Abstract
Fractionalized excitations that emerge from a manybody system have revealed rich physics and concepts, from composite fermions in twodimensional electron systems, revealed through the fractional quantum Hall effect^{1}, to spinons in antiferromagnetic chains^{2} and, more recently, fractionalization of Dirac electrons in graphene^{3} and magnetic monopoles in spin ice^{4}. Even more surprising is the fragmentation of the degrees of freedom themselves, leading to coexisting and a priori independent ground states. This puzzling phenomenon was recently put forward in the context of spin ice, in which the magnetic moment field can fragment, resulting in a dual ground state consisting of a fluctuating spin liquid, a socalled Coulomb phase^{5}, on top of a magnetic monopole crystal^{6}. Here we show, by means of neutron scattering measurements, that such fragmentation occurs in the spin ice candidate Nd_{2}Zr_{2}O_{7}. We observe the spectacular coexistence of an antiferromagnetic order induced by the monopole crystallization and a fluctuating state with ferromagnetic correlations. Experimentally, this fragmentation manifests itself through the superposition of magnetic Bragg peaks, characteristic of the ordered phase, and a pinch point pattern, characteristic of the Coulomb phase. These results highlight the relevance of the fragmentation concept to describe the physics of systems that are simultaneously ordered and fluctuating.
Main
The physics of spin ice materials is intimately connected with the pyrochlore lattice, composed of cornersharing tetrahedra. On the corners of these tetrahedra reside rareearth magnetic moments J_{i}, which, as a consequence of the strong crystal electric field, are constrained to point along their local trigonal axes z_{i}, and behave like Ising spins. The magnetic interactions are composed of nearestneighbour exchange and dipolar interactions between spins i and j separated by a distance r_{ij} (ref. 7):
where , μ_{0} is the permeability of free space, g_{J} is the Landé factor of the magnetic moment, μ_{B} is the Bohr magneton and r_{nn} is the nearestneighbour distance between rareearth ions. The nearestneighbour spin ice Hamiltonian is obtained by truncating the Hamiltonian (1), yielding:
When the effective interaction is positive—that is, when the dipolar term overcomes the antiferromagnetic exchange—a very unusual magnetic state develops, known as the spin ice state. The system remains in a highly correlated but disordered ground state where the local magnetization fulfils the socalled ‘ice rule’: each tetrahedron has two spins pointing in and two spins pointing out (see Fig. 1a), in close analogy with the rule which controls the hydrogen position in water ice^{8}. The extensive degeneracy of this ground state results in a residual entropy at low temperature which is well approximated by the Pauling entropy for water ice^{9}.
Such highly degenerate states, where the organizing principle is dictated by a local constraint, belong to the class of Coulomb phases^{5,10,11}: the constraint (the ice rule for spin ice) can be interpreted as a divergencefree condition of an emergent gauge field. This field has correlations that fall off with distance like the dipolar interaction^{12,13}. In reciprocal space, this powerlaw character leads to bowtie singularities, called pinch points, in the magnetic structure factor. They form a key experimental signature of the Coulomb phase physics. They have been observed by neutron diffraction in the spin ice materials Ho_{2}Ti_{2}O_{7} and Dy_{2}Ti_{2}O_{7}, in excellent agreement with theoretical predictions^{14,15}.
Classical excitations above the spin ice manifold are defects that locally violate the ice rule and so the divergencefree condition: by reversing the orientation of a moment, ‘three in–one out’ and ‘one in–three out’ configurations are created (see Fig. 1b). Considering the Ising spins as dumbbells with two opposite magnetic charges at their extremities, such defects result in a magnetic charge in the centre of the tetrahedron, called a magnetic monopole, that give rise to a nonzero divergence of the local magnetization^{4}.
Recently, theoreticians have introduced the concept of magnetic moment fragmentation^{6}, whereby the local magnetic moment field fragments into the sum of two parts, a divergencefull and a divergencefree part (see Fig. 1c): for example, a monopole in the spin configuration m = {1, 1, 1, −1} on a tetrahedron can be written m = 1/2{1, 1, 1, 1} + 1/2{1, 1, 1, −3}. In this decomposition, the first term carries the total magnetic charge of the monopole. If the monopoles organize as a crystal of alternating magnetic charges, the fragmentation leads to the superposition of an ordered ‘all in–all out’ configuration (Fig. 1d) and of an emergent Coulomb phase associated with the divergencefree contribution (Fig. 1e).
This monopole crystallization occurs when the monopole density is high enough such that the Coulomb interaction between monopoles (which originates in the dipolar interaction between magnetic moments) is minimized through charge ordering, whereas the remaining fluctuating divergencefree part provides a gain in entropy.
The necessary conditions for an experimental realization of this physics are severe: in pyrochlore systems, the fragmentation is expected in the case of strong Ising anisotropy combined with effective ferromagnetic interactions, and for a specific ratio between the dipolar and exchange interactions to form the crystal of monopoles. If fragmentation occurs, the theory predicts that the magnetic structure factor should exhibit both Bragg peaks characteristic of the ‘all in–all out’ structure and a pinch point pattern typical of a Coulomb phase^{6}. The pyrochlore system Nd_{2}Zr_{2}O_{7} is a good candidate in the search for such a system. Previous studies have provided evidence for the strong Ising character of the Nd^{3+} ion, and for ferromagnetic interactions, inferred from the positive Curie–Weiss temperature θ_{CW} = 195 mK (ref. 16). Moreover, Nd_{2}Zr_{2}O_{7} orders below T_{N} = 285 mK in an ‘all in–all out’ state carrying a reduced ordered magnetic moment of about one third of the total Nd^{3+} magnetic moment μ_{eff} = 2.4 μ_{B} (ref. 17) (see Supplementary Information).
To demonstrate that the fragmentation occurs in Nd_{2}Zr_{2}O_{7}, it is essential to observe signatures of the Coulomb phase. To this end, neutron scattering experiments have been carried out as a function of temperature and magnetic field on a large single crystal. As shown in Fig. 2a, the key point here is that the neutron data at 60 mK do exhibit armlike features along the (00ℓ) and (hhh) directions, with pinch points at the (002) and (111) positions, expected in the Coulomb phase^{6}. This pinch point pattern is observed simultaneously with the ‘all in–all out’ Bragg peaks^{17,18} at (220) and (113), which we interpret as evidence for fragmentation and monopole crystallization.
Importantly, this structured neutron scattering signal appears as a flat mode at finite energy around E_{o} = 70 μeV (see Figs 3 top and 4a, b). In addition, above this flat mode, collective dispersive excitations stem from the pinch points and not from the antiferromagnetic ‘all in–all out’ wavevectors (see Fig. 3 top). They are characterized by a spin gap Δ ≍ E_{o} and reach a maximum energy of about 0.25 meV.
When increasing the temperature, the pinch point pattern and the collective modes persist up to 600 mK, far above the antiferromagnetic ordering (T_{N} = 285 mK) (see Supplementary Information). Whereas the energy gap, and the intensity of these features, decrease as the temperature increases (see Fig. 4a), the energy range of the dispersion remains unaffected up to 450 mK. This temperature dependence suggests a scenario in which the fragmentation takes place well above T_{N}: at the temperature where the ferromagnetic correlations start to develop, a Coulomb phase arises in coexistence with a liquid of monopoles. The latter finally crystallizes on cooling in an ‘all in–all out’ phase at T_{N}, leaving the Coulomb phase unchanged. The field dependence is consistent with this scenario (see Fig. 4b, c): at an applied field of 0.15 T, where magnetization measurements show that the ‘all in–all out’ state is replaced by a fieldinduced ordered state^{17}, the Coulomb phase characteristics remain, albeit with less intensity. This observation further confirms the fragmentation scenario in which the divergencefree and divergencefull parts of the magnetic moment field behave independently.
This peculiar spin dynamics, and especially the existence of dispersive modes, are puzzling in an Isinglike system. They call for the existence of additional transverse terms in the Hamiltonian given in equation (2). To address this point, the magnetic moments should not be considered as Ising variables, but as pseudospin half σ_{i} spanning the ↑↓〉 crystalline electric field (CEF) doublet states.
Considering the very peculiar ‘dipolar–octopolar’ nature of the Kramers Nd^{3+} doublet^{17,19,20,21}, such transverse terms arise from a coupling between octopolar moments. Indeed, although 〈↑J↓〉 ≡ 0 because of those CEF properties, it can be shown using the explicit wavefunctions determined in ref. 17 that the octopole is the relevant operator, because . Introducing an octopole–octopole coupling , where κ denotes the strength of the octopolar coupling, and projecting it onto the pseudospin 1/2 subspace leads to:
where σ^{y, z} are the pseudospin components in the local coordinates, is an effective exchange interaction and . For Nd_{2}Zr_{2}O_{7}, g_{J} = 8/11, g_{x} = g_{y} = 0 and g_{z} = 4.5.
The Hamiltonian parameters and can be estimated by fitting the inelastic neutron scattering spectra. From calculations in the random phase approximation (RPA) ^{22,23,24,25} (see Supplementary Information), it is found that the bandwidth of the collective modes is related to whereas the shift of the pinch point pattern up to E_{o} is induced by the transverse term . This is reminiscent of the role of the antisymmetric Dzyaloshinskii–Moriya interaction which lifts the ‘weathervane’ flat mode in kagome systems up to finite energy^{26}. Such transverse terms might also be at the origin of the inelastic pattern observed in the quantum spin ice candidate Pr_{2}Zr_{2}O_{7} (ref. 27). The best agreement is obtained for K and K (see Figs 2b and 3 bottom). For these values, the RPA ground state is an ordered octopolar phase. It is worth noting that, although this RPA calculation accounts for the behaviour of the divergencefree part of the magnetic moment, it is unable to capture the fragmentation mechanism.
We have thus shown that the predicted fragmentation process^{6} exists in the spin ice material Nd_{2}Zr_{2}O_{7}. Below 700 mK, the magnetic moment field fragments into two parts: a divergencefull part which crystallizes at T_{N} = 285 mK, and a divergencefree part for which transverse terms induce gapped and dispersive excitations. Our results highlight that the two fragments behave independently as a function of field and temperature, which opens the appealing possibility of manipulating them separately.
Beyond the classical fragmentation theory described in ref. 6, the importance of transverse terms to describe our observations emphasizes the need for considering quantum effects in further theoretical studies. Indeed, in the classical scheme, the crystallization occurs when the energy required to create the assembly of fragmented monopoles is balanced by the repulsive energy among them, and thus depends on the competition between exchange and dipolar interactions. In the present case, transverse octopolar couplings might enhance the interactions between monopoles, thus promoting their crystallization. We thus anticipate that our experiment will pave the way towards a quantum theory of fragmentation, involving such transverse terms.
In a broader context, the fragmentation theory relies on the Helmholtz decomposition of a charged field, widely used to describe continuous fluid media in a wide variety of fields, from fluid mechanics to robotics^{28}. This decomposition allows one to identify new relevant degrees of freedom, which could not have been separated otherwise. Our results in Nd_{2}Zr_{2}O_{7} indicate its applicability to describe, more generally, localized moment systems where fluctuating and ordered phases coexist. This might cover the case of the pyrochlore compound Yb_{2}Ti_{2}O_{7}, a system showing a strongly reduced ferromagnetic ordering^{29} and a peculiar fluctuation spectrum, and whose physics is probably governed by competing phases^{25,30}. In spin ice, the Helmholtz decomposition is applied at a microscopic level on the emergent gauge field of the Coulomb phase and on its charges, the monopoles. Our experimental findings give a concrete form to these concepts. The observation of fragmentation in Nd_{2}Zr_{2}O_{7} will thus stimulate new conceptual approaches in physical systems where such emergent fields exist.
Methods
Single crystals of Nd_{2}Zr_{2}O_{7} were grown by the floatingzone technique using a fourmirror xenon arc lamp optical image furnace^{16,31}.
Inelastic neutron scattering experiments were carried out at the Institute Laue Langevin (ILL, France) on the IN5 disk chopper timeofflight spectrometer operated at λ = 8.5 Å or λ = 6 Å. The Nd_{2}Zr_{2}O_{7} singlecrystal sample was attached to the cold finger of a dilution insert and the field was applied along [1 −1 0]. The data were processed with the Horace software, transforming the recorded time of flight, sample rotation and scattering angle into energy transfer and Qwavevectors.
The neutron diffraction data were taken at the D23 singlecrystal diffractometer (CEACRG, ILL France) with a copper monochromator and using λ = 1.28 Å. Here the field was applied along the [111] direction. Refinements were carried out with the Fullprof software suite^{32} (http://www.ill.eu/sites/fullprof).
The magnetic diffuse scattering was measured on the D7 diffractometer installed at the ILL, with λ = 4.85 Å, using a standard polarization analysis technique with the guiding field along the vertical axis [1 −1 0].
Calculations are carried out on the basis of a mean field treatment of a Hamiltonian, taking into account the dipolar exchange as well as an octopolar coupling between the crystalline electric field (CEF) ground doublet states of the Nd^{3+} ion. This Hamiltonian is written in terms of a pseudospin 1/2 spanning these states. The spin dynamics is then calculated numerically in the random phase approximation^{22,23,24,25}.
More details are provided in the Supplementary Information.
References
Stormer, H. L. Nobel lecture: the fractional quantum Hall effect. Rev. Mod. Phys. 71, 875–889 (1999).
Sachdev, S. Quantum Phase Transitions (Cambridge Univ. Press, 2011).
Bolotin, K. I., Ghahar, F., Shulman, M. D., Stormer, H. L. & Kim, P. Observation of the fractional quantum Hall effect in graphene. Nature 462, 196–199 (2009).
Castelnovo, C., Moessner, R. & Sondhi, S. L. Magnetic monopoles in spin ice. Nature 451, 42–45 (2008).
Henley, C. L. The Coulomb phase in frustrated systems. Annu. Rev. Condens. Matter. Phys. 1, 179–210 (2010).
BrooksBartlett, M. E., Banks, S. T., Jaubert, L. D. C., HarmanClarke, A. & Holdsworth, P. C. W. Magneticmoment fragmentation and monopole crystallization. Phys. Rev. X 4, 011007 (2014).
den Hertog, B. C. & Gingras, M. J. P. Dipolar interactions and origin of spin ice in Ising pyrochlore magnets. Phys. Rev. Lett. 84, 3430–3433 (2000).
Harris, M. J., Bramwell, S. T., McMorrow, D. F., Zeiske, T. & Godfrey, K. W. Geometrical frustration in the ferromagnetic pyrochlore Ho2Ti2O7 . Phys. Rev. Lett. 79, 2554–2557 (1997).
Ramirez, A. P., Hayashi, A., Cava, R. J., Siddharthan, R. & Shastry, B. S. Zeropoint entropy in spin ice. Nature 399, 333–335 (1999).
Huse, D. A., Krauth, W., Moessner, R. & Sondhi, S. L. Coulomb and liquid dimer models in three dimensions. Phys. Rev. Lett. 91, 167004 (2003).
Bergman, D. L., Fiete, G. A. & Balents, L. Ordering in a frustrated pyrochlore antiferromagnet proximate to a spin liquid. Phys. Rev. B 73, 134402 (2006).
Isakov, S. V., Gregor, K., Moessner, R. & Sondhi, S. L. Dipolar spin correlations in classical pyrochlore magnets. Phys. Rev. Lett. 93, 167204 (2004).
Henley, C. L. Powerlaw spin correlations in pyrochlore antiferromagnets. Phys. Rev. B 71, 014424 (2005).
Fennell, T. et al. Magnetic Coulomb phase in the spin ice Ho2Ti2O7 . Science 326, 415–417 (2009).
Morris, D. J. P. et al. Dirac strings and magnetic monopoles in the spin ice Dy2Ti2O7 . Science 326, 411–414 (2009).
Ciomaga Hatnean, M. et al. Structural and magnetic investigations of singlecrystalline neodymium zirconate pyrochlore Nd2Zr2O7 . Phys. Rev. B 91, 174416 (2015).
Lhotel, E. et al. Fluctuations and allin–allout ordering in dipoleoctopole Nd2Zr2O7 . Phys. Rev. Lett. 115, 197202 (2015).
Ferey, G., de Pape, R., Leblanc, M. & Pannetier, J. Ordered magnetic frustration: VIII. Crystal and magnetic structures of the pyrochlore form of FeF3 between 2.5 and 25 K from powder neutron diffraction. Comparison with the other varieties of FeF3 . Rev. Chim. Miner. 23, 474–484 (1986).
Abragam, A. & Bleaney, B. Electron Paramagnetic Resonance of Transition Ions (Oxford Classic Texts in the Physical Sciences, Oxford Univ. Press, 1970).
Watahiki, M. et al. Crystalline electric field study in the pyrochlore Nd2Ir2O7 with metalinsulator transition. J. Phys. Conf. Ser. 320, 012080 (2011).
Huang, Y.P., Chen, G. & Hermele, M. Quantum spin ices and topological phases from dipolaroctupolar doublets on the pyrochlore lattice. Phys. Rev. Lett. 112, 167203 (2014).
Jensen, J. & Mackintosh, A. R. Rare Earth Magnetism (International Series of Monographs on Physics, Clarendon, 1991).
Kao, Y. J., Enjalran, M., Del Maestro, A., Molavian, H. R. & Gingras, M. J. P. Understanding paramagnetic spin correlations in the spinliquid pyrochlore Tb2Ti2O7 . Phys. Rev. B 68, 172407 (2003).
Petit, S. et al. Order by disorder or energetic selection of the ground state in the XY pyrochlore antiferromagnet Er2Ti2O7. An inelastic neutron scattering study. Phys. Rev. B 90, 060410 (2014).
Robert, J. et al. Spin dynamics in the presence of competing ferromagnetic and antiferromagnetic correlations in Yb2Ti2O7 . Phys. Rev. B 92, 064425 (2014).
Matan, K. et al. Spin Waves in the Frustrated Kagomé Lattice Antiferromagnet KFe3(OH)6(SO4)2 . Phys. Rev. Lett. 96, 247201 (2006).
Kimura, K. et al. Quantum fluctuations in spinicelike Pr2Zr2O7 . Nature Commun. 4, 1934 (2013).
Bhatia, H., Norgard, G., Pascucci, V. & Bremer, P.T. The Helmholtz–Hodge decomposition—a survey. IEEE Trans. Vis. Comput. Graphics 19, 1386–1404 (2013).
Chang, L.J. et al. Higgs transition from a magnetic Coulomb liquid to a ferromagnet in Yb2Ti2O7 . Nature Commun. 3, 992 (2012).
Jaubert, L. D. C. et al. Are multiphase competition and order by disorder the keys to understanding Yb2Ti2O7? Phys. Rev. Lett. 115, 267208 (2015).
Ciomaga Hatnean, M., Lees, M. R. & Balakrishnan, G. Growth of singlecrystals of rareearth zirconate pyrochlores, Ln2Zr2O7 (with Ln = La, Nd, Sm, and Gd) by the floating zone technique. J. Cryst. Growth 418, 1–6 (2015).
RodríguezCarvajal, J. Recent advances in magnetic structure determination by neutron powder diffraction. Physica B 192, 55–69 (1993).
Acknowledgements
We acknowledge the ILL for the beam time. We also thank J. Robert, P. C. W. Holdsworth, V. Simonet and Y. Sidis for fruitful discussions. M.C.H., M.R.L. and G.B. acknowledge financial support from the EPSRC, UK, Grant No. EP/M028771/1.
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Crystal growth and characterization were performed by M.C.H., M.R.L. and G.B. Inelastic neutron scattering experiments were carried out by S.P., E.L., J.O. and H.M. Diffraction experiments were carried out by S.P., E.L., A.R.W. and E.R. The data were analysed by S.P. and E.L., with input from B.C., A.R.W., E.R. and J.O. RPA calculations were carried out by S.P. The paper was written by E.L. and S.P., with feedback from all authors.
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Petit, S., Lhotel, E., Canals, B. et al. Observation of magnetic fragmentation in spin ice. Nature Phys 12, 746–750 (2016). https://doi.org/10.1038/nphys3710
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DOI: https://doi.org/10.1038/nphys3710
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