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# Experimental Insights into Quantum Spin Ice Physics in Dipole-Octupole Pyrochlore Magnets

Evan M Smith, Elsa Lhotel, Sylvain Petit, Bruce D Gaulin

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Evan M Smith, Elsa Lhotel, Sylvain Petit, Bruce D Gaulin. Experimental Insights into Quantum Spin Ice Physics in Dipole-Octupole Pyrochlore Magnets. Annual Review of Condensed Matter Physics, 2025, 16, pp.387-415. 10.1146/annurev-conmatphys-041124-015101. hal-05393908

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*Annual Review of Condensed Matter Physics*

# Experimental Insights into Quantum Spin Ice Physics in Dipole–Octupole Pyrochlore Magnets

Evan M. Smith,<fnref n="1" /> Elsa Lhotel,<fnref n="2" /> Sylvain Petit,<fnref n="3" /> and Bruce D. Gaulin<fnref n="1" />,<fnref n="4" />,<fnref n="5" />

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Annu. Rev. Condens. Matter Phys. 2025. 16:387–415

First published as a Review in Advance on
December 3, 2024

The *Annual Review of Condensed Matter Physics* is
online at conmatphys.annualreviews.org

https://doi.org/10.1146/annurev-conmatphys-041124-015101

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Keywords

geometric frustration, pyrochlores, dipole–octupole pseudospins, quantum spin liquid, spin ice, neutron scattering

Abstract

We review a key subset of the experimental studies that have recently focused on cubic pyrochlore magnets whose pseudospin-1/2 degrees of freedom have mixed dipolar and octupolar character. We discuss how this comes about and how the character of the pseudospin-1/2 can be experimentally determined. The minimal spin Hamiltonian for such magnetic insulators is known to give rise to a rich phase diagram with both disordered U(1) quantum spin ice (QSI) states and all-in-all-out (AIAO) noncollinear ordered states, each with dipolar and octupolar character. We focus primarily on experimental studies on two such single crystal systems, the $J = 5/2$ Ce$_2$Zr$_2$O$_7$ and the $J = 9/2$ Nd$_2$Zr$_2$O$_7$. We make the case that Ce$_2$Zr$_2$O$_7$ is an excellent QSI ground-state candidate material, close to the border between QSIs with dipolar and octupolar symmetry. Nd$_2$Zr$_2$O$_7$ exhibits an AIAO ordered phase, featuring an order parameter consisting of dipolar and octupolar magnetic moments. It is found to reside close to a QSI phase boundary and features dynamic fragmentation in its excitation spectrum.

<footnote n="1">¹Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada; email: evansmithphysics@gmail.com, gaulin@mcmaster.ca</footnote>

<footnote n="2">²Institut Néel, CNRS, Université Grenoble Alpes, Grenoble, France; email: elsa.lhotel@neel.cnrs.fr</footnote>

<footnote n="3">³LLB, CEA, CNRS, Université Paris-Saclay, CEA Saclay, Gif-sur-Yvette, France; email: sylvain.petit@cea.fr</footnote>

<footnote n="4">⁴Brockhouse Institute for Materials Research, McMaster University, Hamilton, Ontario, Canada</footnote>

<footnote n="5">⁵Canadian Institute for Advanced Research, Toronto, Ontario, Canada</footnote>
Rare-earth transition metal pyrochlores are a family of cubic materials in which the magnetic ions can occupy a three-dimensional network of corner-sharing tetrahedra, which are well known to be prone to the effects of geometric frustration. These materials have been very topical over the past ~30 years due in large part to the likelihood and realization of exotic magnetic phases arising from their frustrated lattices (1-8).

Although cubic pyrochlores can be metallic (9, 10), sometimes with superconducting ground states (11, 12), much interest has focused on magnetic insulators in which short-range intersite interactions allow the geometric frustration to dominate the physics, at least at low temperatures. Among these insulating cubic pyrochlores are the well-studied rare-earth pyrochlores with the chemical formula $R_2B_2O_7$ ($Fd\bar{3}m$ space group), where $R^{3+}$ is a magnetic rare-earth ion and $B^{4+}$ is usually a nonmagnetic transition metal ion (5). In **Figure 1a**, we show the network of corner-sharing tetrahedra formed by the $R^{3+}$ ions in the $R_2B_2O_7$ pyrochlores, which can be magnetic if the $R^{3+}$ ion itself is magnetic. The local environment of the rare-earth site, which has $D_{3d}$ point-group symmetry, is shown in **Figure 1b**.

The study of these insulating compounds has revealed rich physics, highlighting and bringing to the fore concepts of frustrated magnetism. In particular, the case of classical spin ice has been studied extensively (13). The corresponding model considers spins or pseudospins with an energetic preference to point along the local (111) z axes. In the presence of dominant antiferromagnetic nearest-neighbor interactions, the classical ground-state spin configurations for a single tetrahedron are those in which the spins on the tetrahedron all point directly toward the center of the tetrahedron or all point directly away (**Figure 1c**) (14). The resulting noncollinear antiferromagnetic order is referred to as an all-in-all-out (AIAO) phase (15). Interestingly, when ferromagnetic nearest-neighbor interactions are heavily dominant, the classical ground-state configurations for a single tetrahedron are those in which two of the spins on the tetrahedron point directly toward the center of the tetrahedron and the other two spins on the tetrahedron point directly away (**Figure 1c**) (14, 16). This local rule for the energy-minimizing spin configurations is called the 2-in-2-out rule or ice rule (16-18).

Materials governed by the 2-in-2-out rule are magnetically frustrated with a degeneracy of six per isolated tetrahedron. The 2-in-2-out rule maps directly onto the rule describing proton disorder in water ice (14, 16-20), and this equivalence led to the branding of this magnetic frustration

<figure><img src="image_7.png" /><figcaption>**Figure 1**</figcaption></figure>

(a) The sublattice of magnetic $R^{3+}$ ions in $R_2B_2O_7$ pyrochlores. The gray lines show the conventional cubic unit cell. Each $R^{3+}$ ion is at the corners of two connected tetrahedra. (b) A pair of $R^{3+}$-tetrahedra in the $R_2B_2O_7$ pyrochlore crystal structure, showing the local environment around the center-most $R^{3+}$ ion in the figure. The arrows in panel *b* show the local y and z axes for the center-most $R^{3+}$ ion, where $\mathbf{x} = \mathbf{y} \times \mathbf{z}$. (c) A pair of corner-sharing tetrahedra with spins (red) that satisfy the 2-in-2-out rule for conventional spin ices (left) and with spins displaying conventional all-in-all-out (AIAO) magnetic order (right).

**Frustration:** the condition that exists when the simultaneous minimization of all pairwise interactions between coupled degrees of freedom is not possible
problem as spin ice. This magnetic state is beautifully realized in Ho₂Ti₂O₇ and Dy₂Ti₂O₇, being the first spin ices to be discovered and characterized. Both display disordered spin ice phases at low temperature (below 1–2 K) (16, 17). Due to the lack of quantum effects and the fact that the ferromagnetic coupling results from magnetic dipole–dipole interactions between the large (~10 µB) (21–23) dipole moments of Ho³⁺ and Dy³⁺, these materials are called classical dipolar spin ices (18, 24, 25). Theoretical work, following the discovery of spin ice physics in these materials, generated much interest by reframing the problem into a magnetostatics description, with emergent magnetic monopoles associated with 3-in–1-out and 1-in–3-out violations of the local 2-in–2-out ice rules on tetrahedra (26–28).

An important question that immediately arose is how quantum mechanics affects this spin ice state. This has been the subject of numerous theoretical and experimental studies (6). Theoretical work on quantum versions of the spin ice problem recognized quantum spin ice (QSI) as a U(1) quantum spin liquid that maps onto an emergent quantum electrodynamics (QED) (6, 29–33). The elementary excitations associated with QSI are expected to be two sets of gapped excitations referred to as electric charges (or spinons), and magnetic monopoles (or visons). At lower energies still, gapless emergent photons are expected to disperse linearly into magnetic zone centers, but with a spectral weight that vanishes with decreasing energy (32)—the exact complement of what is expected for acoustic phonons, for example. In this electrodynamics description, the spinons arising from 3-in–1-out and 1-in–3-out tetrahedra correspond to emergent electric monopoles (charges), as opposed to the correspondence between spinons and magnetic monopoles relevant for the magnetostatics description used in classical spin ice literature. The magnetic monopoles and photons of this QED description have no classical analogs.

The main strategy employed in the experimental search for this new QSI phase has been to study other pyrochlore materials, where the rare-earth ion also possesses an Ising character, but which are more prone to quantum effects (34). This has been the subject of intense studies on pyrochlore systems with non-Kramers ions such as Tb³⁺ and Pr³⁺, for instance (34–38). More recently, a family of Kramers rare-earth ions that possess dipole–octupole pseudospin-1/2 degrees of freedom has been of interest, and they do indeed exhibit these particularities. These dipole–octupole pyrochlores host a wide variety of ground-state selection, ranging from antiferromagnetic ordering to quantum spin liquid phases akin to QSI. In the following sections, we describe how their dipole–octupole pseudospin-1/2 degrees of freedom arise from their crystal electric field (CEF) properties, as well as the general form of their theoretically expected phase diagram, starting from a minimal Hamiltonian that we then restrict to the dipolar–octupolar case. We then present a summary of experimental results obtained on two dipole–octupole pyrochlore materials, Ce₂Zr₂O₇ and Nd₂Zr₂O₇. These systems are growable as relatively large single crystals (39–41), thus making possible extensive experimental studies, which result in their placement on a common general phase diagram for dipole–octupole pyrochlores. These two compounds exhibit clearly different ground-state properties, with Ce₂Zr₂O₇ being an excellent QSI candidate system, whereas Nd₂Zr₂O₇ exhibits a remarkable noncollinear ordered state whose unconventional excitations reveal its dipole–octupole character.

## 2. CRYSTAL ELECTRIC FIELD EFFECTS IN RARE-EARTH PYROCHLORES

As is usual for 4f electrons, the spin–orbit coupling and CEF are the two largest energy scales in rare-earth pyrochlore insulators, apart from the Coulomb interactions that dictate the filling of atomic levels. Furthermore, there typically exists a clear energy hierarchy, with the spin–orbit coupling dominating over the effects of the CEF (5–8), and both of these subsequently dominate over weak exchange and other intersite interactions.

**Magnetic dipole–dipole interaction:** the long-ranged direct interaction between two magnetic dipole moments due to the magnetic fields created by the dipole moments

**Quantum spin liquid:** a disordered magnetic phase, ostensibly at zero temperature, in which the spins or pseudospins fluctuate in a quantum entangled manner

**Pseudospin-1/2:** pseudospin-1/2 operators are defined within a crystal electric field ground-state doublet and mimic typical spin-1/2 algebra

**Crystal electric field (CEF):** the electric field at a particular ion location within a crystalline material, which arises due to the surrounding charge distribution

**Spin–orbit coupling:** arises in ions because of the coupling between electron spin and the relative motion of the ion’s charged nucleus
To determine the magnetic state of the rare-earth ion, we first consider Hund's rules, which take into account the Pauli principle, the Coulombic repulsion, and the spin-orbit coupling. These rules predict the spin, orbital, and total angular momentum quantum numbers of the ground-state multiplet, *S*, *L*, and *J*, respectively. In pyrochlores, the spin-orbit coupling, which lifts the degeneracy between different multiplets with a given total angular momentum *J*, induces a separation on the order of 200 meV between the ground and first excited multiplets. The CEF can then be treated as a perturbation to the spin-orbit coupling. This leads to the so-called CEF states that are linear combinations of states with constant total angular momentum *J* and constant total angular moment projection *m*<sub>*J*</sub> (denoted as |*J*, *m*<sub>*J*</sub>〉).

Despite this energy hierarchy between the CEF and spin-orbit coupling, one can take into account the complexity of the electronic structure of the 4f shell. This goes beyond the usual approximation (that focuses on the ground multiplet only) to include a possible mixing between the ground and the first excited multiplets, which results in more complex wave functions (42–45).

Focusing on the low-energy crystal field states, a feature of most of the R<sub>2</sub>B<sub>2</sub>O<sub>7</sub> pyrochlores is the energy gap between the CEF ground state and the first excited state, which is typically ~10 to ~50 meV (21, 23, 40, 46–49). These rare-earth pyrochlores display CEF ground-state doublets, except for the case of the non-Kramers ion *R* = Tm<sup>3+</sup>, which possesses a nonmagnetic singlet ground state. In addition, the three-fold rotational symmetry of the D<sub>3d</sub> point-group at the R<sup>3+</sup> site restricts the composition of the wave functions making up the CEF states to those with *m*<sub>*J*</sub> values separated by Δ*m*<sub>*J*</sub> = 3.

We illustrate the CEF scheme for the R<sup>3+</sup> ions in the R<sub>2</sub>B<sub>2</sub>O<sub>7</sub> pyrochlores using Ce<sub>2</sub>Zr<sub>2</sub>O<sub>7</sub> as an example in **Figure 2a**. **Figure 2b** shows the inelastic neutron scattering spectra measured from Ce<sub>2</sub>Zr<sub>2</sub>O<sub>7</sub> and the low-lying CEF excitations in this spectra that were used to refine the CEF Hamiltonian (40). The first excited state lies ~56 meV above the CEF ground-state doublet. The level-scheme for the refined CEF Hamiltonian of Ce<sub>2</sub>Zr<sub>2</sub>O<sub>7</sub> is shown in **Figure 2c** (40), with the |*J* = 5/2, *m*<sub>*J*</sub> = ±3/2〉 doublet as the ground state.

<figure><img src="image_8.png" /><figcaption>Figure 2</figcaption></figure>

CEF states of the 4f¹ Ce³⁺ ions in Ce₂Zr₂O₇. (a) A schematic diagram of the energy hierarchy and level-splittings associated with spin-orbit coupling and the CEF for the single-ion energy levels of the Ce³⁺ ion. The leftmost collection of lines represents the degenerate ground states of the free ion. The center collection of lines represents the two spin-orbit coupling manifolds, with *J* = |*L* − *S*| = 5/2 in the ground-state manifold and with *J* = *L* + *S* = 7/2 in the excited-state manifold (*S* = 1/2, *L* = 3). The rightmost collection of lines represents the different eigenstates obtained accounting for the CEF. (b) Inelastic neutron scattering spectra measured from a powder sample of Ce₂Zr₂O₇ at *T* = 5 K in Reference 40. The arrows indicate excitations from the CEF ground-state doublet to the first and second excited CEF doublets at ~56 and 111 meV. These CEF excitations are easily identified by their lack of dispersion and the fact that their scattering intensity decreases with increasing |*Q*| as expected for the magnetic form factor.

(c) Lowest-lying CEF eigenstates for Ce³⁺ in Ce₂Zr₂O₇ (40). Panels *b* and *c* adapted from Reference 40 with permission; copyright 2019 American Physical Society. Abbreviation: CEF, crystal electric field.
In Nd-based pyrochlores, the situation is similar. The Nd³⁺ ion has three electrons in its 4f shell, and Hund's rules give L = 6, S = 3/2, and J = |L - S| = 9/2 in the ground-state multiplet. The first excited multiplet (J = 11/2) is located around ~200 meV higher in energy, which is much greater than the splitting due to the crystal field. The ground multiplet is split by the CEF into five doublets around 0, 23, 34, 35, and 107 meV (42, 43, 50, 51). The wave function of the ground-state doublet is a linear combination of m<sub>J</sub> = ±3/2 and ±9/2 states.

These properties have important consequences. First, the large gap allows the approximation that the CEF ground state is the only level occupied at low temperatures and makes it legitimate to use pseudospin-1/2 degrees of freedom, corresponding to the ground-state doublet (8, 44, 52–62). This pseudospin is defined locally because the natural quantization axis, along a (111) direction in the case of the pyrochlores, changes from site to site in the lattice. The pseudospin-1/2 components are related to the actual magnetic dipole moment through the following equation (8, 31):

$$ \hat{\mu}_i = -\mu_B[g_{xy}(\mathbf{x}_i \cdot \hat{S}_i^x + \mathbf{y}_i \cdot \hat{S}_i^y) + g_z \mathbf{z}_i \cdot \hat{S}_i^z], \quad 1. $$

where $\hat{\mu}_i$ is the magnetic dipole moment operator and $\mathbf{x}_i$, $\mathbf{y}_i$, and $\mathbf{z}_i$ are unit vectors along the x, y, and z directions in the local coordinate system for ion i (see Figure 1b). The anisotropic g factors, $g_z$ and $g_{xy}$, are given by $g_z = 2g_J \langle \pm | \hat{J}_z | \pm \rangle$ and $g_{xy} = g_J \langle \pm | \hat{J}_\pm | \mp \rangle$, where $\hat{J}_\pm = \hat{J}_x \pm i\hat{J}_y$, $g_J$ is the Landé g factor, and $|\pm\rangle$ denote the states in the CEF ground-state doublet (63). The anisotropic g factors contain information on the single-ion anisotropy of the dipole moment, with three scenarios arising: (a) When $g_z \gg g_{xy}$, each dipole moment is nearly parallel or antiparallel to the local z direction, and the anisotropy is referred to as Ising-like; (b) when $g_{xy} \gg g_z$, each dipole moment lies primarily in the local xy plane, and the anisotropy is referred to as XY-like; and (c) when $g_{xy} \sim g_z$, the magnetic moments display near-three-dimensional isotropy and are referred to as Heisenberg-like.

This being said, there are three possible symmetry classifications that arise in the rare-earth pyrochlores, based on how the pseudospin-1/2 operators $\hat{S}^x$, $\hat{S}^y$, and $\hat{S}^z$ transform under the $D_{3d}$ symmetry operations and time-reversal symmetry (8, 64–68). In each case, $\hat{S}^z$ transforms like a magnetic dipole, and the classifications are drawn according to the symmetry of $\hat{S}^x$ and $\hat{S}^y$. The non-Kramers ground-state doublet is relevant for magnetic ions with an even number of electrons, and for these doublets, $\hat{S}^x$ and $\hat{S}^y$ both transform like electric quadrupoles. Note that in this case, $g_{xy} = 0$. For Kramers ions, with an odd number of electrons, there are two possibilities: The first possibility is that $\hat{S}^x$ and $\hat{S}^y$ operators both transform like magnetic dipoles, and this is called an effective spin-1/2 doublet. The second possibility for Kramers ions is that $\hat{S}^x$ transforms like a magnetic dipole under the $R^{3+}$ site symmetries and time-reversal symmetry, and $\hat{S}^y$ transforms like a magnetic octupole. This type of CEF doublet is called a dipole–octupole doublet, and it is the focus of this article.

The different symmetries and multipolar magnetic moments for the pseudospin-1/2 components in the dipole–octupole pyrochlores are summarized in Table 1. All dipole–octupole pyrochlores have Ising single-ion anisotropy, with magnetic dipole moments pointing along the local z axes ($g_{xy} = 0$ in Equation 1). $\hat{S}^x$ and $\hat{S}^y$ correspond to octupolar magnetic moments, but with different symmetries. Indeed, $\hat{S}^y$ has dipolar symmetry but physically corresponds to an octupolar magnetic moment.

One should remember that measurements of magnetic susceptibility, muon spin rotation, and neutron scattering are mainly sensitive to the dipolar magnetic moment and, therefore, mainly sensitive to the z components of pseudospin. In the case of neutron scattering, two additional comments can be made: The neutron spin is, strictly speaking, sensitive to the distribution of magnetic field in a given sample. Therefore, the x and y components of the pseudospin must have a

Kramers ion: an ion with an odd number of electrons, which must have an even numbered degeneracy for each energy level
<table><caption>**Table 1** The symmetry and multipolar magnetic moment associated with $\hat{S}^x$, $\hat{S}^y$, and $\hat{S}^z$ for the dipole–octupole pyrochlores</caption><thead><tr><th>Component</th><th>Symmetry<sup>a</sup></th><th>Magnetic Moment<sup>b</sup></th></tr></thead><tbody><tr><td>$\hat{S}^x$</td><td>Dipolar<br>$\mathcal{T}: -\hat{S}^x$,<br>$C_3: \hat{S}^x, \mathcal{I}: \hat{S}^x, \mathcal{M}: -\hat{S}^x$</td><td>Octupolar<br>$\hat{\mu}_{Oct}^x \propto \mathcal{P}(\hat{J}_x^3 - \frac{1}{3}(\hat{J}_x\hat{J}_y^2 + \hat{J}_y\hat{J}_x\hat{J}_y + \hat{J}_y^2\hat{J}_x))\mathcal{P} = C_x\hat{S}^x$</td></tr><tr><td>$\hat{S}^y$</td><td>Octupolar<br>$\mathcal{T}: -\hat{S}^y$,<br>$C_3: \hat{S}^y, \mathcal{I}: \hat{S}^y, \mathcal{M}: \hat{S}^y$</td><td>Octupolar<br>$\hat{\mu}_{Oct}^y \propto \mathcal{P}(\hat{J}_y^3 - \frac{1}{3}(\hat{J}_y\hat{J}_x^2 + \hat{J}_x\hat{J}_y\hat{J}_x + \hat{J}_x^2\hat{J}_y))\mathcal{P} = C_y\hat{S}^y$</td></tr><tr><td>$\hat{S}^z$</td><td>Dipolar<br>$\mathcal{T}: -\hat{S}^z$,<br>$C_3: \hat{S}^z, \mathcal{I}: \hat{S}^z, \mathcal{M}: -\hat{S}^z$</td><td>Dipolar<br>$\hat{\mu}_{Dip}^z \propto \mathcal{P}\hat{J}_z\mathcal{P} = C_z\hat{S}^z$</td></tr></tbody></table>

<sup>a</sup>The time-reversal symmetry operation is denoted as $\mathcal{T}$ here. The symmetry elements corresponding to the $D_{3d}$ point group can be generated by the operations, $C_3$, $\mathcal{I}$, and $\mathcal{M}$, where $C_3$ is a three-fold rotation along the $z$ direction, $\mathcal{I}$ is the inversion operation, and $\mathcal{M}$ is the mirror plane that takes the $+x$ and $+y$ directions to the $-y$ and $-x$ directions, respectively (66).

<sup>b</sup>The operator $\mathcal{P} = |-\mathcal{X}| - |+\mathcal{X}| + |+\mathcal{Y}| + |-\mathcal{Y}|$ is the projection operator that projects into the basis of the crystal electric field ground-state doublet (68, 69), and $C_x$, $C_y$, and $C_z$ are constants with values that depend on the composition of the crystal electric field ground-state doublet (70–72).

neutron scattering signature, as they correspond to octupolar moments. Although weak, this signal is expected to exist at large wave vector $\mathbf{Q}$ (44, 73). However, weak scattering from magnetic octupole moments can be masked relatively easily by more-intense scattering from magnetic dipoles, even at high $\mathbf{Q}$ (44, 73). Additionally, the octupolar components can be seen indirectly by inelastic neutron scattering. Imagine, for example, that the pseudospin, in its ground state, is ordered and aligned in the local $x$ or $y$ direction. In a naive view, the low-lying excitations are spin waves, with a precession of the pseudospin around the $x$ or $y$ axis (see **Figure 3a**). This precessional motion

<figure><img src="image_9.png" /><figcaption>**Figure 3**</figcaption></figure>

(a) A sketch of a precession of the pseudospin (blue arrow) around the local $x$ axis, resulting in a finite and oscillating projection onto the $z$ axis (red arrow). (b) A hexagonal plaquette formed by six corner-sharing tetrahedra in the pyrochlore lattice, shown in a plane perpendicular to the [1, 1, 1] direction. This hexagon permits quantum fluctuations between different 2-plus–2-minus configurations for U(1) quantum spin ices. The positive and negative signs denote ions that are in the $|S^\alpha = +1/2\rangle$ and $|S^\alpha = -1/2\rangle$ states, respectively, in the $|S^\alpha = \pm 1/2\rangle$ basis of the pseudospin component associated with the 2-plus–2-minus rule. We show this for a collection of tetrahedra for which the pseudospin states alternate between $|S^\alpha = +1/2\rangle$ and $|S^\alpha = -1/2\rangle$ around the hexagon, permitting pseudospin-flipping by the ring-exchange Hamiltonian $H_{\text{ring}} = 2g_{\text{eff}} \sum_{\mathcal{O}} \cos(\nabla_{\mathcal{O}} \times \mathbf{A})$. Panel a adapted with permission from Reference 38; copyright 2016 American Physical Society.
gives an oscillating projection of pseudospin along the z axis, at the frequency of the wave, and this can be seen by inelastic neutron scattering.

### 3. THE LOW-ENERGY SPIN HAMILTONIAN

We now discuss the magnetic couplings within the ground-state doublet. The minimal Hamiltonian that can be written for the pyrochlore lattice involves nearest-neighbor, symmetry-allowed, bilinear couplings in the local x, y, and z components of the pseudospin (8, 52, 64, 74):

$$ \sum_{<ij>} \left[ J_{z} \hat{S}_{i}^{z} \hat{S}_{j}^{z} - J_{\pm} (\hat{S}_{i}^{+} \hat{S}_{j}^{-} + \hat{S}_{i}^{-} \hat{S}_{j}^{+}) + J_{\pm\pm} (\gamma_{ij} \hat{S}_{i}^{+} \hat{S}_{j}^{+} + \gamma_{ij}^{*} \hat{S}_{i}^{-} \hat{S}_{j}^{-}) \right. \\ \left. + J_{z\pm} \left[ \zeta_{ij} (\hat{S}_{i}^{z} \hat{S}_{j}^{+} + \hat{S}_{i}^{+} \hat{S}_{j}^{z}) + \zeta_{ij}^{*} (\hat{S}_{i}^{z} \hat{S}_{j}^{-} + \hat{S}_{i}^{-} \hat{S}_{j}^{z}) \right] \right], \qquad 2. $$

where $\hat{S}^{\pm} = \hat{S}^{x} \pm i\hat{S}^{y}$ and $\hat{S}_{i}^{\alpha}$ ($\alpha = x, y, z$) are the pseudospin-1/2 components of rare-earth ion i in the local $\{x, y, z\}$ coordinate frame. The parameters $J_{z}$, $J_{\pm}$, $J_{\pm\pm}$, and $J_{z\pm}$ depend on the details of the magnetic interactions at a microscopic level (8, 52, 54, 74) and are often estimated for a given material via fits to experimental data (52, 53, 60–62, 75–82).

The symmetry classifications of the CEF ground-state doublet (see Section 2) restrict $\gamma_{ij}$ and $\zeta_{ij}$. All terms are present for effective spin-1/2 pyrochlores, with $\gamma_{ij} = -\zeta_{ij}^{*}$; the values of $\zeta_{ij}$ for each distinct bond are given in References 52 and 74. This Hamiltonian is appropriate to Yb-based (52, 76, 80) or Er-based (53, 75, 77, 83–86) pyrochlores, for example. For the case of a non-Kramers CEF ground-state doublet, $J_{z\pm} = 0$ (31, 64). This Hamiltonian is appropriate to Tb$^{3+}$-based (87, 88) or Pr$^{3+}$-based (34, 38, 78) pyrochlores, for example. In the case of a dipole–octupole CEF ground-state doublet, $\gamma_{ij} = \zeta_{ij} = 1$, and the appropriate Hamiltonian can be rewritten as (8, 66)

$$ \mathcal{H}_{\text{DO}} = \sum_{<ij>} [J_{x} \hat{S}_{i}^{x} \hat{S}_{j}^{x} + J_{y} \hat{S}_{i}^{y} \hat{S}_{j}^{y} + J_{z} \hat{S}_{i}^{z} \hat{S}_{j}^{z} + J_{xz} (\hat{S}_{i}^{x} \hat{S}_{j}^{z} + \hat{S}_{i}^{z} \hat{S}_{j}^{x})], \qquad 3. $$

which applies to Ce$^{3+}$-based (41, 44, 60–62, 81, 82), Nd$^{3+}$-based (43, 79, 89, 90), and Sm$^{3+}$-based pyrochlores (91, 92).

It is worth noting that Ho$^{3+}$ and Dy$^{3+}$ ions, known to stabilize classical spin ice, are non-Kramers with $J = 8$ and dipolar–octupolar with $J = 15/2$, respectively, and their ground-state doublets are predominately made up of $|m_J = \pm J\rangle$ (23). This results in couplings between the transverse $\hat{S}^x$ and $\hat{S}^y$ components that are very small (54), with effective couplings $J_{\pm}$ and $J_{\pm\pm}$ for Ho$^{3+}$ and $J_x$, $J_y$, and $J_{xz}$ for Dy$^{3+}$ that are negligible compared to $J_z$. In addition, due to the large dipole moments in these classical spin ices, it is necessary to go beyond the nearest-neighbor couplings to include long-range dipole–dipole interactions to describe the experiments (18, 25, 93).

Another form of the dipole–octupole Hamiltonian (Equation 3) can be arrived at by applying a rotation of local axes that removes the $J_{xz}$ coupling term (66, 67). This requires a rotation of each local frame about its y axis by the angle $\theta$ (57, 66):

$$ \theta = \frac{1}{2} \tan^{-1} \left( \frac{2J_{xz}}{J_{x} - J_{z}} \right). \qquad 4. $$

This rotation yields new local coordinate frames that are commonly denoted as the $\{\tilde{x}, \tilde{y}, \tilde{z}\}$ coordinate frames. The new Hamiltonian in the $\{\tilde{x}, \tilde{y}, \tilde{z}\}$ coordinate frames is referred to as the XYZ
Hamiltonian and, in the presence of an external magnetic field **h**, is given by (66, 67)

$$ \mathcal{H}_{XYZ} = \sum_{<ij>} [J_{\vec{x}} \hat{S}_{i}^{\vec{x}} \hat{S}_{j}^{\vec{x}} + J_{\vec{y}} \hat{S}_{i}^{\vec{y}} \hat{S}_{j}^{\vec{y}} + J_{\vec{z}} \hat{S}_{i}^{\vec{z}} \hat{S}_{j}^{\vec{z}}] - g_{z} \mu_{B} \sum_{i} (\mathbf{h} \cdot \hat{\mathbf{z}}_{i}) (\hat{S}_{i}^{\vec{z}} \cos \theta + \hat{S}_{i}^{\vec{y}} \sin \theta). \quad 5. $$

Note that $\hat{\mathbf{z}}_i$ is the local z axis for ion i (see **Figure 1b**). ($\hat{S}^{\vec{x}}$, $\hat{S}^{\vec{y}}$, $\hat{S}^{\vec{z}}$) are related to the pseudospin-1/2 components in $\mathcal{H}_{\text{DO}}$ through the transformations given in Reference 57 (for example), and ($J_{\vec{x}}$, $J_{\vec{y}}$, $J_{\vec{z}}$) are related to the nearest-neighbor coupling constants in $\mathcal{H}_{\text{DO}}$ through

$$ J_{\vec{x}} = \frac{1}{2} (J_{x} + J_{z} + \sqrt{4J_{xz}^{2} + (J_{x} - J_{z})^{2}}), \quad J_{\vec{y}} = J_{y}, \quad J_{\vec{z}} = \frac{1}{2} (J_{x} + J_{z} - \sqrt{4J_{xz}^{2} + (J_{x} - J_{z})^{2}}), \quad 6. $$

corresponding to $J_x = J_{\vec{x}} \cos^2 \theta + J_{\vec{z}} \sin^2 \theta$, $J_z = J_{\vec{x}} \sin^2 \theta + J_{\vec{z}} \cos^2 \theta$, and $J_{xz} = \sin(2\theta)(J_{\vec{x}} - J_{\vec{z}})/2$. Note that the z direction is associated with the dipole magnetic moment, and the x and y directions with octupole moments, with different symmetry. When $\theta$ is nonzero, $\hat{S}^{\vec{x}}$ and $\hat{S}^{\vec{z}}$ are each associated with both an octupole and a dipole moment, but the dipolar symmetry of each is preserved. The different symmetries and multipolar magnetic moments for the pseudospin-1/2 components in the dipole-octupole pyrochlores are summarized in **Table 1** and illustrated in **Figure 4a,b** using the local {x, y, z} coordinate systems.

<figure><img src="image_10.png" /><figcaption>Figure 4</figcaption></figure>

(a) A representation of the symmetry of the dipolar and octupolar pseudospin components in dipole-octupole pyrochlores using their associated magnetic charge distributions. For pseudospins in the yz plane, a pseudospin along the $\pm z$ ($\pm y$) direction has pure dipolar (octupolar) symmetry, and pseudospins that are not directly along $\pm y$ or $\pm z$ have mixed dipolar and octupolar symmetry. (b) The magnetic charge distributions for an assortment of $R^{3+}$ dipoles (right) and octupoles (left) on the pyrochlore lattice. (c) The magnetic ground-state phase diagram for the XYZ Hamiltonian (Equation 5), along with illustrations of the different ground-state phases using magnetic charge distributions to represent the symmetry of the pseudospins (94). Panel c also shows the exchange parameters estimated for Ce$_2$Zr$_2$O$_7$ and Nd$_2$Zr$_2$O$_7$ in References 57 and 60–62. Panels a and b adapted with permission from Reference 81 (CC BY 4.0). Abbreviation: QSI, quantum spin ice.
4. THEORY OF DIPOLE–OCTUPOLE PYROCHLORES

The magnetic ground-state phase diagram for the XYZ Hamiltonian (Equation 5) in zero applied magnetic field (Figure 4c) is known to contain at least six distinct phases, two of which are ordered phases and four of which are U(1) QSI phases (66, 70, 71, 94–96). The QSI states occur when the dominant exchange parameter is positive (ferromagnetic) and much stronger than the largest neg- ative exchange parameter. The ordered ground states occur when a negative (antiferromagnetic) exchange parameter exists that is not strongly dominated by some positive exchange parame- ter. Classically, the boundary between these disordered and ordered ground states corresponds to a positive exchange parameter that is three times the largest negative exchange parameter, in absolute value (70, 97).

For the QSI phases (ordered phases), the ground states with $J_{\bar{x}}$ and $J_{\bar{z}}$ as the largest (most-
negative) exchange parameters are smoothly connected to one another by the parameter $\theta$, so
they do not represent distinct phases from the point of view of symmetry. Accordingly, we do not
distinguish between the $J_{\bar{x}}$-dominated and $J_{\bar{z}}$-dominated spin ice phases, and we do not distinguish
between the ordered phase with $J_{\bar{x}}$ as the most-negative exchange parameter and the ordered phase
with $J_{\bar{z}}$ as the most-negative exchange parameter.

The two ordered phases are distinguishable by the dipolar or octupolar symmetry of their order
parameter, and the four QSI phases are distinguishable by the dipolar or octupolar symmetry of
their emergent electric field as well as by the emergent flux through a hexagonal plaquette of
the pyrochlore lattice, which can be 0 [denoted as U(1)$_0$ QSI] or $\pi$ [denoted as U(1)$_\pi$ QSI] as
discussed below. Following the above discussion, a phase with dipolar symmetry can thus be made
of octupolar moments associated with $\hat{S}^v$, which is going to affect its experimental signatures.

A QSI phase (ordered phase) has octupolar symmetry if $|J_{\bar{y}}| > |J_{\bar{x}}|$, $|J_{\bar{z}}|$ (if $J_{\bar{y}} < J_{\bar{x}}, J_{\bar{z}}$) and has dipolar symmetry if $|J_{\bar{x}}| > |J_{\bar{y}}|$ or $|J_{\bar{z}}| > |J_{\bar{y}}|$ (if $J_{\bar{x}} < J_{\bar{y}}$ or $J_{\bar{z}} < J_{\bar{y}}$). A dipolar (octupolar) spin ice has $\hat{S}^{\bar{x}}$ or $\hat{S}^{\bar{z}}$ (has $\hat{S}^{\bar{y}}$) associated with the emergent electric field and governed by a 2-plus-2-minus rule (66, 70, 94, 95), analogous to the conventional 2-in-2-out rule (Figure 1c) for $\hat{S}^z$ in classical dipolar spin ices. Specifically, the 2-plus-2-minus rule for $\hat{S}^{\alpha}$ ($\alpha = \bar{x}, \bar{y},$ or $\bar{z}$) dictates that, in the ground state, there are two ions on each tetrahedron in $|S^{\alpha} = +1/2\rangle$ states and two in $|S^{\alpha} = -1/2\rangle$ states, in the $|S^{\alpha} = \pm 1/2\rangle$ basis. We refer to these U(1) quantum spin liquids as QSI phases due to the similarity between the 2-plus-2-minus rule and the 2-in-2-out rule. In the special case of a spin ice with $\theta = 0$ and $J_{\bar{z}}$ as the dominant exchange parameter, the 2-plus-2-minus rule is the conventional 2-in-2-out rule for magnetic dipole moments.

A dipolar (octupolar) ordered phase has $\hat{S}^{\bar{x}}$ or $\hat{S}^{\bar{z}}$ (has $\hat{S}^{\bar{y}}$) associated with the order parameter and exhibits an all-plus-all-minus order (70, 94, 95). This ordered state is conventional AIAO antiferromagnetic order (Figure 1c) for the case in which $\hat{S}^{\bar{z}}$ is associated with the order parameter. We refer to these ordered phases as AIAO phases due to the similarity between all-plus-all-minus order and conventional AIAO order.

The magnetic ground-state phase diagram for the XYZ Hamiltonian, as determined from Reference 94, is shown in **Figure 4c**. The estimated locations of Ce<sub>2</sub>Zr<sub>2</sub>O<sub>7</sub> and Nd<sub>2</sub>Zr<sub>2</sub>O<sub>7</sub> in this ground-state phase diagram are shown according to the values of the nearest-neighbor exchange parameters estimated in References 57, 60–62, and we discuss these estimates further in Sections 5.1.3 and 6.2. **Figure 4c** also shows representations of the ground-state phases using magnetic charge distributions to illustrate the dipolar and octupolar symmetries of the pseudospin components. Of course, the representation using magnetic charge distributions on a single tetrahedron offers no distinction between 0-flux and $\pi$-flux QSIs of the same symmetry. The distinction between the 0-flux and $\pi$-flux QSI phases arises through details of quantum fluctuations on hexagonal plaquettes in the pyrochlore lattice, as we now discuss.
If the system is in a QSI phase and the dominant exchange parameter is $J_α$ ($α = \tilde{x}, \tilde{y}, \tilde{z}$), then the XYZ Hamiltonian in zero field can be separated into a classical spin ice term $\mathcal{H}_{\text{CSI}} = \sum_{<ij>} J_α \hat{S}_i^\alpha \hat{S}_j^\alpha$, with ground states that correspond to the degenerate, classical manifold of 2-plus-2-minus states, and a quantum term $\mathcal{H}_{\beta\gamma} = \sum_{<ij>} (J_\beta \hat{S}_i^\beta \hat{S}_j^\beta + J_\gamma \hat{S}_i^\gamma \hat{S}_j^\gamma)$, where $\beta, \gamma = \tilde{x}, \tilde{y}, \tilde{z}$ with $\beta \neq \gamma$ and $\beta, \gamma \neq \alpha$. The U(1) gauge theory for QSIs is described on the diamond lattice formed by the centers of tetrahedra in the pyrochlore lattice (6, 29-33). The QED mapping takes $\hat{S}_j^\alpha$ into the emergent electric field E via $\hat{S}_j^\alpha = \pm E_{rr'}$, where r and r' denote the diamond lattice sites nearest to the rare-earth ion j. The raising and lowering operators, $\hat{S}_j^\pm = \hat{S}_j^\beta \pm i\hat{S}_j^\gamma$, are taken to the magnetic potential A via $\hat{S}_j^\pm = e^{\pm i\phi_{rr'}}$, where $\phi_{rr'} = \pm A_{rr'}$. Here, the plus and minus signs for $E_{rr'}$ and $A_{rr'}$ vary depending on the orientation of the two tetrahedra surrounding the diamond lattice sites r and r' (further details in 6 and 29, for example).

Degenerate perturbation theory can be applied on the quantum term $\mathcal{H}_{\beta\gamma}$ to yield an effective Hamiltonian that acts within the ground-state manifold of 2-plus-2-minus states (29, 31, 32, 52, 61, 96, 99). The leading order term from this degenerate perturbation theory is a ring-exchange Hamiltonian, $\mathcal{H}_{\text{ring}} = g_{\text{eff}} \sum_{\circ} (\hat{S}_1^{\dagger} \hat{S}_2^{-} \hat{S}_3^{\dagger} \hat{S}_4^{-} \hat{S}_5^{\dagger} \hat{S}_6^{-} + \hat{S}_6^{\dagger} \hat{S}_5^{-} \hat{S}_4^{\dagger} \hat{S}_3^{-} \hat{S}_2^{\dagger} \hat{S}_1^{-})$, which flips the six pseudospins on hexagons, as sketched in **Figure 3b**. Here, the sum is over the different hexagonal plaquettes in the pyrochlore lattice with the index $j = \{1, 2, 3, 4, 5, 6\}$ denoting the six ions forming a hexagon; the labeling is assigned by going around the hexagon in an arbitrary direction while labeling spins consecutively. Note that pseudospin-flipping by the ring-exchange Hamiltonian is only permissible on a hexagon if the pseudospin states of the ions on that hexagon show an alternation between $|S^\alpha = +1/2\rangle$ and $|S^\alpha = -1/2\rangle$ around the hexagon, in the $|S^\alpha = \pm 1/2\rangle$ basis. The ring-exchange Hamiltonian can be rewritten as $\mathcal{H}_{\text{ring}} = 2g_{\text{eff}} \sum_{\circ} \cos(\nabla_{\circ} \times \mathbf{A})$, where $\nabla_{\circ} \times \mathbf{A} = \sum_{rr' \in \circ} A_{rr'}$ is the lattice curl of $A_{rr'}$ around a hexagon and corresponds to the emergent magnetic flux through a hexagonal loop in the pyrochlore lattice (31, 32, 61, 66, 67, 100). The sum over $rr' \in \circ$ is taken over all neighboring pairs of diamond lattice sites r and r' at the tetrahedral centers of the six tetrahedra forming the hexagon (29).

For $g_{\text{eff}} < 0$ ($g_{\text{eff}} > 0$), $\mathcal{H}_{\text{ring}}$ is minimized when $\nabla_{\circ} \times \mathbf{A} = 0$ ($\pi$) for each flippable hexagon, giving the distinction between 0- and $\pi$-flux QSIs. Evaluation of $\mathcal{H}_{\text{ring}}$ to the lowest nonvanishing and nonconstant order of perturbation theory results in the approximation, $g_{\text{eff}} = -12J_{\pm}^3/J_{\alpha}^2$, where $J_{\pm} = -\frac{1}{4}(J_{\beta} + J_{\gamma})$ (32, 52). This approximation can be improved using a variational extension to cluster mean-field theory (94), which leads to the 0- and $\pi$-flux phase boundary shown in **Figure 4c**. Further details on the 0- and $\pi$-flux boundary and calculations used to investigate it can be found in References 29, 66, 70, 72, 94, and 96, for example.

Spinon excitations composed of 3-in-1-out and 1-in-3-out pairs of tetrahedra correspond to emergent electric monopoles (charges) in the electrodynamics description and have an energy gap on the order of $J_α$. At lower energies are the vison (emergent magnetic monopole) excitations, which are topological defects associated with dissatisfaction of the ring Hamiltonian $\mathcal{H}_{\text{ring}}$, with an energy gap that scales approximately as $J_{\pm}^3/J_{\alpha}^2$. The lowest-energy excitations are gapless photons, arising from coherent fluctuations in the 2-in-2-out manifold that can be described as electric field fluctuations.

## 5. EXPERIMENTAL SIGNATURES OF QUANTUM SPIN ICES IN DIPOLE-OCTUPOLE PYROCHLORES

In this section, we focus our discussion on Ce$_2$Zr$_2$O$_7$, for which all evidence points to a QSI ground state. We then briefly discuss experimental results on other Ce-based pyrochlores.
## 5.1. Main Experimental Features in Ce₂Zr₂O₇

We begin this subsection on Ce₂Zr₂O₇ by discussing the air-sensitivity of Ce₂Zr₂O₇ samples and highlighting sample preparation techniques used to combat this effect. We then outline some key measurements of Ce₂Zr₂O₇'s magnetic behavior and summarize previous successes in fitting experimental data from Ce₂Zr₂O₇ to estimate the magnetic interaction parameters for Ce₂Zr₂O₇ and the corresponding magnetic ground state. We end this subsection with a discussion that further connects measurements on Ce₂Zr₂O₇ to relevant theoretical work.

### 5.1.1. Sample preparation and stoichiometry issues with Ce₂Zr₂O₇

Ce₂Zr₂O₇ has attracted much experimental attention, as it was relatively straightforward to grow as large single crystals. However, at least in powder form, it was found to be sensitive to oxidation, as occurs with simple exposure to the ambient environment. As a result, annealing procedures in a reducing environment, such as hydrogen, were important to avoid unwanted nonmagnetic-Ce⁴⁺ content and related oxygen defects (40, 60, 62, 101). This instability is manifest by the powder X-ray diffraction data of Reference 40, showing a progressive shift of Bragg peak positions to higher angle with time in air (102). Furthermore, Ce₂Zr₂O₇ samples change in color from light greenish-yellow to dark gray in air, indicating the importance of proper storage procedures. This effect seems to be specific, or stronger, in Ce₂Zr₂O₇ compared to other Ce-based pyrochlores, as Ce₂Zr₂O₇'s sister pyrochlore Ce₂Hf₂O₇ shows no evidence for oxidation on exposure to ambient environment.

### 5.1.2. Advanced characterization of Ce₂Zr₂O₇

The complete body of experimental work on Ce₂Zr₂O₇ in zero magnetic field (40, 41, 60, 62, 73, 101, 103) is consistent with the absence of both long-ranged order and spin freezing above ~0.02 K. Those data, combined with the snowflake-like pattern of diffuse neutron scattering measured from single crystal Ce₂Zr₂O₇ in zero field at low temperature (40, 41, 60, 73), are consistent with a low-temperature quantum spin liquid state, and these form the basis for the QSI candidacy of Ce₂Zr₂O₇.

Two of these experimental results on Ce₂Zr₂O₇ are shown in **Figure 5a,b**. In **Figure 5a**, we show zero-field µSR measurements on Ce₂Zr₂O₇ (41, 101), where the absence of oscillations in

<figure><img src="image_11.png" /><figcaption>**Figure 5**</figcaption></figure>

<footnote>(a) The muon polarization measured from single crystal Ce₂Zr₂O₇ (black) (101) and from powder Ce₂Zr₂O₇ (maroon) (41) in zero-field at *T* = 0.02 K. (b) The magnetic contribution to the heat capacity (*C*<sub>mag</sub>) measured from Ce₂Zr₂O₇ in zero magnetic field. The line in panel *b* shows the calculated *C*<sub>mag</sub> according to seventh-order NLC calculations using nearest-neighbor exchange parameters estimated for Ce₂Zr₂O₇ by fitting the measured *C*<sub>mag</sub> and magnetic susceptibility (60). (c) The measured *C*<sub>mag</sub> from Ce₂Zr₂O₇ in zero-field, along with an interpolation that connects the NLC calculation in panel *b* at high temperature (T ≳ 0.5 K) to a cubic *T*³ form of *C*<sub>mag</sub> at low temperature (60). (d) The entropy recovered in zero-field via *S*<sub>mag</sub> = ∫<sub>0</sub><sup>*T*</sup> (C<sub>mag</sub>/T) d*T* using the interpolation of *C*<sub>mag</sub> in panel *c* below the lowest-temperature data point (60). Panel *a* adapted with permission from Reference 101; copyright 2023 American Physical Society. Panels *b–d* adapted with permission from Reference 60 (CC BY 4.0). Abbreviation: NLC, numerical linked cluster.</footnote>
<figure><img src="image_12.png" /><figcaption>Figure 6</figcaption></figure>

The spin-flip (a) and non-spin-flip (b) diffuse neutron diffraction signals measured from Ce₂Zr₂O₇ at *T* = 0.045 K (60), and the spin-flip (c) and non-spin-flip (d) diffuse neutron diffraction signals measured from Ho₂Ti₂O₇ at *T* = 1.7 K (104). The gray lines in panel *b* show the face-centered cubic Brillouin zone boundaries. The Ce₂Zr₂O₇ data in panels *a* and *b* each include the subtraction of a dataset taken at *T* = 10 K. Figure adapted with permission from Reference 60 (CC BY 4.0) and Reference 104 with permission from AAAS.

the muon polarization as a function of time is consistent with a lack of static magnetic fields and, thus, a lack of both magnetic order and spin freezing of magnetic dipoles down to the lowest measured temperature of *T* = 0.02 K (41, 101). These independent results, on two different single crystal samples of Ce₂Zr₂O₇, are qualitatively similar, but the Gao et al. measurements (41) show stronger relaxation of the muon signal at low temperatures suggesting an additional source of relaxation, likely originating from small levels of defects.

The magnetic contribution to the heat capacity ($C_{mag}$) of Ce₂Zr₂O₇ in zero field is shown in **Figure 5b** and **Figure 5c** (60, 62) and is consistent with that reported in a separate study on Ce₂Zr₂O₇ (41), showing no signs of a phase transition, involving either magnetic dipoles or octupoles, above *T* = 0.058 K. Instead, a broad, Schottky-like hump is observed, which indicates the onset of spin ice correlations.

The onset of the broad hump in the $C_{mag}$ of Ce₂Zr₂O₇ coincides with the appearance and growth of inelastic magnetic spectral weight below ~0.2 meV in the neutron scattering signal with decreasing temperature (60). This signal falls within the approximate energy range from *E* = 0 to 0.15 meV (40, 60). Polarized neutron diffraction measurements, integrating in energy over this inelastic spectral weight, were carried out, and **Figure 6** shows the snowflake-like patterns of diffuse scattering measured in the spin-flip channel of such measurements for both Ce₂Zr₂O₇ (**Figure 6a**) and Ho₂Ti₂O₇ (**Figure 6c**). The polarized spin-flip diffraction from Ho₂Ti₂O₇, with so-called pinch point diffuse scattering evident along (0, 0, *L*) and (*H*, *H*, *H*), established Ho₂Ti₂O₇ as having a classical spin ice state at low temperature (104, 105). Pinch points are expected to be suppressed in QSI phases (32, 96). The non-spin-flip diffuse scattering in both Ce₂Zr₂O₇ (**Figure 6b**) and Ho₂Ti₂O₇ (**Figure 6d**) shows similar zone-boundary scattering; Brillouin zone boundaries are drawn in the reciprocal space of **Figure 6b**. We further discuss the origins of this diffuse scattering in Section 5.1.4.

5.1.3. **Estimations of the spin Hamiltonian for Ce₂Zr₂O₇.** The ground state of any material that lacks long-range order is best established by estimating the terms in its microscopic spin Hamiltonian and attempting to locate it on a general phase diagram, such as **Figure 4c**. The preferred method by which microscopic exchange parameters are estimated in the rare-earth pyrochlores is by fitting measured spin-wave spectra obtained in inelastic neutron scattering experiments (see 52, 53, 75, 76, 80 for examples). This, of course, requires that sharp spin-wave
dispersions can be measured, which is only possible in zero magnetic field if the ground state possesses long-range order, but may require the application of a magnetic field in cases in which the zero-field magnetic ground state is disordered and shows no well-defined spin waves. In the case of Ce₂Zr₂O₇, no well-defined spin waves have yet been detected in either zero or nonzero magnetic field (62). Although this may be telling in and of itself in such a dipole-octupole system, it does mean that an alternate protocol must be followed using the experimental data that are available. As discussed in the previous section and reported in References 60–62, for Ce₂Zr₂O₇, that includes low-temperature heat capacity, magnetic susceptibility, and magnetization, as well as diffuse neutron scattering and its temperature dependence.

A variety of theoretical methods can be used to calculate physical properties of dipole-octupole pyrochlores in different regions of the XYZ phase diagram. In broad terms, exact quantum Monte Carlo techniques can be successfully applied, without a sign problem, in limited regions of the phase diagram only. However, the sign problem is at play in part of the ordered region and much of the quantum disordered region of the phase diagram. In further detail, letting $J_α$ ($α = \tilde{x}, \tilde{y}, \tilde{z}$) be the largest exchange parameter in absolute value with $J_β$ and $J_γ$ ($β, γ = \tilde{x}, \tilde{y}, \tilde{z}, β ≠ γ$ and $β, γ ≠ α$) as the two nondominant exchange parameters, the XYZ Hamiltonian (Equation 5) in zero field suffers from the sign problem when $J_β + J_γ > 0$ (30, 66, 94). This means that the U(1)$_π$ QSI regime is almost entirely free from the sign problem, whereas significant portions of the ordered regime and the full U(1)$_π$ QSI regime each experience the sign problem (94). For that reason, quantum Monte Carlo methods cannot fairly be used to explore how well physical measurements may be fit to parameters across the full XYZ phase diagram.

This sign problem of the quantum Monte Carlo method prompts the use of other approaches. Numerical linked cluster (NLC) techniques and the finite temperature Lanczos (FTL) method were used to fit the magnetic heat capacity, the magnetic susceptibility, and the magnetization of Ce₂Zr₂O₇ (60–62) and, thereby, to estimate the values of ($J_{\tilde{x}}$, $J_{\tilde{y}}$, $J_{\tilde{z}}$). These are fully quantum mechanical treatments and can be applied to determine exchange parameters in any part of the XYZ phase diagram equally well. The actual calculations are performed on finite-sized systems, and so they are accurate only above a certain cutoff temperature at which the relevant correlation lengths are less than the system size. Thus, these theoretical descriptions are expected to be good at these high temperatures and become less accurate at lower temperatures, due to the finite size of the calculation, and to the fact that weaker terms in the Hamiltonian, not included in the nearest-neighbor XYZ Hamiltonian, may become relevant at low temperatures. **Figure 5b** shows a comparison of the $C_{mag}$ data from Ce₂Zr₂O₇ and seventh-order NLC calculations for $C_{mag}$ using the best-fit nearest-neighbor parameters from Reference 60. The fit above $T ∼ 0.5$ K is excellent, but deviations are observed, as expected, at lower temperatures, where weak interactions beyond the nearest-neighbor, XYZ model-Hamiltonian begin to become more relevant. The seventh-order NLC calculation is converged with the sixth-order NLC calculation over the full temperature range that the calculation is shown for in **Figure 5b**, suggesting that the reduced goodness-of-fit below $T ∼ 0.5$ K is not due to finite-size effects.

This protocol has now been followed three times in total for Ce₂Zr₂O₇, in zero and nonzero magnetic field on two different single crystal samples, by two independent groups. The nearest-neighbor exchange parameters estimated from References 60–62 agree well, as can be seen from their placement on the phase diagram for the XYZ Hamiltonian in **Figure 4c**. Specifically, Reference 60 uses the NLC method and estimates that ($J_{\tilde{x}}$, $J_{\tilde{y}}$, $J_{\tilde{z}}$) = (0.064, 0.063, 0.11) meV or (0.063, 0.064, 0.11) meV; Reference 62 also uses the NLC method and estimates that ($J_{\tilde{x}}$, $J_{\tilde{y}}$, $J_{\tilde{z}}$) = (0.063, 0.062, 0.11) meV or (0.062, 0.063, 0.11) meV; and Reference 61 uses the FTL method and estimates that ($J_{\tilde{x}}$, $J_{\tilde{y}}$, $J_{\tilde{z}}$) = (0.05 ± 0.02, 0.08 ± 0.01, 0.02 ± 0.01) meV. Each of the three parameter sets shows Ce₂Zr₂O₇ to lie well within the U(1)$_π$ QSI ground-state region of the phase
diagram. The exchange parameters suggested in References 60 and 62, with $J_{\vec{x}} \approx J_{\vec{y}}$, reside almost exactly on the boundary between QSIs with octupolar and dipolar symmetry of the emergent electric field, whereas those suggested in Reference 61 with $J_{\vec{y}} > J_{\vec{x}}$, suggest an octupolar QSI ground state also near the boundary to the $\pi$-flux dipolar QSI. Both sets of analyses yield $J_{\vec{x}}, J_{\vec{y}} > J_z$ with $\theta \approx 0$, and the corresponding $\pi$-flux QSI ground state is expected to obey local 2-plus-2-minus spin ice rules associated with the configurations of magnetic octupole moments ($\hat{S}^{\text{z}}$ or $\hat{S}^{\text{y}}$ in terms of pseudospin).

Importantly, the lack of sharp spin waves in inelastic neutron scattering measurements on Ce₂Zr₂O₇ is consistent with expectations based on the exchange parameters suggested in References 60–62. This absence of sharp spin waves is associated with the partially octupolar character of the Ce³⁺ pseudospins and is borne out in detailed calculations using estimates for the nearest-neighbor exchange parameters of Ce₂Zr₂O₇ in Reference 62.

In this pseudospin description, the dipole-dipole interactions are absorbed into the exchange parameters describing the interactions between $\hat{S}^z$ components of pseudospins. A reasonable upper limit on the size of the nearest-neighbor dipole-dipole interaction in Ce₂Zr₂O₇ at low temperature is given by taking the CEF moment size of 1.29 $\mu_B$, which is the moment size arising when pseudospins point along the z direction. With this, the part of $J_z$ arising from the dipole-dipole interaction is at most 5.4 μeV. For the case of Ce₂Zr₂O₇, the dipole-dipole interaction and the parameter $J_z$ are much smaller than $J_x$ and $J_y$, which are each ~60 μeV (60–62) and carry no contribution from the dipole-dipole interaction. Similarly, the nearest-neighbor exchange parameters for Nd₂Zr₂O₇ (discussed in Section 6) have energy scale on the order of 100 μeV (57, 79, 89, 98), which is ~4 times greater than the upper limit for the contribution to $J_z$ from nearest-neighbor dipole-dipole interactions. This is in contrast to the classical spin ices Ho₂Ti₂O₇ and Dy₂Ti₂O₇, for example, where the dipole-dipole interactions are responsible for the majority of the magnetic interaction between nearest neighbors (25).

### 5.1.4. Discussion.
As discussed in Reference 40, the diffuse snowflake-like signal from unpolarized neutron scattering measurements on Ce₂Zr₂O₇ was compared with predictions from Reference 32 for emergent photon excitations in a QSI. However, owing to the estimated exchange parameters outlined in the previous section, the emergent photon excitations in Ce₂Zr₂O₇ are not expected to give a significant neutron scattering signal within the *Q* range of these measurements (73). This is because neutrons have a very weak coupling to magnetic octupoles and the corresponding $\hat{S}^z$ and $\hat{S}^y$ components of the pseudospins. Furthermore, the magnetic form factor for octupoles peaks at high *Q* and decreases toward zero as *Q* approaches zero (73, 106). For these reasons, the dominant inelastic neutron scattering signal at low *Q* arises from the magnetic dipole moments associated with the $\hat{S}^z$ components of the pseudospins. According to the exchange parameters estimated for Ce₂Zr₂O₇ (60–62), this means that spinon excitations should dominate the signal and generate the snowflake-like pattern of diffuse scattering (60, 61). In Table 2, we outline the excitations arising from the $\hat{S}^z$ pseudospin components for different regions of phase space of the XYZ Hamiltonian (Equation 5) for zero field. These are the excitations expected at low *Q* in inelastic neutron scattering measurements.

Although the comparison of polarized diffuse neutron scattering from Ce₂Zr₂O₇ and Ho₂Ti₂O₇ (Figure 6) is a natural comparison to make, it should be noted that the diffuse scattering reported for Ce₂Zr₂O₇ (40, 41, 60) is inelastic and thought to arise from spinons (as outlined in the previous paragraph), whereas the diffuse scattering from the classical spin ice phase in Ho₂Ti₂O₇ (104, 107) arises from static correlations within the ground-state manifold. Interestingly, the diffuse scattering from other QSI candidates, such as Pr₂Zr₂O₇ (34, 38, 58) and Pr₂Hf₂O₇ (78), also displays snowflake-like patterns of diffuse scattering in the (*H*, *H*, *L*) scattering plane similar to Ce₂Zr₂O₇.
<table><caption>**Table 2** The excitations probed at low Q by inelastic neutron scattering for various ground states of the XYZ Hamiltonian<sup>a,b</sup></caption><thead><tr><th>Phase</th><th>Dominant parameter</th><th>θ = 0</th><th>0 &lt; θ &lt; π/2</th></tr></thead><tbody><tr><td>Dipolar QSI</td><td>J<sub>x</sub></td><td>Spinons</td><td>Photons, spinons, magnetic monopoles</td></tr><tr><td>Octupolar QSI</td><td>J<sub>y</sub></td><td>Spinons</td><td>Spinons</td></tr><tr><td>Dipolar QSI</td><td>J<sub>z</sub></td><td>Photons, magnetic monopoles</td><td>Photons, spinons, magnetic monopoles</td></tr></tbody></table>

<sup>a</sup>These expectations arise through the fact that the emergent electric field of a QSI is associated with the component of pseudospin with the dominant exchange parameter (referred to as $\hat{S}^\alpha$ in Section 4), and due to the fact that at low Q, neutrons only couple significantly to the $\hat{S}^z$ components of pseudospin. Photons and magnetic monopoles are generated through the $\hat{S}^\alpha$ components of pseudospin, and spinons are generated from the components of pseudospin transverse to $\hat{S}^\alpha$ (referred to as $\hat{S}^\beta$ and $\hat{S}^Y$ in Section 4).

<sup>b</sup>We omit a full presentation of the excitations for the ordered phases of the XYZ Hamiltonian but discuss here which regions of phase space are expected to show well-defined spin waves (single magnon modes) in inelastic neutron scattering at low Q. In the case in which $\theta = 0$, the single magnon modes are probed by inelastic neutron scattering at low Q for the AIAO phases with dominant $J_x$ or $J_y$, as the most-negative exchange parameter, but not for those with $J_z$ as the most-negative exchange parameter. In addition, field-polarized phases are not expected to show single magnon modes in inelastic neutron scattering at low Q when $\theta = 0$. For $0 < \theta < \pi/2$, the single magnon modes in all AIAO phases and in field-polarized phases are probed by inelastic neutron scattering at low Q.

Abbreviations: AIAO, all-in-all-out; QSI, quantum spin ice.

Note that the zone-boundary diffuse neutron scattering observed in the non-spin-flip channel of polarized neutron diffraction measurements on Ce<sub>2</sub>Zr<sub>2</sub>O<sub>7</sub> (Figure 6b) is not a prediction of the third-order, $T = 0.5$ K NLC calculations in Reference 60, whereas calculations using an extension to gauge mean-field theory (72, 108), with better access to low temperatures and larger system sizes, do indeed predict zone-boundary scattering using the nearest-neighbor XYZ Hamiltonian. Similar scattering in the classical spin ice Ho<sub>2</sub>Ti<sub>2</sub>O<sub>7</sub> (Figure 6d) was interpreted in terms of further-than-nearest-neighbor interactions (104, 109). The analysis of Ce<sub>2</sub>Zr<sub>2</sub>O<sub>7</sub> by Bhardwaj et al. (61) used the self-consistent Gaussian approximation and semiclassical molecular dynamics calculations to show that a weak second-nearest-neighbor interaction between magnetic dipole moments is able to account for finer details in the measured unpolarized neutron scattering signals from References 41 and 40. However, Hosoi et al. (96) have shown, using exact diagonalization, that these finer features also appear at the nearest-neighbor level.

The full form of $C_{\text{mag}}$ is inaccessible to experiment due to the relatively low energy scale of the exchange parameters and the minimum temperature at which practical measurements can be made, ~0.058 K in the case of the single-crystal Ce<sub>2</sub>Zr<sub>2</sub>O<sub>7</sub> measurements performed to date. Although $C_{\text{mag}}$ can be theoretically modeled using NLC techniques at intermediate and high temperatures, these techniques cannot be applied at the lowest temperatures. At these low temperatures, quantum Monte Carlo calculations, where possible within the 0-flux QSI regime, predict a broad Schottky-like hump in $C_{\text{mag}}$ at “high temperature,” corresponding to the entrance into the classical spin ice manifold, followed by a sharper peak at lower temperatures, which corresponds to the entrance into the true QSI regime (95, 99, 110). Below the sharper peak, a $T^3$ law is expected, accounting for the gapless emergent photon excitations associated with U(1) QSIs (67, 95). However, signatures of a gapped Z<sub>2</sub> quantum spin liquid state have also been found in quantum Monte Carlo calculations using parameters near the boundary between the ordered regime and 0-flux QSI regime—specifically, a $C_{\text{mag}}$ that drops exponentially to zero below the low-temperature peak (95). This picture requires three temperature regimes, with an intermediate Pauling entropy plateau at $\frac{R}{2} \ln(\frac{1}{2})$ [which is the “missing” part of the recovered entropy in the classical spin ices Ho<sub>2</sub>Ti<sub>2</sub>O<sub>7</sub> and Dy<sub>2</sub>Ti<sub>2</sub>O<sub>7</sub> (17, 111)] before entrance into the quantum regime at lower temperature. In
contrast to this, Desrochers & Kim (108) use a self-consistent exclusive boson representation (an extension to gauge mean-field theory) to show that the broad hump in $C_{\text{mag}}$ and the sharp peak in $C_{\text{mag}}$ have temperature scales that are on the same order of magnitude over a significant portion of the $\pi$-flux QSI regime relevant for Ce$_2$Zr$_2$O$_7$, which can lead to these two features in $C_{\text{mag}}$ combining with one another to result in a lack of entropy plateau. In any case [gapless U(1) QSI or gapped $Z_2$ quantum spin liquid], the full $R\ln(2)$ entropy, expected for the CEF ground-state doublet, would be recovered.

Experimental work on Ce$_2$Zr$_2$O$_7$ (60) showed that a pure $T^3$ form does not connect smoothly to the $C_{\text{mag}}$ data at the lowest measured temperature. Instead, these authors employed Padé-approximant methods to interpolate between a high-temperature regime (where $C_{\text{mag}}$ can be modeled with NLC down to $T \gtrsim 0.5$ K), and a low-temperature $T^3$ regime, as shown in **Figure 5c**. With this extrapolation method, the $R\ln(2)$ entropy is accounted for within 5%, as shown in **Figure 5c,d**. The lack of a “Pauling entropy” plateau may be puzzling at first, but it is generally consistent with the large quantum fluctuations in the $\pi$-flux QSI regime (108) implied by estimates of the exchange parameters for Ce$_2$Zr$_2$O$_7$, which have $|J_{\vec{x}}| \sim |J_{\vec{y}}| \gg |J_{\vec{z}}|$ (60–62).

## 5.2. The Cases of Ce$_2$Sn$_2$O$_7$ and Ce$_2$Hf$_2$O$_7$

A second example of a cerium-based dipole-octupole pyrochlore is Ce$_2$Sn$_2$O$_7$, which was the first cerium-based dipole-octupole pyrochlore to be discovered (44, 112). Low-temperature measurements on powder samples of Ce$_2$Sn$_2$O$_7$ grown through solid-state synthesis techniques show no signs for either magnetic order or spin freezing to a base temperature of 0.02 K (44, 81, 112, 113). Interestingly, a temperature-dependent neutron diffraction signal was reported at high Q, which was interpreted as evidence of octupolar QSI correlations (44). However, recent measurements on powder and single crystal samples of Ce$_2$Sn$_2$O$_7$ grown via hydrothermal methods also show a lack of both magnetic order and spin freezing down to the lowest-measured temperature but with a temperature-dependent neutron diffraction signal that is observed at low Q only; that is, it is absent at high Q (81). Fits of NLC theory and quantum Monte Carlo to Ce$_2$Sn$_2$O$_7$'s $C_{\text{mag}}$, magnetic susceptibility, and diffuse neutron diffraction permit an estimate of its spin Hamiltonian. This study places Ce$_2$Sn$_2$O$_7$'s ground state in a dipolar AIAO ordered phase, relatively close to a dipolar QSI phase. In contrast to the case of Ce$_2$Zr$_2$O$_7$, a discrepancy then exists between these works on Ce$_2$Sn$_2$O$_7$ samples, grown by different techniques, which remains unresolved to date (44, 81).

A third cerium-based dipole-octupole pyrochlore, Ce$_2$Hf$_2$O$_7$, has also been the subject of recent experimental investigations (45, 82). Some structural disorder was reported (45), but initial results available to date show similarity to those on Ce$_2$Zr$_2$O$_7$ and Ce$_2$Sn$_2$O$_7$, with no obvious magnetic ordering or spin freezing observed. A quantum spin liquid ground state in Ce$_2$Hf$_2$O$_7$ was suggested from fits to the magnetization and heat capacity (82), with these fits suggesting that the dominant exchange parameter is either $J_{\vec{x}}$ or $J_{\vec{y}}$.

# 6. DIPOLE-OCTUPOLE ANTIFERROMAGNETS

There are an additional two families of dipole-octupole pyrochlores where, in contrast to Ce$_2$Zr$_2$O$_7$, the ground state is not located in one of the QSI regions of the general phase diagram shown in **Figure 4c** but in the ordered region with dipolar symmetry. These are the Sm$^{3+}$-based pyrochlore insulators with $J = 5/2$, such as Sm$_2$Ti$_2$O$_7$, and the Nd$^{3+}$-based pyrochlore insulators with $J = 9/2$, such as Nd$_2$Zr$_2$O$_7$. Both Nd$_2$Zr$_2$O$_7$ and Sm$_2$Ti$_2$O$_7$ indeed exhibit phase transitions to AIAO ordered states with the appearance of neutron Bragg peaks at low temperatures (42, 51, 92). The study of Sm$_2$Ti$_2$O$_7$, and all Sm-based materials, using neutron scattering techniques is
difficult owing to both the very high absorption cross section of neutrons for Sm and its small magnetic moment. For that reason, we do not consider this family of Sm-based pyrochlores further in this review. However, this is not the case for Nd, where Nd₂Zr₂O₇ and related pyrochlores have been extensively studied using neutron scattering techniques—indeed these studies predated those discussed earlier in this review on Ce₂Zr₂O₇.

## 6.1. Long-Range All-In-All-Out Dipole Order in Nd-Based Pyrochlores

A large variety of Nd-based pyrochlores have been studied. This includes Nd₂Sn₂O₇ (114), Nd₂Zr₂O₇ (42, 50, 51), Nd₂Hf₂O₇ (115), and the disordered systems Nd₂ScNbO₇ (116, 117) and Nd₂GaSbO₇ (118). A change in the lattice parameter, from ~10.34 Å in Nd₂GaSbO₇ to ~10.68 Å in Nd₂Zr₂O₇, is observed (118, 119), and crystallographic studies show little variation of the other structural parameters along the series. The existence of a low-temperature long-ranged ordered phase in Nd-based pyrochlores was clear from early bulk characterization of these materials, including specific heat (120, 121), magnetization, and AC susceptibility measurements on Nd₂Zr₂O₇ (42, 51), Nd₂Hf₂O₇ (115), and Nd₂Sn₂O₇ (114). The critical temperatures range from ~0.3 K for Nd₂Zr₂O₇ to ~1.1 K for Nd₂GaSbO₇ (51, 118).

Neutron diffraction measurements confirmed the antiferromagnetic nature of the low-temperature phase and its $k = 0$ AIAO magnetic structure in all Nd-based pyrochlores samples (42, 51, 114–118). The AIAO magnetic structure corresponds to the $\Gamma_3$ irreducible representation and has all dipole moments pointing into or out of a given tetrahedron (analogous to the spins in Figure 1c). It is easily identified in neutron diffraction measurements, with magnetic intensity appearing at the (2, 2, 0) and (1, 1, 3) Bragg positions, in relative proportions that leave no doubt as to the result. $\Gamma_3$ is the sole configuration that also leads to zero magnetic intensity for both the (1, 1, 1) and (0, 0, 2) Bragg peaks, as observed in measurements. The ordered magnetic moment shows a surprisingly large range for the Nd-based pyrochlores, from ~1.7 $\mu_B$ in Nd₂Sn₂O₇ down to ~0.6 $\mu_B$ in Nd₂Hf₂O₇ (114, 115).

This observed ordered dipole moment is reduced compared to the 2.5 $\mu_B$ associated with the CEF ground-state doublet and pseudospins along their local $z$ directions. It was originally interpreted in Reference 89 as the consequence of magnetic fragmentation (122). However, it was subsequently shown by Benton (57) that this reduction is related to the $J_{xz}$ coupling term in the Hamiltonian of Equation 3, or equivalently, the mixing angle $\theta$ in the XYZ Hamiltonian (Equations 4 and 5). Consequently, the ordered component of each pseudospin points within the local $(x, z)$ plane and is rotated by the angle $\theta$ with respect to the natural local $z$ axes (see Section 3). As the physical magnetic moment is proportional to the projection of the pseudospin along the $z$ axis, the latter appears reduced, by a factor of $\cos\theta$ in dipole–octupole pyrochlores. Taking into account the reduction resulting from quantum fluctuations, the magnetic moment size satisfies $\mu < (g_z/2)\cos\theta$ with Reference 57 estimating $\mu = 0.87(g_z/2)\cos\theta$ for Nd₂Zr₂O₇. In addition to this magnetic dipole moment, the ordered component of pseudospin in Nd₂Zr₂O₇ has a component along the $x$ axis that corresponds to an ordered octupolar component within its unconventional AIAO ground state. This ordered octupolar component remains to be observed experimentally.

## 6.2. The Spin Hamiltonian for Nd₂Zr₂O₇

Extensive inelastic neutron scattering measurements were carried out on Nd₂Zr₂O₇ and Nd₂Hf₂O₇ to examine the low-energy spin dynamics associated with the AIAO ordered state in these dipole–octupole pyrochlore systems. We focus on the results obtained for Nd₂Zr₂O₇, which are the most comprehensive of the two, but we note that Nd₂Hf₂O₇ exhibits very similar features

Magnetic fragmentation: when the magnetic moment has divergence-full and divergence-free components with the behavior of each component able to be described separately
<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /><figcaption>Figure 7</figcaption></figure>

(a,b) Constant energy slices of neutron scattering in the (H, H, L) plane, where each panel is divided into an upper part featuring the experimental data from Nd₂Zr₂O₇ and a lower part showing spin-wave calculations. (a) The spectral weight at E = 0.05 meV of the flat mode at E = 0.06 meV, which is that of the diffuse scattering in spin ice. (b) Dispersing branches that develop above the energy of the flat band. The bottom of the dispersive band touches the flat band at pinch-point positions such as (1, 1, 1) and (0, 0, 2), for instance, with spectral weight forming arcs or half-moons. (c) Sketch of the classical phase diagram for dipole–octupole pyrochlores materials, for J⃗z < 0 and assuming J⃗f = 0. The positions of several Nd-based compounds are indicated by colored rectangles (123). For powder samples, J⃗f/J⃗z was estimated combining the energy gap of the flat mode and the Curie–Weiss temperature, whereas for single crystals, this ratio was obtained by a direct analysis of the spin-wave dispersions. Panels a and b adapted with permission from Reference 89. Panel c adapted with permission from Reference 123; copyright 2024 American Physical Society. Abbreviation: AIAO, all-in–all-out.

(125). These results contrast strongly with the inelastic scattering from Ce₂Zr₂O₇, with its QSI ground state, which shows no sharp, dispersive spin excitations at any wave vector, both in zero magnetic field and with a magnetic field applied along the [1, ¯1, 0] or [0, 0, 1] directions (60, 62).

The spectrum of spin excitations out of the AIAO state in Nd₂Zr₂O₇ presents remarkable features. As revealed by several sets of inelastic neutron scattering measurements, the lowest-energy excitation is a flat, nondispersive mode with a gap of Δ ~ 0.06 meV above the elastic scattering of the AIAO ordered state (79, 89) (Figure 7a). This gapped, inelastic signal from Nd₂Zr₂O₇ exhibits diffuse, arm-like features along the (0, 0, L) and (H, H, H) directions, which strongly resemble the diffuse magnetic scattering seen in classical spin ice (Figure 6c). In addition, above this flat mode, dispersive collective magnetic excitations stem from the pinch points at the (0, 0, 2) and (1, 1, 1) wave vector positions. These dispersive excitations have a bandwidth of about 0.2 meV and reach their maximum energy at the (2, 2, 0) and (1, 1, 3) wave vectors (Figure 7b). Remarkably, all of this complex inelastic and elastic scattering can be fit and understood in terms of a linear spin-wave theory (as reported in 57, 89). This traditional analysis of using spin-wave measurements and theory to estimate microscopic terms in the spin Hamiltonian gives rise to the following estimates for the coupling constants (57): J⃗x = 0.103 meV, J⃗f = 0, J⃗z = -0.047 meV, and θ = 0.83 rad. These parameters were further refined (79, 90, 98), and all studies give a positive J⃗x and a negative J⃗z with J⃗x/J⃗z |~ 2. Calculations using the so-obtained Hamiltonian can reproduce the magnetic ground state with its ordered moment, the magnetic excitations, and also a previously puzzling feature: the Curie–Weiss temperature, which is positive (and thus indicative of effective ferromagnetic interactions) despite the observed AIAO order. Indeed, the Curie–Weiss
temperature ($\Theta_{CW}$) for dipole-octupole pyrochlores has contributions from both $J_{\tilde{z}}$ and $J_{\tilde{z}}$, given as $\Theta_{CW} = \frac{1}{2k_B}(J_{\tilde{z}} \cos^2 \theta + J_{\tilde{x}} \sin^2 \theta)$ (57).

These nearest-neighbor coupling parameters estimated for Nd$_2$Zr$_2$O$_7$ allow us to also locate this dipole-octupole pyrochlore within the phase diagram appropriate for the XYZ Hamiltonian (Equation 5). These are shown in Figure 4c and clearly place Nd$_2$Zr$_2$O$_7$ to lie in the region corresponding to AIAO order with dipolar symmetry.

Nevertheless, as stated above, $J_{\tilde{z}}$ and $J_{\tilde{x}}$ have different signs and, thus, compete. $J_{\tilde{z}} < 0$ favors an AIAO phase with $\hat{S}_{\tilde{z}}$ associated with the order parameter, and $J_{\tilde{x}} > 0$ favors a U(1) QSI with $\hat{S}_{\tilde{x}}$ associated with the emergent electric field. On cooling, the latter is the origin of an elastic signal with a spin ice structure factor observed below about 1 K and stabilized by entropy effects. When further decreasing the temperature, a gap to these spin ice correlations opens and the AIAO state becomes more favorable. This scenario is captured by classical Monte Carlo calculations (90, 124). Note that scattering from both the $\tilde{x}$ and the $\tilde{z}$ components of the pseudospin-1/2 can be measured in a neutron experiment because each possesses a finite projection on the magnetic dipole moment (along $z$) due to the nonzero $\theta$ angle.

## 6.3. Dynamic Fragmentation

The measured spectral weight of the dispersive part of the spin excitation spectrum is also unconventional, as was clear from the constant energy Q-maps of the inelastic neutron scattering on Nd$_2$Zr$_2$O$_7$ shown in Figure 7 (89). For the more conventional case of a cone of spin-wave excitations dispersing out of Bragg positions, such constant energy maps would show rings of scattering around the magnetic zone centers (the Bragg positions). The radii of these rings would increase with energy, and their spectral weight would be uniformly distributed about these rings. In contrast for Nd$_2$Zr$_2$O$_7$, intense spectral weight is present along two symmetrical arcs that face each other on either side of the pinch points. The higher-energy arcs are bright where the spectral weight of the flat mode at $\Delta \sim 0.06$ meV is smallest (79, 89, 125).

This complex inelastic scattering is a consequence of a specific property of the model, known as dynamic fragmentation (57). As described by Benton (57) and following from Brooks-Bartlett et al. (122), the equations of motion describing the dynamics of the $\tilde{x}$ and $\tilde{y}$ components of the pseudospin decompose into two independent fragments, described as divergence-full and divergence-free. On one hand, the divergence-free fragment corresponds to the dispersionless inelastic spectral weight, with a spin ice pattern and pinch points (Figure 7a) resulting from the zero divergence condition (126). Within spin-wave theory, the associated energy gap of $\Delta \sim 0.06$ meV in Nd$_2$Zr$_2$O$_7$ depends on the XYZ Hamiltonian parameters as (57)

$$ \Delta = \sqrt{(3|J_{\tilde{z}}| - J_{\tilde{x}})(3|J_{\tilde{z}}| - J_{\tilde{y}})}. \quad 7. $$

On the other hand, the divergence-full fragment is responsible for the dispersive excitations above the flat spin ice-like band, which, in the $(H, H, L)$ plane, appear as the arcs of inelastic diffuse scattering observed in the experiment above $\sim 0.1$ meV (Figure 7b). As mentioned above for the static components, the existence of a nonzero $\theta$ angle means that both divergence-full and divergence-free fragments of the dynamical magnetization can be observed in neutron scattering experiments. Note that in this language, the ground state, i.e., the AIAO order of the $\tilde{z}$ component, is entirely divergence-full, as the magnetic structure corresponds to sources and sinks of magnetic flux on alternate tetrahedra.

As shown in Reference 57, the transition between the AIAO ground state and a U(1) QSI state corresponds to the gap $\Delta$ going to zero energy. The relative proximity of Nd$_2$Zr$_2$O$_7$'s AIAO phase to a QSI ground state is given by the smallness of the gap, which closes at a critical value of
$J_{\bar{x}}/|J_{\bar{z}}| = 3$ for $J_{\bar{z}} < 0 < J_{\bar{x}}$ (relevant to $Nd_2Zr_2O_7$) in this semiclassical approximation, whereas the experimental estimate is $J_{\bar{x}}/|J_{\bar{z}}| \sim 2$ for $Nd_2Zr_2O_7$ (57, 79, 90, 98).

## 6.4. The Influence of Quenched Disorder on the Dipole-Octupole All-In-All-Out States

The influence of quenched disorder, as is always present at some level in real materials, is generally important in condensed matter physics. For both dipole-octupole pyrochlores with QSI ground states, such as $Ce_2Zr_2O_7$, and those with AIAO ground states, such as $Nd_2Zr_2O_7$, it is already experimentally clear that disorder is present in the materials studied. As discussed in Section 5.1.1, powder $Ce_2Zr_2O_7$ samples show oxidation via exposure to ambient atmosphere in a matter of hours, whereas $Nd_2Zr_2O_7$ shows variation in the lattice parameter between crystals and powders, which has been attributed to nonmagnetic Zr substitutions on the Nd site (119).

Members of this Nd-based pyrochlore family appear to display some sample variation in the details of their magnetic ordering, which is usually ascribed to uncontrolled disorder for certain samples. For $Nd_2Zr_2O_7$, for example, a peak is observed in the real and imaginary parts of its susceptibility as a function of temperature at a temperature that is greater in powder samples (about 0.4 K) than in single-crystal samples (0.285 K) (51). This discrepancy was recently shown to be related to the presence of disorder in single crystals (119). This is not so unusual as single crystals are grown from the melt, whereas powders are typically synthesized by solid-state synthesis at lower temperatures. Yet for both powder and single-crystal samples, AIAO ordering and dynamic fragmentation are realized, albeit with different $T_N$ and with different ordered magnetic moments along (111).

Quenched disorder in Nd-based dipole-octupole pyrochlores was first addressed using compounds in which the *B* site of the $R_2B_2O_7$ pyrochlore lattice is occupied by two different cations whose oxidation states average to 4+, $Nd_2ScNbO_7$ (116, 117) and $Nd_2GaSbO_7$ (118). A related strategy is to synthesize Nd-based pyrochlores with several different *B*-site cations, as was done for the high-entropy system $Nd_2Ti_{0.4}Zr_{0.4}Hf_{0.4}Sc_{0.4}Nb_{0.4}O_7$ (NdMOX) (123, 127). These strategies induce local variations in the Nd crystal field due to the composition disorder and distributions of the *B* cation charges (or oxidation states 3+ and 5+). Site-substitution disorder has also been introduced into Nd-based pyrochlores, including nonmagnetic substitution on the Nd site in $(Nd_{1-y}La_y)_2Zr_2O_7$ ($y$ = 10, 20, and 40%), resulting in a dilution on the Nd pyrochlore lattice (123) and substitution on the nonmagnetic site in $Nd_2(Zr_{1-x}Ti_x)_2O_7$ for $x$ = 2.5 and 10% (128). Generally, such quenched disorder away from the Nd site is expected to affect the superexchange paths, or magnetic bonds in the system, as well as the CEF scheme of the $Nd^{3+}$ ions.

Crystallographic studies of these disordered Nd-based pyrochlores show that, except for the lattice parameters, the chemical structure is not significantly perturbed by any of these forms of quenched disorder. The measured CEF spectra and associated schemes are similar to those in pure $Nd_2Zr_2O_7$, albeit broadened somewhat. These Nd CEF states appear to be less affected by disorder than $Tb^{3+}$ in solid solutions of $Tb_2Ti_{2-x}Sn_xO_7$ (129). This is probably due to the fact that $Nd^{3+}$ is a Kramers ion; hence, the doublet nature of its CEF states is symmetry protected, whereas $Tb^{3+}$ is a non-Kramers ion. In any case, the CEF eigenfunctions and, hence, the dipole-octupole nature of the Nd pseudospin-1/2 degree of freedom appear to persist in the presence of quenched disorder in these Nd pyrochlores.

In all the disordered Nd pyrochlores, except those with a magnetically dilute Nd sublattice, the AIAO order is enhanced by means of a larger ordering temperature. Concomitantly, the ratio $J_{\bar{x}}/|J_{\bar{z}}|$ decreases, this effect being apparently related to the lattice parameter decrease. In contrast, in compounds for which the Nd sublattice is diluted by nonmagnetic ions, $T_N$ decreases, as expected for magnetic dilution, and the associated $J_{\bar{x}}/|J_{\bar{z}}|$ ratio is greater than in the
pure compounds, pushing these compounds closer to the boundary with a QSI ground state (see **Figure 7c**).

# 7. DIPOLE-OCTUPOLE PYROCHLORES IN MAGNETIC FIELD

The phase diagram of the dipole-octupole pyrochlores in the presence of an applied magnetic field is rich, with a multitude of field-induced phases (70, 97). Here, we focus on recent results on Ce₂Zr₂O₇ and Nd₂Zr₂O₇. The magnetic field dependencies of Ce₂Zr₂O₇ and Nd₂Zr₂O₇ at low temperature are expected to be quite different as the zero field low-temperature state in Nd₂Zr₂O₇ is an ordered AIAO state, and the field-induced states are obtained through metamagnetic-like transitions. With a QSI disordered ground state in Ce₂Zr₂O₇, we can expect a smooth deformation of the ground state by the magnetic field.

For Nd₂Zr₂O₇, the metamagnetic transitions at low temperature are characterized by an in-flection point in the magnetization as a function of field strength and, likewise, by both a peak in its derivative dM/db versus b, as we discuss below, and an abrupt change of the magnetic structure as seen in neutron diffraction (128). In contrast, in Ce₂Zr₂O₇ the field-induced states are reached smoothly (62). For a [0, 0, 1] field, the ground state is expected to evolve toward an ordered spin ice state (130), with the 2-in-2-out ice rule satisfied by the magnetic moments on each tetrahedron and a net magnetization along the field direction. This is indeed what is observed in both Ce₂Zr₂O₇ and Nd₂Zr₂O₇ (62, 103, 128) and in the classical spin ices Ho₂Ti₂O₇ and Dy₂Ti₂O₇ (131, 132). The results for a field applied along the [1, 1, 0] and [1, 1, 1] directions are where the richer physics emerges.

The application of a magnetic field along the [1, 1, 0] direction is particularly interesting in the pyrochlores, as the magnetic R³⁺ lattice can be separated into chains along the [1, 1, 0] magnetic field direction (called α chains) and chains perpendicular to the [1, 1, 0] field direction (called β chains), with the β chains lacking significant coupling to both the magnetic field and the α chains (97, 133). At saturation, similar to the behavior of classical spin ices Ho₂Ti₂O₇ and Dy₂Ti₂O₇ (16, 107, 131, 132, 134), the dipole-octupole pyrochlores Ce₂Zr₂O₇ and Nd₂Zr₂O₇ (62, 128, 135) display a partially polarized spin ice structure with the α chains showing a field-induced polarization that is noncollinear due to the Ising single-ion anisotropy, whereas the β chains exhibit noncollinear, short-ranged ferromagnetic order (**Figure 8a**). The short-ranged ferromagnetic order of the β chains is consistent with estimates of the pseudospin interaction parameters in each case (62, 135), and together with the polarization of the α chains, these measurements establish that the dipole moments in each of these materials obey the 2-in-2-out ice rule in a [1, 1, 0] field. For Ce₂Zr₂O₇ and Nd₂Zr₂O₇, no inter-β chain correlations were detected (62, 135), in contrast to Ho₂Ti₂O₇ and Dy₂Ti₂O₇ where antiferromagnetic interchain correlations were observed (16, 107, 131, 132, 134) and attributed to dipole-dipole interactions (133).

The elastic neutron scattering signal measured from Ce₂Zr₂O₇ in the (H, H, L) scattering plane at T = 0.03 K is shown for zero magnetic field in **Figure 8b** and for an *b* = 1.5 T magnetic field along the [1, 1, 0] direction in **Figure 8c**. These datasets show the field-induced appearance of magnetic Bragg scattering at Q = (0, 0, 2), which arises from the polarization of the α chains (62). The field-strength and temperature dependence of magnetic Bragg scattering at Q = (0, 0, 2) is shown in **Figure 8d**. The lines in this figure show semiclassical and quantum calculations for the field-dependence of the magnetic Bragg intensity at T = 0.3 K (62), with the quantum calculation fitting much better than its semiclassical counterpart, appearing to emphasize the quantum nature of the α chains.

The [1, 1, 0]-field-induced distinction of α and β chains can also be seen in the magnetic heat capacity (62, 133, 136), with the zero-field hump in Ce₂Zr₂O₇'s heat capacity being separated by a [1, 1, 0] field into two separate features. In further detail, the magnetic heat capacity in a [1, 1, 0]
<figure><img src="image_17.png" /><figcaption>Figure 8</figcaption></figure>

(a) The partially polarized spin ice structure of dipole moments relevant to Ce₂Zr₂O₇ and Nd₂Zr₂O₇ for magnetic fields along the [1, ¯1, 0] direction (62, 128, 135). The α chains (red) along the [1, ¯1, 0] magnetic field direction are noncollinearly polarized, whereas the β chains (blue) perpendicular to the field direction show short-ranged ferromagnetic order consistent with local adherence to the 2-in-2-out spin ice rule for dipoles. (b,c) The elastic neutron scattering in the (H, H, L) plane measured from Ce₂Zr₂O₇ at T = 0.03 K in zero magnetic field and in an *h* = 1.5 T magnetic field along [1, ¯1, 0], respectively. (d) The magnetic field strength and temperature dependence of the **Q** = (0, 0, 2) magnetic Bragg peak of Ce₂Zr₂O₇ for a field along [1, ¯1, 0]. The lines show one-dimensional quantum calculations (solid green line) and semiclassical molecular dynamics calculations (dotted red line) using estimates for the exchange parameters for Ce₂Zr₂O₇ from Reference 62. (e) The heat capacity (C<sub>P</sub>) of Ce₂Zr₂O₇ in a [1, ¯1, 0] magnetic field and of La₂Zr₂O₇ in zero magnetic field (62). The La₂Zr₂O₇ data provide an estimate of the phonon contribution to the heat capacity for Ce₂Zr₂O₇. The lines show the temperature dependence of C<sub>mag</sub> according to NLC calculations using the exchange parameters estimated for Ce₂Zr₂O₇ obtained by fitting the C<sub>mag</sub> data shown in this figure (62). Figure adapted with permission from Reference 62; copyright 2023 American Physical Society.

field shows a Schottky anomaly whose peak center increases with field strength (attributed to the field-polarized α chains) and a low-temperature hump that appears to remain at the same temperature with increasing field strength (attributed to the field-decoupled β chains). This is shown for the case of Ce₂Zr₂O₇ in **Figure 8e** (62), with an analogous splitting of the heat capacity also reported for Dy₂Ti₂O₇ (136). Fits to the magnetic heat capacity measured from Ce₂Zr₂O₇ in a [1, ¯1, 0] field (see **Figure 8e**) yield an anisotropic *g* factor of *g*<sub>*z*</sub> = 2.24, which is reduced from the value of 2.57 that is expected for a |*m*<sub>*J*</sub> = ±3/2〉 doublet and is similar to the various values estimated in References 61 and 101.

Interestingly, the estimated exchange parameters for Ce₂Zr₂O₇ from References 60 and 62, with *J*<sub>*x*</sub> ≈ *J*<sub>*y*</sub>, imply that the β-chain excitations are nearly gapless. More generally, the β chains in Ce₂Zr₂O₇ and Nd₂Zr₂O₇ in a [1, ¯1, 0] magnetic field are intriguing owing to the fact that they possess the rare combination of quasi-one-dimensionality, field decoupling, and a high degree of quantum fluctuations (62, 135).
<figure><img src="image_18.png" /><figcaption>Figure 9</figcaption></figure>

(a) dM/db of Nd₂Zr₂O₇ at T = 0.09 K when ramping the field from negative values for a field applied along [0, 0, 1], [1, 1̄, 0], and [1, 1, 1]. (b) The field dependence of the magnetic moments of the apical and kagome spins in Nd₂Zr₂O₇ for a field applied along [1, 1, 1], obtained from neutron diffraction measurements at T = 0.06 K (98). The measured magnetic moments show a hysteresis similar to the magnetization measurements in panel a. The critical field b_c is marked with an arrow and “DKI” and “DSI” correspond to the regions in which the dynamic kagome ice and dynamic spin ice behaviors are observed in the excitations. The bottom part of panel b shows the observed ground-state configurations of magnetic dipoles in Nd₂Zr₂O₇ and the corresponding field-regimes, for a field applied along [1, 1, 1] when sweeping the field from negative to positive values: 3-out–1-in, all-out, and eventually 3-in–1-out.
(c) The T = 0.06 K inelastic diffuse scattering in the (−H − K, −H + K, 2H) plane measured from Nd₂Zr₂O₇ for E = 0.05 meV with an b = 0.5 T field applied along [1, 1, 1], showing the star pattern expected in a kagome ice phase. Panels b and c are adapted with permission from Reference 98.

In Nd₂Zr₂O₇, when applied along different high-symmetry directions ([1, 1, 0], [0, 0, 1], and [1, 1, 1] directions), the magnetic field induces metamagnetic-like transitions, identified in magnetization measurements as a peak in dM/db just below 0.1 T (Figure 9a). These magnetization data are all taken sweeping the magnetic field strength from relatively strong negative values, thus preparing the Nd₂Zr₂O₇ sample in a single domain, and moving to higher and ultimately positive field strengths. The metamagnetic-like transitions are identified for positive magnetic fields while ramping the field up. The data are asymmetric as a function of b, which indicates metastable states due to the existence of domains in the AIAO structure.

This hysteresis, also observed in neutron diffraction experiments, is especially pronounced when the magnetic field is applied along the [1, 1, 1] direction. Along this direction, it is convenient to view the cubic pyrochlore lattice as a stacking of edge-sharing-triangular lattice planes and kagome lattice planes (see **Figure 3b**). Each tetrahedron is then comprised of one rare-earth ion at the apex of the tetrahedra whose magnetic moment is easily polarized by a [1, 1, 1] magnetic field, and three rare-earth ions within a kagome plane. Due to the dipole–octupole nature of the ions, the value of the dipolar magnetic moment along the local *z* direction is not constant as a function of field, as shown in **Figure 9b**. For this reason, the transitions are less sharp than in conventional Ising systems, and the apical and kagome dipolar magnetic moments do not necessarily have the same value. When sweeping the field from negative to positive values, for a given tetrahedron, the magnetic structure goes from 3-out–1-in to all-out, and eventually to 3-in–1-out at *b*<sub>c</sub> (98, 137). The hysteresis is reproduced qualitatively by Monte Carlo and mean-field calculations (79, 128).

No signature of the field-induced transition is observed in the excitation spectrum of Nd₂Zr₂O₇ when the magnetic field is applied along the [1, 1, 1] direction. The flat mode observed in zero field shown in **Figure 7a** persists up to 1 T, but its intensity modulation as a function of Q is strongly modified when increasing the magnetic field. When the apex spin is nearly saturated,

**Kagome lattice:**
a two-dimensional
lattice formed by
corner-sharing
triangles, prone to
magnetic frustration
typically for $h > 0.25$ T (i.e., well above the critical field $h_c \sim 0.1$ T), the spectral weight of the flat mode changes into a star pattern in the $(-H - K, -H + K, 2H)$ plane (Figure 9c). This pattern is reminiscent of the diffuse scattering signal observed in the kagome ice phase (138–140) stabilized in moderate [1, 1, 1] field in classical spin ice, where it is characteristic of the local kagome ice rules obeyed by the spins in the kagome planes, with one spin pointing into the triangles making up the kagome net and two pointing out, or vice versa (130, 141, 142). In $Nd_2Zr_2O_7$, the change in Q-dependence of the inelastic flat mode is associated to a similar dimensional reduction, which manifests in the excitations, and these excitations feature a dynamic kagome ice state. Dispersive modes are also observed above the flat mode and the overall spectrum can be described in the dynamic fragmentation picture (described in Section 6.3) applied to the kagome planes (98).

## 8. CLOSING REMARKS

Significant progress has been made in characterizing the low-temperature states for dipole-octupole Ce- and Nd-based pyrochlores. However, that is not to say that all of the relevant measurements are quantitatively understood. This could likely be improved by consideration of more realistic microscopic Hamiltonians. Although the XYZ Hamiltonian is an appropriate starting point for $Ce_2Zr_2O_7$, and captures much of the salient dipole-octupole QSI physics, weak further-than-nearest-neighbor interactions are almost certainly relevant.

Several current puzzles for $Ce_2Zr_2O_7$ include the reduced value of $g_z < 2.57$ suggested in References 60–62 and 101, the finer features in the diffuse neutron scattering at low temperatures (40, 41, 60, 62), the failure of heat capacity calculations to capture the lowest-temperature form of the corresponding experimental measurements (60–62), the detailed nature of the diffuse inelastic scattering from the $\beta$ chains in a $[1, \bar{1}, 0]$ magnetic field, and a peculiar field-strength dependence of the inelastic neutron scattering signal at high fields in a $[0, 0, 1]$ magnetic field (62).

Somewhat of a detail, but an interesting one, is that Gaudet et al. (40) identified a weak dispersionless excitation at $E \sim 100$ meV with intensity that decreases with increasing $Q$, attributed to a possible vibronic bound state between CEF excitation and phonon. This feature is also present at $E \sim 100$ meV in the inelastic neutron scattering data reported for $Ce_2Zr_2O_7$ in Reference 41, and a similar feature is present for $Ce_2Hf_2O_7$ in Reference 45. In contrast to this, the high-energy inelastic neutron scattering data reported for $Ce_2Sn_2O_7$ in Reference 44 show no signs for a potential vibronic bound state near $E \sim 100$ meV. Significant magnetoelastic coupling leading to the formation of a clear vibronic bound state has been reported for other rare-earth pyrochlore materials such as $Ho_2Ti_2O_7$ (143, 144), $Tb_2Ti_2O_7$ (145), and $Pr_2Zr_2O_7$ (146), for example, as well as other cerium-based magnets (147–153). Interestingly, magnetoelastic coupling is known to have an effect on the dynamics of spinon excitations in spin ice materials (144). These spinons are referred to as magnetic monopoles in the naming convention of Reference 144 but correspond to electric monopoles (charges) in the convention used here. Investigation of the magnetoelastic coupling in cerium-based pyrochlores could well be important and, at a minimum, provide an explanation for the putative vibronic bound state in $Ce_2Zr_2O_7$ and $Ce_2Hf_2O_7$ (40, 41, 45).

Perhaps the most important gap in our understanding of the dipole-octupole pyrochlore families is the relative absence of direct experimental information available on the behavior of the octupolar magnetic moments in $Ce_2Zr_2O_7$, $Nd_2Zr_2O_7$, or other dipole-octupole pyrochlore systems. The exception to this are the reports of high-Q diffuse scattering in $Ce_2Sn_2O_7$ (44) and $Ce_2Hf_2O_7$ (82). However, there is no clear consensus on the visibility of octupolar diffuse scattering at high Q for cerium pyrochlores (73). The unambiguous demonstration of the existence of the octupolar component of AIAO order in $Nd_2Zr_2O_7$, for example, would be an achievement.

Of course, the behavior of the octupolar magnetic moments can be inferred from the values of the exchange parameters and the behavior of dipole moments. For example, the octupolar
magnetic moments in the *β* chains of Ce<sub>2</sub>Zr<sub>2</sub>O<sub>7</sub> in [1, 1, 0] are expected to show ferromagnetic interchain correlations (*J*<sub>*x*</sub>, *J*<sub>*y*</sub> > 0) with correlation lengths longer than the 10-Å correlation length of the dipole moments due to the fact that *J*<sub>*x*</sub>, *J*<sub>*y*</sub> ≫ |*J*<sub>*z*</sub>| for Ce<sub>2</sub>Zr<sub>2</sub>O<sub>7</sub> (60–62). Nonetheless, much insight into the nature of the dipole-octupole QSI state of Ce<sub>2</sub>Zr<sub>2</sub>O<sub>7</sub> would be gained by direct sensitivity to the octupolar magnetic moments.

The other outstanding issue is that emergent QED photons remain a theoretical construct. The experimental challenges to their observation are considerable. Foremost among them are their very small energy scale and the fact that their spectral weight is predicted to vanish at the lowest energies (32). It is relatively easy to understand why no such observation has been reported to date, and such a discovery may itself require the discovery of QSI candidate materials with larger exchange energy scales, which will likely not be based on insulating 4f electrons. Nonetheless the unambiguous direct observation of emergent photons would be a major accomplishment.

## SUMMARY POINTS

1. Ce<sup>3+</sup> and Nd<sup>3+</sup> ions at the *R* sites of the cubic pyrochlore structure both possess dipole-octupole pseudospin-1/2 degrees of freedom, associated with their Kramers crystal electric field ground-state doublets, which have dipole-octupole symmetry.

2. Different components of the pseudospin-1/2 degrees of freedom transform as magnetic dipoles and octupoles under the symmetry operations of the point group at the rare-earth site and time-reversal symmetry.

3. The general form of the phase diagram corresponding to the XYZ Hamiltonian for dipole-octupole pyrochlores is understood and possesses both disordered quantum spin ice (QSI) and ordered all-in-all-out (AIAO) ground states, each with both dipolar and octupolar character.

4. Theoretical modeling of detailed experimental data on Ce<sub>2</sub>Zr<sub>2</sub>O<sub>7</sub> places it in the QSI region of the XYZ phase diagram, close to the boundary between QSIs with octupolar and dipolar symmetry.

5. Two components to the inelastic neutron scattering spectral weight can be identified for Nd<sub>2</sub>Zr<sub>2</sub>O<sub>7</sub> and modeled using linear spin-wave theory to place Nd<sub>2</sub>Zr<sub>2</sub>O<sub>7</sub> in the region of the XYZ phase diagram corresponding to AIAO order with dipolar symmetry.

6. The inelastic spectral weight in Nd<sub>2</sub>Zr<sub>2</sub>O<sub>7</sub> possesses both a gapped, dispersionless component, whose *Q* dependence resembles divergence-free spin ice, and a dispersive component, corresponding to the divergence-full part of the excitations, thus featuring the existence of dynamic fragmentation in the excitations.

## DISCLOSURE STATEMENT

The authors are not aware of any affiliations, memberships, funding, or financial holdings that might be perceived as affecting the objectivity of this review.

## ACKNOWLEDGMENTS

The authors appreciate helpful conversations and interactions with O. Benton, B. Placke, R. Schäfer, R. Moessner, J. Gaudet, A. D. Bianchi, D. R. Yahne, K. A. Ross, J. Dudemaine, A. Fitterman, A. R. Wildes, N. P. Butch, E. Kermarrec, G. M. Luke, and R. Sibille. This work was supported by the Natural Sciences and Engineering Research Council of Canada.
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