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.

[1] J. E. Greedan, Geometrically frustrated magnetic materials, J. Mater. Chem. 11, 37-53, (2001)

[2] J. S. Gardner, M. J. P. Gingras and J. E. Greedan, Magnetic Pyrochlore Oxides, Rev. Mod. Phys. 82, 53-107, (2010)

[3] Y. Shimizu, K. Miyagawa, K. Kanoda, M. Maesato, and G. Saito, Spin Liquid State in an Organic Mott Insulator with a Triangular Lattice, Phys. Rev. Lett. 91, 107001 (2003)

[4] T. H. Han, J. S. Helton, S. Y. Chu, D. G. Nocera, J. A. Rodriguez-Rivera, C. Broholm and Y. S. Lee, Fractionalized excitations in the spin-liquid state of a kagome-lattice antifer-romagnet Nature 492, 406-410 (2012)

[5] K. Kimura, S. Nakatsuji, J. J. Wen, C. Broholm, M. B. Stone, E. Nishibori and H. Sawa, Quantum fluctuations in spin-ice like Pr2Zr2O7, Nature Commun. 4, 2914 (2013)

[6] R. Sibille, E. Lhotel, V. Pomjakushin, C. Baines, T. Fennell and M. Kenzelmann, Candidate Quantum Spin Liquid in the Ce3+Pyrochlore Stannate Ce2Sn2O7, Phys. Rev. Lett. 115, 097202 (2015)

[7] L. Balents, Spin liquids in frustrated magnets, Nature 464, 199-208 (2010)

[8] X.-G. Wen, Quantum orders and symmetric spin liquids, Phys. Rev. B 65, 165113 (2002).

[9] A. Kitaev, Anyons in an exactly solved model and beyond, An-nals of Physics 321, 2, (2006)

[10] C. Castelnovo, R. Moessner and S. L. Sondhi, Magnetic monopoles in spin ice, Nature 451, 42 (2008)

[11] R. Moessner and S. L. Sondhi, Irrational charge from topolog-ical order, Phys. Rev. Lett. 105, 166401 (2010).

[12] C. L. Henley, The “Coulomb phase” in frustrated systems, Annu. Rev. Condens. Matter Phys. 1, 1 (2010).

[13] S. M. Yan, D. A. Huse and S. R. White, Spin-Liquid Ground State of the S=1/2 Kagome Heisenberg Antiferromagnet, Sci-ence 332, 1173-1176, (2011).

[14] M. J. P. Gingras and P. A. McClarty, Quantum spin ice: a search for gapless quantum spin liquids in pyrochlore magnets, Rep. Prog. Phys. 77, 056501 (2014).

[15] O. Benton, L. D. C. Jaubert, H. Yan and N. Shannon, A spin-liquid with pinch-line singularities on the pyrochlore lattice, Nature Commun. 7, 11572 (2016)

[16] L. Savary and L. Balents, Coulombic quantum liquids in spin-1/2 pyrochlores, Phys. Rev. Lett. 108, 037202 (2012).

[17] M. E. Brooks-Bartlett, S. T. Banks, L. D. C. Jaubert, A. Harman-Clarke and P. C. W. Holdsworth, Magnetic Moment Fragmentation and Monopole Crystallization, Phys. Rev. X 4, 011007, (2014).

[18] S. Powell, Ferromagnetic Coulomb phase in classical spin ice, Phys. Rev. B 91, 094431 (2015).

[19] L. D. C. Jaubert, Monopole Holes in a Partially Ordered Spin Liquid, Spin 5, 1540005 (2015).

[20] B. Canals, I.-A. Chioar, V.-D. Nguyen, M. Hehn, D. Lacour, F. Montaigne, A. Locatelli, T. Onur Mentes, B. Santos Burgos