Quantum Versus Classical Spin Fragmentation in Dipolar Kagome Ice ${\mathrm{Ho}}_3{\mathrm{Mg}}_2{\mathrm{Sb}}_3{\mathrm{O}}_{14}$

A promising route to realize entangled magnetic states combines geometrical frustration with quantum-tunneling effects. Spin-ice materials are canonical examples of frustration, and Ising spins in a transverse magnetic field are the simplest many-body model of quantum tunneling. Here, we show that the tripod-kagome lattice material ${\mathrm{Ho}}_3{\mathrm{Mg}}_2{\mathrm{Sb}}_3{\mathrm{O}}_{14}$ unites an icelike magnetic degeneracy with quantum-tunneling terms generated by an intrinsic splitting of the ${\mathrm{Ho}}^{3+}$ ground-state doublet, which is further coupled to a nuclear spin bath. Using neutron scattering and thermodynamic experiments, we observe a symmetry-breaking transition at $T^{*}\approx 0.32\mathrm{K}$ to a remarkable state with three peculiarities: a concurrent recovery of magnetic entropy associated with the strongly coupled electronic and nuclear degrees of freedom; a fragmentation of the spin into periodic and icelike components; and persistent inelastic magnetic excitations down to $T\approx 0.12\mathrm{K}$. These observations deviate from expectations of classical spin fragmentation on a kagome lattice, but can be understood within a model of dipolar kagome ice under a homogeneous transverse magnetic field, which we survey with exact diagonalization on small clusters and mean-field calculations. In ${\mathrm{Ho}}_3{\mathrm{Mg}}_2{\mathrm{Sb}}_3{\mathrm{O}}_{14}$, hyperfine interactions dramatically alter the single-ion and collective properties, and suppress possible quantum correlations, rendering the fragmentation with predominantly single-ion quantum fluctuations. Our results highlight the crucial role played by hyperfine interactions in frustrated quantum magnets and motivate further investigations of the role of quantum fluctuations on partially ordered magnetic states.

Popular Summary

Quantum spin ice is a highly entangled state that researchers have long tried to achieve in the lab because of its theoretical simplicity and its close relation to real pyrochlore minerals. But it is very difficult to endow the underlying spins with robust quantum dynamics on the pyrochlore lattice without introducing additional complications and departing from the necessarily simplified theoretical models. Our work exploits a new concept for realizing a 2D quantum spin ice by using a low-symmetry crystal structure to increase quantum fluctuations, an approach that will help others find and design new quantum materials.

Specifically, we study the material ${\mathrm{Ho}}_3{\mathrm{Mg}}_2{\mathrm{Sb}}_3{\mathrm{O}}_{14}$, a “tripod kagome magnet” derived from the pyrochlore lattice. This system offers the geometrical frustration of spin ice and a material-based approach to introducing a homogenous transverse magnetic field that make it ideal for this kind of work. Using neutron-scattering experiments, we find a spin-fragmented state at low temperature with persistent quantum dynamics, which is strongly affected by hyperfine interactions. This discovery highlights how nuclear spins contribute to the dynamics of frustrated magnets and exemplifies how low-symmetry crystal structures can be used to increase quantum fluctuations for spin-ice-like materials in general.

With these findings, we hope that researchers will now consider it a practical option to introduce quantum fluctuations in frustrated magnets and pay close attention to the role of nuclear spins in their quest to create quantum spin ices.

Figure 1

Classical spin fragmentation (CSF) process in a model of dipolar kagome ice displaying emergent-charge order, based on Ref. [39]. The expectation values of spins $⟨{\sigma }_i^z⟩$ are represented by black arrows. Each triangle has one spin pointing “in” (toward its center) and two pointing “out” (away from its center), or vice versa. The emergent magnetic charge of a triangle is defined as the number of spins pointing in minus the number pointing out. Positive ($Q_j=+$) and negative ($Q_j=-$) emergent charges are represented as red and blue triangles, respectively, and form a staggered arrangement. Three distinct spin configurations are possible for a given emergent charge, which yields a macroscopic number of degenerate spin configurations associated with emergent-charge ordering. Spin fragmentation decomposes each unit-length spin into “divergence-full” and “divergence-free” channels (center and right-hand images, respectively). The fragmented spins are shown as orange circles with diameter proportional to the length of the fragmented spin. The green hexagon represents the flipping of six spins around a closed loop: this is the simplest process that connects two distinct spin configurations within the degenerate CSF manifold.

Figure 2

(a) Simplified partial crystal structure of ${\mathrm{Ho}}_3{\mathrm{Mg}}_2{\mathrm{Sb}}_3{\mathrm{O}}_{14}$, showing alternating ${\mathrm{Ho}}^{3+}$ kagome layers (large blue spheres) and ${\mathrm{Mg}}^{2+}$ triangular layers (small orange spheres). (b) Local environment of ${\mathrm{Ho}}^{3+}$ ions, showing eight oxygen ions (small pink spheres), two ${\mathrm{Mg}}^{2+}$ ions, and four nearest-neighbor ${\mathrm{Ho}}^{3+}$ ions around a central ${\mathrm{Ho}}^{3+}$ ion. The tripodlike arrangement of the local Ising axes is enforced by the oxygen in the center of each ${\mathrm{MgHo}}_3$, which yields a canting angle of 22.3° with respect to the kagome plane. Spins are labeled by black arrows while $Q_j=+1$ and $Q_j=-1$ magnetic charges are illustrated by red and blue spheres, respectively. The coordinate system depicts the local Ising, transverse field, and mean-field directions. (c) Crystal-field excitations measured by inelastic neutron-scattering experiments on the SEQUOIA spectrometer. Open black circles and blue squares indicate intensities measured with incident neutron energies of 60 and 120 meV, respectively. Five crystal-field levels can be directly resolved at energies of 190(2), 238(2), 388(9), 627(5), and 743(9) K, whose peak positions are extracted from Lorentzian fits to the data (black dashed lines). The overall fit, including a modeled phonon background, is shown as a red line. Detailed analysis of the crystal-field excitations can be found in a separate study [53]. (d) Comparison of the single-ion response expected for a ${\mathrm{Ho}}^{3+}$ ion in a pyrochlore versus a tripod-kagome geometry. Crystal-field singlets $|0⟩$ and $|1⟩$ in ${\mathrm{Ho}}_3{\mathrm{Mg}}_2{\mathrm{Sb}}_3{\mathrm{O}}_{14}$ can be effectively described as a transverse field $h^x$ acting on the non-Kramers doublet $|\pm ⟩$ of ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$. While hyperfine interactions split the electronic-nuclear manifold into eight levels in both systems, the transverse field $h^x$ transforms the classical single-ion picture of ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ into a quantum picture in ${\mathrm{Ho}}_3{\mathrm{Mg}}_2{\mathrm{Sb}}_3{\mathrm{O}}_{14}$, where each eigenstate is a quantum superposition of $|+,I^z⟩$ and $|-,I^z⟩$. Red dashed arrows mark the allowed transitions between eigenstates of the same $I^z$. (e) Calculated scattering intensity of single-ion excitations as a function of $h^x$ at a temperature of 1.6 K. Dashed lines present the transition energies whose intensities are calculated and convoluted with a Lorentzian function with $\mathrm{FWHM}=0.35\mathrm{K}$. (f) Low-energy inelastic neutron-scattering response measured on the OSIRIS spectrometer for pure ${\mathrm{Ho}}_3{\mathrm{Mg}}_2{\mathrm{Sb}}_3{\mathrm{O}}_{14}$ (SG sample) and the dilute Ho-tripod magnet $({\mathrm{Ho}}_{0.01}{\mathrm{La}}_{0.99}{)}_3{\mathrm{Mg}}_2{\mathrm{Sb}}_3{\mathrm{O}}_{14}$ at a temperature of 1.6 K. The two datasets are normalized to the same total spectrum weight. Solid lines represent the single-ion spectrum calculated using $h^x=0.57\mathrm{K}$, which yields the best agreement with the experimental data.

Figure 3

(a) Magnetic contribution to the specific heat in ${\mathrm{Ho}}_3{\mathrm{Mg}}_2{\mathrm{Sb}}_3{\mathrm{O}}_{14}$ ($C_m$, shown as blue squares for the SS sample and red dots for the SG sample). The specific heat contains coupled electronic and nuclear spin contributions. The navy dashed line shows the single-ion nuclear specific heat calculated assuming an isolated ${\mathrm{Ho}}^{3+}$ ion with $h_x=0.85\mathrm{K}$ and $A_{\mathrm{hf}}=0.319\mathrm{K}$. Green solid lines show a mean-field Monte Carlo calculation including spin-spin interactions ($J_{\mathrm{NN}}=-0.64\mathrm{K}$, $D=1.29\mathrm{K}$). (b) Magnetic entropy change for $C_m$ from 20 K to the lowest measurement temperature of 76 mK (colors and symbols as above). Our measurements on the SS sample render a residual entropy of $1.5(5)\mathrm{J}{\mathrm{K}}^{-1}{\mathrm{mol}}_{\mathrm{Ho}}^{-1}$ at 87 mK, and our mean-field Monte Carlo simulations predict a zero-point entropy of $0.910\mathrm{J}{\mathrm{K}}^{-1}{\mathrm{mol}}_{\mathrm{Ho}}^{-1}$.

Figure 4

Low-energy neutron-scattering response of ${\mathrm{Ho}}_3{\mathrm{Mg}}_2{\mathrm{Sb}}_3{\mathrm{O}}_{14}$ at $T=4.2$, 1.6, 0.4, and 0.12 K (top to bottom panels). Data are from two distinct measurements on different spectrometers and samples. Black circles correspond to measurements on the SS sample using the DCS spectrometer with an incident neutron wavelength of 5 Å; blue squares correspond to measurements on the SG sample using the OSIRIS spectrometer with a final neutron wavelength of 6.67 Å. (a) Dependence of the inelastic magnetic scattering intensity $I(Q,\omega )$ on wave-vector transfer $Q$ and energy transfer $\omega$. (b) Energy-integrated ($-20\le \omega \le 20\mathrm{K}$) magnetic neutron-scattering intensity $I_{\mathrm{sub}}(Q)$ at four temperatures, showing experimental data (black circles), fits using the paramagnetic effective field theory (dashed red lines), and calculations using the mean-field theory (solid olive lines). The correlated magnetic scattering has been isolated by subtracting high-temperature data or simulations taken at $T=30\mathrm{K}$, which leads to negative values for the subtracted intensity. (c) Energy dependence of magnetic neutron-scattering intensity $I(\omega )$ integrated over wave-vector transfers $0.4\le Q\le 1.6{\AA }^{-1}$ at four different temperatures. An empty container subtraction was used to remove background.

Figure 5

Results of exact diagonalization of the effective Hamiltonian [Eq. (10)] on an $N=12$ cluster. (a) and (b) are obtained with $J_{\mathrm{NN}}=0$ and $A_{\mathrm{hf}}=0$. (a) Second derivative of the ground-state energy as a function of $h^x/D$ (black squares) and energy gap between the ground state and first excited state (red circles), indicating possible quantum phase transitions. Blue lines represent linear fits to the data. (b) Projection of the ground-state wave function into the CSF basis (black squares). (c) Nearest-neighbor spin correlation (blue) and emergent charge correlation (red) of the ground states as a function of hyperfine coupling strength where $Q_l$ and $Q_m$ are the emergent charges defined as in Fig. 1. Parameters appropriate for ${\mathrm{Ho}}_3{\mathrm{Mg}}_2{\mathrm{Sb}}_3{\mathrm{O}}_{14}$ ($D=1.29\mathrm{K}$, $J_{\mathrm{NN}}=-0.64\mathrm{K}$, $h^x=0.85\mathrm{K}$) are used in (c). The two dashed lines indicate nuclear spin sectors with $I_i^z=\pm 1/2$ and $\pm 7/2$ on each site, respectively.

Figure 6

Mean-field Monte Carlo (MFMC) simulation results of the mean-field Hamiltonian, Eq. (12). (a) Phase diagram as a function of $J_{\mathrm{NN}}/D$ and $h^x/D$ with $A_{\mathrm{hf}}=0$. The color represents the ordered moment of the all in, all out (AIAO) structure at 0.12 K. Four distinct phases are observed: AIAO, paramagnetic, one in, one out (OIOO), and moment-modulated spin fragmented (MMSF). The boundary (white dashed line) between the paramagnetic phase and the OIOO phase is obtained from the phase diagram of the static moment (see Fig. 11). (b) Representative spin configurations of the MMSF phase and the OIOO phase, with $h^x$ chosen to be 0.85 K [white dot in (a)] and 2 K [white square in (a)], respectively. The exchange interaction $J_{\mathrm{NN}}=-0.64\mathrm{K}$ and the dipolar interaction $D=1.29\mathrm{K}$ are the same for all subsequent panels in this figure. The sizes of arrows are scaled with static spins $⟨{\sigma }_i^z⟩$ on each site. Cyan and white arrows denote long ($|⟨{\sigma }_i^z⟩|>0.9$) and short static spins ($|⟨{\sigma }_i^z⟩|<0.9$), respectively. (c) Fourier transform of static spin correlations $⟪{\sigma }_i^z⟩⟨{\sigma }_j^z⟫$ in the MMSF phase (top) and the OIOO phase (bottom). (d) Ordered moment as a function of $h^x$ with and without hyperfine coupling (green squares and black circles, respectively). The orange line corresponds to $h^x=0.85\mathrm{K}$ for ${\mathrm{Ho}}_3{\mathrm{Mg}}_2{\mathrm{Sb}}_3{\mathrm{O}}_{14}$. (e) Normalized distribution of static spin lengths and (f) normalized distribution of magnetic charges for the MMSF phase with and without hyperfine couplings (green and black bars, respectively).

Figure 7

Rietveld refinements to diffraction data for the SS sample. Corefinements of the crystal structure to neutron and x-ray diffraction data are shown in (a) and (b), respectively. No obvious difference was observed between the x-ray diffraction pattern of SG and SS samples (not shown here). Refinement of the average magnetic structure to low-temperature magnetic diffraction data is shown in (c). In all panels, experimental data are shown as red circles, Rietveld fits as black lines, and difference (data-fit) as blue lines. Inset: illustrations of the all in, all out average magnetic structure within a unit cell and a single kagome layer.

Figure 8

Isothermal magnetization measurements (open symbols) on the SG sample of ${\mathrm{Ho}}_3{\mathrm{Mg}}_2{\mathrm{Sb}}_3{\mathrm{O}}_{14}$ at various temperatures. Solid lines were calculated according to Eq. (c3) with fixed values of $\alpha =7.79$, $h_x=0.85\mathrm{K}$, $A_{\mathrm{hf}}=0.319\mathrm{K}$, and a fitted Van Vleck term ${\chi }_{\mathrm{VV}}/{\mu }_0=0.0148{\mu }_B/T$.

Figure 9

(a) Different clusters used for the exact diagonalization (ED) calculations of size $N=12$, 18. ED results of Eq. (10) with $J_{\mathrm{NN}}=0$, and $A_{\mathrm{hf}}=0$, showing (b) second derivative of the ground-state energy, and (c) energy gap between the ground state and first excited state. “Exchange model” denotes a transverse Ising model with nearest-neighbor exchange couplings only.

Figure 10

The relative difference of total energy between the Ewald summation and direct summation as a function of the splitting parameter $\alpha$ and reciprocal space cutoff $n_k$. The real space cutoff is taken to be $r_c=\min (a,b,c)/2$, where $a$, $b$, and $c$ are the dimension of $5\times 3\times 2$ orthorhombic supercell. The real-space cutoff in the direct sum is $r_c=1000r_{\mathrm{NN}}$. The calculation is averaged over 50 random spin configurations. We take the reciprocal-space cutoff $n_k=10$ and the best splitting parameter $\alpha =7.78/\min (a,b,c)$. The same procedure is performed to find the best splitting parameter $\alpha =8.10/\min (a,b,c)$ for the $10\times 6\times 4$ simulation box with the same reciprocal-space cutoff.

Figure 11

Comparison of the ordered moment (a),(d), the static moment (b),(e), and the specific heat (c),(f) without the hyperfine coupling (top three panels) and with the hyperfine coupling $A_{\mathrm{hf}}=0.319\mathrm{K}$ (bottom three panels) from our MFMC simulations. The dipolar interaction strength is 1.29 K. The temperature of (a), (b), (d), and (e) is 0.12 K. The exchange interaction strength in (c) and (f) is $-0.64\mathrm{K}$. The OIOO phase is destroyed by including the hyperfine coupling as shown by comparing (a) and (d).