in origin at $E\sim 56$ and $\sim 112\mathrm{meV}$, as their intensity decreases as a function of $|\mathbf{Q}|$, consistent with the ${\mathrm{Ce}}^{3+}$ magnetic form factor. (c) The energy eigenvalues corresponding to the CEF states belonging to the spin-orbit ground-state manifold of ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$. The composition of the CEF eigenfunctions are also presented in (c), revealing the DO nature of the ground-state doublet—that is, it corresponds to pure $m_J=\pm 3/2$ states.

Figure 2

(a) The onset of dynamic spin ice correlations with decreasing temperature in an annealed ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ powder sample. (b) The powder-averaged difference neutron scattering spectra for an annealed single crystal sample of ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$. A dataset at $T=2\mathrm{K}$ has been subtracted from that at $T=0.06\mathrm{K}$. (c) A cut along $|\mathbf{Q}|$ through this difference spectra showing that the dominant quasielastic signal, integrated in energy between 0 and 0.15 meV, is centered on $|\mathbf{Q}|=|(001)|$ ($\sim 0.59{\AA }^{-1}$) and (d) a comparison of two cuts in energy through the difference spectra shown in (b), with one of these cuts taken with a $|\mathbf{Q}|$ integral centered on $|(001)|$ ($0.35–0.85{\AA }^{-1}$), and one removed from $|(001)|$, integrating between 1.3 and $1.8{\AA }^{-1}$. For all these panels, the error bars correspond to 1 standard deviation.

Figure 3

Comparison of the measured low-energy inelastic neutron scattering from (a) an annealed single crystal sample of ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ with the calculated quasielastic neutron scattering for (b) the classical near-neighbor spin ice model at $T=0\mathrm{K}$ and (c) a quantum spin ice at finite $T$. Data in (a) are the symmetrized difference between inelastic scattering at $T=0.06\mathrm{K}$ and $T=2\mathrm{K}$, integrated between 0 and 0.15 meV. (b),(c) Simulations taken from Benton et al. [34]. The lack of intensity around $\mathbf{Q}=(000)$ and the fact that the ring of diffuse inelastic scattering peaks along $(00L)$ provides evidence for ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ displaying a dynamic quantum spin ice state at these low temperatures. Also, the observed diffuse inelastic scattering at $\mathbf{Q}=(003)$ is more pronounced than that at $\mathbf{Q}=(\frac{3}{2}\frac{3}{2}\frac{3}{2})$, again consistent with the expectations of quantum spin ice, and not consistent with classical near-neighbor spin ice. Note the extra features centered at the Bragg peak positions such as (111) likely originate from leakages of the structural Bragg peaks, due to the subtraction of two large intensities.