where \( k _ { B } \) is Boltzmann’s constant. If we take \( \vartheta \approx \) 0.83 this reproduces the Curie-Weiss temperature \( T _ { \mathrm { C W } } \approx \, 0 . 2 \mathrm { K } \) mea-sured in Ref. [27, 28, 30]. As a final consistency check, we can then calculate the magnitude of the ordered moment \( m ^ { \mathsf { o r d } } \) which we expect to find in the ground state. At our level of approximation, the ratio of \( m ^ { \mathsf { o r d } } \) to the full, saturated, moment \( m ^ { \mathsf { s a t } } \) is given by

\[ \frac { m ^ { \mathrm { o r d } } } { m ^ { \mathrm { s a t } } } = \cos ( \vartheta ) \left( \frac { S - \langle a _ { i } ^ { \dagger } a _ { i } \rangle } { S } \right) . \eqno ( 2 0 ) \]

The spin wave calculation gives us \( \begin{array} { r } { \left( \frac { S - \langle a _ { i } ^ { \dagger } a _ { i } \rangle } { S } \right) \approx \, 0 . 8 7 . } \end{array} \) The fact that this number is close to unity is a good indica-tor that the linear spin wave approach is valid. Combining this with the estimated value of \( \vartheta \) gives an ordered moment fraction \( \frac { m ^ { \mathrm { o r d } } } { m ^ { \mathrm { s a t } } } \approx 0 . 5 9 \). This is close to the value \( \begin{array} { r } { \frac { m ^ { \mathrm { o r d } } } { m ^ { \mathrm { s a t } } } \approx 0 . 5 } \end{array} \) ob-tained in Ref. [27], but a bit higher than the value \( \frac { m ^ { \mathrm { o r d } } } { m ^ { \mathrm { s a t } } } \approx 0 . 3 3 \) obtained in Ref. [28]. It is interesting to note that most of this moment reduction comes from the pseudospin rotation \( \vartheta \), not from the zero-point fluctuations, as one might typically expect.

The theory presented in this article is thus the first to present a consistent treatment of the ground state and the finite energy spectrum in \( \mathrm { N d _ { 2 } Z r _ { 2 } O _ { 7 } } \). At the same time it is also able to ac-count for the apparent contradiction between the Curie-Weiss temperature and the antiferromagnetic ordering and gives rea-sonable agreement with the strongly reduced ordered moment measured in experiments [27].

Moment fragmentation– Having established that a theory based on a linear spin wave treatment of Eq. (4) correctly reproduces the experimental phenomenology, we can now ask what this theory tells about the proposed “magnetic moment fragmentation”. In particular, is this a true example of moment fragmentation, as proposed in Ref. [17], and if so, what is its origin?

The proposal of Brooks-Bartlett et al. in Ref. [17] is based on the Helmholtz decomposition of the magnetisation density

\[ \mathbf { m } = \mathbf { m } _ { m } + \mathbf { m } _ { d } = \nabla \psi + \nabla \times \mathbf { A } \qquad \qquad ( 2 1 ) \]

where \( \mathbf { m } _ { d } \; = \; \nabla \times \mathbf { A } \) is divergence-free \( ( \nabla \cdot \mathbf { m } _ { d } \ = \ 0 \)) and \( \mathbf { m } _ { m } = \nabla \psi \) is “divergence-full”. The fragmentation phe-nomenon is observed when magnetic order occurs in \( \mathbf { m } _ { m } \), but \( \mathbf { m } _ { d } \) remains fluctuating quasi-independently of \( \mathbf { m } _ { m } \). Since \( \mathbf { m } _ { d } \) obeys \( \nabla \cdot \mathbf { m } _ { d } = 0 \) this gives rise to the pinch point corre-lations associated with a Coulomb phase [12, 22–24].

We can understand the magnetic fragmentation phe-nomenon in \( \mathrm { N d _ { 2 } Z r _ { 2 } O _ { 7 } } \) by defining fields \( \mathbf { m } _ { i } ^ { ( \tilde { \alpha } ) } \) for each pseu-dospin component \( \tilde { \tau } _ { i } ^ { \tilde { \alpha } } \), according to Eq. (7) and then applying the Helmholtz decomposition to each one individually

\[ \mathbf { m } _ { i } ^ { ( \tilde { \alpha } ) } = \nabla \psi ^ { ( \tilde { \alpha } ) } + \nabla \times \mathbf { A } ^ { ( \tilde { \alpha } ) } . \qquad \qquad ( 2 2 ) \]

FIG. 3: Closing of the flat band gap \( \Delta _ { \mathrm { f l a t } } \) with increasing \( \tilde { J } _ { x } \) and the proximity of a \( U ( 1 ) \) quantum spin liquid phase, shown for \( \tilde { J } _ { y } = 0 \). The gap to the flat band which contains the Coulomb-phase-like cor-relations of Fig. 1(b) closes when the ratio \( \begin{array} { r } { \frac { \tilde { J } _ { x } } { | \tilde { J } _ { z } | } = 3 . } \end{array} \) This is a likely indicator of the onset of a \( U ( 1 ) \) spin liquid phase. Based on the pa-rameterisation of Eq. (18) this ratio is \( \begin{array} { r } { \frac { \bar { J } _ { x } } { | \bar { J } _ { z } | } \approx 2 . 1 9 } \end{array} \) in \( \mathrm { N d _ { 2 } Z r _ { 2 } O _ { 7 } } \), suggesting the possibility of observing a \( U ( 1 ) \) spin liquid ground state \( \mathrm { N d _ { 2 } Z r _ { 2 } O _ { 7 } } \) or related Nd based pyrochlores, induced by appli-cation of chemical or physical pressure.

(Note that \( \mathbf { m } _ { i } ^ { ( \tilde { y } ) } \) does not contribute to the physical mag-netisation field \( \mathbf { m } _ { i } \) [Eq. (6)]).

In the all-in-all-out ground state, \( \mathbf { m } _ { i } ^ { ( \tilde { z } ) } \) is completely “divergence-full”, and we may write \( { \bf A } ^ { ( z ) } \; = \; 0 \). The fluc-tuations of \( \mathbf { m } _ { i } ^ { ( \tilde { x } ) } \) and \( \mathbf { m } _ { i } ^ { ( \tilde { y } ) } \), on the other hand, have both divergence-free and “divergence-full” components. The mo-ment fragmentation phenomenon is observed because the equations of motion decouple the dynamics of divergence-free and “divergence-full” components of \( \mathbf { m } _ { i } ^ { ( \tilde { x } ) } , \mathbf { m } _ { i } ^ { ( \tilde { y } ) } \).

Writing down the Heisenberg equations of motion for \( \mathbf { m } _ { i } ^ { ( \tilde { x } ) } \) and \( \mathbf { m } _ { i } ^ { ( \tilde { y } ) } \) and linearising around the all-in-all-out ground state we find

\[ \begin{array} { r l } { \partial _ { t } \mathbf { m } _ { i } ^ { ( \tilde { \alpha } ) } \approx \varepsilon _ { \tilde { \alpha } ^ { \prime } \tilde { \alpha } \tilde { z } } S \big ( \tilde { \mathsf { J } } _ { \alpha ^ { \prime } } \nabla _ { i } ( \nabla \cdot \mathbf { m } ^ { ( \tilde { \alpha } ^ { \prime } ) } ) + } & { { } } \\ { ( 6 | \tilde { \mathsf { J } } _ { z } | - 2 \tilde { \mathsf { J } } _ { \alpha ^ { \prime } } ) \mathbf { m } _ { i } ^ { ( \tilde { \alpha } ^ { \prime } ) } \big ) } & { { } ( 2 3 ) } \end{array} \]

where \( \tilde { \alpha } { } ^ { \prime } \, = \, \tilde { y } \) when \( \tilde { \alpha } \, = \, \tilde { x } \) and vice versa and \( \varepsilon _ { \tilde { x } \tilde { y } \tilde { z } } = \)\( - \varepsilon _ { \tilde { \boldsymbol { \Psi } } \tilde { \boldsymbol { z } } } = 1 \). In Eq. (23), \( \nabla \) and \( \nabla \cdot \)· should be interpreted as the lattice gradient and divergence. This suggestive form for the equations of motion in terms of the lattice gradient and divergence arises because the sites of the pyrochlore lattice can be considered as the bonds of a bipartite (in this case, diamond) lattice [12].

Eq. (23) can be solved in terms of the Helmholtz decompo-sitions [Eq. (22)], by writing

\[ \partial _ { t } \psi ^ { ( \tilde { \alpha } ) } = \varepsilon _ { \tilde { \alpha } ^ { \prime } \tilde { \alpha } \tilde { z } } S \bigl ( \tilde { \mathsf { J } } _ { \alpha ^ { \prime } } \nabla ^ { 2 } \psi ^ { ( \tilde { \alpha } ^ { \prime } ) } + ( 6 | \tilde { \mathsf { J } } _ { z } | - 2 \tilde { \mathsf { J } } _ { \alpha ^ { \prime } } ) \psi ^ { ( \tilde { \alpha } ^ { \prime } ) } \bigr ) \]

\[ \partial _ { t } { \bf A } ^ { ( \tilde { \alpha } ) } = \varepsilon _ { \tilde { \alpha } ^ { \prime } \tilde { \alpha } \tilde { z } } S ( 6 | \tilde { \bf J } _ { z } | - 2 \tilde { \bf J } _ { \alpha ^ { \prime } } ) { \bf A } ^ { ( \tilde { \alpha } ^ { \prime } ) } \eqno ( 2 5 ) \]

The important feature of Eqs. (24)-(25) is that the diver-genceless fluctuations i.e. fluctuations of \( { \bf A } ^ { ( \tilde { \alpha } ) } ) \) are com-pletely decoupled from the “divergence full” fluctuations i.e.

fluctuations of \( \psi ^ { ( \tilde { \alpha } ) } ) \). Fluctuations of \( \mathbf { A } ^ { ( \tilde { \alpha } ) } \) form a flat band at energy \( \Delta _ { \mathrm { f l a t } } = \sqrt { ( 3 | \tilde { \bf J } _ { z } | - \tilde { \bf J } _ { x } ) ( 3 | \tilde { \bf J } _ { z } | - \tilde { \bf J } _ { y } ) } \), while fluctuations of \( \psi ^ { ( \tilde { \alpha } ) } \) form dispersive bands.

The physical magnetisation field in \( \mathrm { N d _ { 2 } Z r _ { 2 } O _ { 7 } } \) [Eq. (6)]thus comprises (i) a static, ordered, “divergence full” compo-nent, (ii) a finite energy divergenceless component exhibiting Coulomb-liquid-like correlations and finally (iii) another “di-vergence full” component corresponding to the finite energy dispersive bands.

The fact that all three components are observable within a magnetisation field which is strictly Ising like (in the sense that \( \mathbf { m } _ { i } \) is always parallel to the local easy axis) is a consequence of the unusual symmetry of dipolar octupolar doublets- specifically that the x-component of the pseudospin transforms like the z-component of a dipole moment. This un-derstanding of the moment fragmentation is fully compatible with the observation that the pinch points remain observable above the ordering transition at \( T _ { N } \), but at lower frequency[25]. Above the transition \( \mathbf { m } _ { i } ^ { ( \tilde { x } ) } \) can fluctuate for little or no energy cost but its correlations will remain ice like due to the positive value of \( \tilde { \bf { J } } _ { x } \).

Conclusions– In conclusion we have explained the quantum origins of the moment fragmentation in \( \mathrm { N d } _ { 2 } / { \bf r } _ { 2 } ( \r ) , \), observed in Ref. [25]. It may be rationalized as the consequence of the symmetry properties of dipolar-octupolar doublets and a de-coupling of divergence-free and divergence-full fluctuations in the equations of motion.

Much of the physics discussed here is generic to systems described by the exchange Hamiltonian \( \mathcal { H } _ { \mathrm { X Y Z } } ^ { \mathrm { D O } } \) which have an all-in-all-out ground state. Specifically, the flat band exhibit-ing pinch points at finite energy is present throughout the all-in-all-out phase of \( \mathcal { H } _ { \mathrm { X Y Z } } ^ { \mathrm { D O } } \), at least at the level of linear spin wave theory. It will therefore be interesting to investigate whether the moment fragmentation phenomenon is also ob-served in other Nd based pyrochlores showing an all-in-all-out ground state such as \( \mathrm { N d _ { 2 } S n _ { 2 } O _ { 7 } } \) [34],\( \mathbf { N d } _ { \mathcal { O } } ] \)\( \mathsf { H r } _ { \mathsf { \Phi } } ( \mathsf { \Lambda } ) , \) 7 [35] and possibily \( \mathrm { N d } _ { 2 } \mathrm { P b } _ { 2 } \mathrm { O } _ { 7 } \) [36].

The parameterisation of the exchange Hamiltonian \( \mathcal { H } _ { \mathrm { X Y Z } } ^ { \mathrm { D O } } \)[Eq. (4)], given in Eq. (18) suggests that \( \mathrm { N d _ { 2 } Z r _ { 2 } O _ { 7 } } \) is proxi-mate to the \( U ( 1 ) \) spin liquid phase which has been long sought amongst “quantum spin ice” pyrochlores [14, 16, 37–42]. As shown in Fig. 3, the closing of the gap to the flat band con-taining the pinch point correlations occurs at \( \begin{array} { r } { \frac { \tilde { \bf J } _ { x } } { | \tilde { \bf J } _ { z } | } = 3 } \\ { \quad \quad \quad \quad \quad \quad \quad } \end{array} \) within linear spin wave theory. Classically, this would signal the for-mation of an extensive ground state manifold with ice-like character, but the mixing of these states by quantum fluctu-ations is known to stablise a \( U ( 1 ) \) spin liquid with dynamic emergent gauge fields [37, 40]. The placement of \( \mathrm { N d _ { 2 } Z r _ { 2 } O _ { 7 } } \) close to the point where this gap vanishes hints at the prox-imity of the \( \bar { U } ( 1 ) \) spin liquid phase. If there is a well formed Coulomb phase above \( T _ { N } \) in \( \mathrm { N d _ { 2 } Z r _ { 2 } O _ { 7 } } \) this may make the observed magnetic ordering a candidate for the observation of a Higgs transition in which the emergent gauge field of the Coulomb phase is gapped by the condensation of emergent gauge charges [44, 45]. We therefore have reason to hope that experiments on \( \mathrm { N d _ { 2 } Z r _ { 2 } O _ { 7 } } \) and related materials may yet re-veal even more exotic phenomena.

Acknowledgements – This work was supported by the The-ory of Quantum Matter Unit of the Okinawa Institute of Sci-ence and Technology Graduate University. The author is grateful to Ludovic Jaubert and Nic Shannon for careful read-ings of the manuscript.

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