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.