As is mentioned in the first chapter, investigations of frustrated magnets concentrate on two core questions: What is the ground state, ordered or disordered? And what are the excitations, confined or fractionalized. Using neutron scattering, we have addressed these two questions for three frustrated spinels: CdEr2X4(X =S, Se), MnSc2S4, and MgCr2O4. Their relatively large magnetic moment sizes (J=15/2 for Er3+,S=5/2 for Mn2+, andS=3/2 for Cr3+) lead to rather weak quantum fluctuations. However, due to the special geometry of the diamond lattice of theA-site and the pyrochlore lattice of theB-site with additional perturbations from the spin-lattice coupling and even thermal fluctuations, very unusual spin correlations have been revealed in these three compounds.
For CdEr2X4(X=S, Se), the spins are found to mimic the behavior of water ice: each tetrahe-dron of the Er3+pyrochlore lattice exhibits a two-in-two-out spin configuration. To achieve this conclusion, inelastic neutron scattering is used to confirm the Ising character of the Er3+spins and neutron diffuse scattering is used to verify the dominant nearest-neighbour ferromagnetic couplings: both are the pre-requisites for the realization of a spin ice state.
Low-energy excitations in CdEr2X4can be viewed as fractional magnetic monopoles, whose extremely fast dynamics have been studied using the neuron spin echo technique together with magnetic susceptibility measurements, and can be explained by a strong increase in the monopole hopping rates.
In MnSc2S4, Mn2+ spins constitute a diamond lattice. Due to strong second-neighbour couplings, the spins can fluctuate collectively as spirals and realize the spiral spin-liquid state.
Using neutron diffuse scattering, we directly observe the (hk0) cut of the continuous spiral surface formed by the propagation vectors and experimentally verify the spiral spin-liquid state in MnSc2S4. Neutron diffraction experiments reveal the importance of the third-neighbour couplingsJ3and the dipolar interactions in determining the ordering behavior of the spiral spin-liquid. And a triple-q vortex-like state is discovered under magnetic field. Using inelastic neutron scattering, we study the spin wave excitations in MnSc2S4, and our results further confirm the importance of theJ3term.
MgCr2O4realizes a highly frustrated antiferromagnetic Heisenberg pyrochlore model, whose ground state degeneracy is however relieved by the spin-lattice coupling. Our synchrotron X-ray diffraction experiments at low temperatures reveal the distorted structure to be either F ddd orI41/amd. And the magnetic structure is solved by combining neutron diffraction and spherical neutron polarimetry. The spin dynamics in MgCr2O4is very unusual in that both resonance-like and spin wave excitations are observed. Although a complete understanding of these excitations is still unavailable at the moment, we found that the linear spin wave theory can qualitatively capture the observed spin wave dispersions.
With the classical spin ice state established in CdEr2X4(X=S, Se), it is tempting to look for the quantum spin ice state in the CdRE2X4series. Until now, four other compounds with RE = Dy, Ho, Tm, and Yb have been successfully synthesized [112, 229–231]. Their crystal electric field parameters can be approximated using the scaled values of CdEr2Se4[75]:
Aml (R0)= al+1(R)
al+1(R0)Aml (R) , (6.1)
whereRandR0represent different rare-earth ions,ais the size of the unit cell and the values for the CdRE2Se4compounds are listed in Tab. 6.1, andAml are the Hutchings crystal electric field parameters that can be transformed from the Stevens crystal electric field parameters Bml following the relation:
Blm=Aml 〈rl〉θl. (6.2)
In this expression,〈rl〉is the expectation value of thel-th power of thef-electron radius and can be found in Ref. [232];θl is the Stevens factor that can be found in Ref. [73].
Tab. 6.1 lists the scaled Hutchings and Wybourne crystal electric field parameters for the CdRE2Se4compounds. The calculated CEF levels are shown in Fig. 6.1, and the ground states are:
Dy3+|±〉 =0.1756|7.5,±6.5〉 ∓0.3498|7.5,±3.5〉 +0.2730|7.5,±0.5〉 (6.3)
±0.5545|7.5,∓2.5〉 +0.6817|7.5,∓5.5〉, Ho3+|±〉 =0.1567|8,±8〉 ∓0.4564|8,±5〉 +0.2518|8,±2〉
±0.0960|8,∓1〉 +0.7434|8,∓4〉 ±0.3768|8,∓7〉, Tm3+|φ〉 =0.6594|6,±6〉 ±0.2004|6,±3〉 +0.2235|6,0〉
Yb3+|±〉 = −0.2557|3.5,±3.5〉 ±0.3895|3.5,±0.5〉 +0.8848|3.5,∓2.5〉.
Firstly, it is clear that the ground state of Tm3+is a singlet due to its non-Kramers character, while for spin ice, a doublet ground state is required. For the remaining compounds where a doublet CEF ground state is realized, spins in CdDy2Se4and CdYb2Se4exhibit Heisenberg-like character, with theg-factors ofg⊥=5.69,g∥=6.38 for Dy3+andg⊥=2.16,g∥=3.97 for Yb3+.
Table 6.1 – The Hutchings and Wybourne crystal electric field parameters for the CdRE2Se4 compounds scaled from the measured CdEr2Se4values. The sizes of the unit cell used in the scaling calculations are also listed.
CdDy2Se4 CdHo2Se4 CdEr2Se4 CdTm2Se4 CdYb2Se4 unit cell (Å) 11.467 11.638 11.5959 11.560 11.528
Hutchings CEF parameters (meV)
A02 −17.84 −17.86 −18.07 −18.23 −18.36 A04 −10.37 −10.41 −10.60 −10.75 −10.91 A34 −222.74 −224.57 −227.66 −230.87 −234.23
A06 0.32 0.32 0.33 0.34 0.34
A36 −4.89 −4.97 −5.07 −5.17 −5.28
A66 1.81 1.84 1.88 1.91 1.95
Wybourne CEF parameters (meV)
L02 −27.88 −26.594 −25.70 −24.82 −23.96 L04 −124.91 −114.88 −107.73 −100.99 −95.04 L34 −113.33 −104.23 −97.74 −91.63 −86.23
L06 30.68 27.70 25.31 23.28 21.53
L36 −23.10 −20.85 −19.06 −17.53 −16.21
L66 11.53 10.40 9.51 8.75 8.09
For the Ho3+spin, an Ising character withg⊥=0,g∥=4.43 is observed, which satisfies the local Ising condition to realize the spin ice state. Therefore, from the CEF point of view, CdHo2Se4 might be the most promising compound to realize the quantum spin liquid state. However, it should be noted that the CEF excited states in CdDy2Se4, CdTm2Se4, and CdHo2Se4are lying at energies below∼2 meV. Such low-lying excited levels might renormalize the spin couplings and make our single-ion analysis inappropriate [229, 233].
E (meV)
0 5 10 15 20 25 30 35 40 45 50 55
Dy3+ Ho3+ Tm3+ Yb3+
Figure 6.1 – Crystal electric field levels for CdRE2Se4with RE = Dy, Ho, Tm, and Yb using the CEF parameters listed in Tab. 6.1.
In MnSc2S4, the spiral spin-liquid state enters a long-range ordered state due to the third-neighbour J3term instead of thermal fluctuations. Therefore, it will be very interesting to look for similarA-site spinel compounds where the further-neighbour couplings are not so strong and thus the order-by-disorder transition might be realized. Also, the study of the spiral spin-liquid state should not be restricted to theA-site spinels or even to the diamond lattice.
The primary constituent to realize spiral spin-liquids is the frustratedJ1–J2model, which can be constructed for most bipartite lattices. One recent example is the honeycomb lattice, where a two-dimensional analogue of the spiral spin-liquid state and a subsequent order-by-disorder transition have been predicted [234].
Another interesting prospect that is related to MnSc2S4is the realization of the skyrmion lattice in theA-site spinels. As is discussed in Chapter 4, one difference between the vortex lattice in MnSc2S4and the skyrmion lattice lies in the dimension of their basis vectors. For the skyrmion lattice, the basis vectors are two-dimensional, resulting in a helical structure for each arm; While for the vortex lattice, the basis vectors are one-dimensional, resulting in a sinusoidally modulated collinear structure for each arm. This means that in theA-site spinels, if the helical structure can be further stabilized byi.e.the Dzyaloshinskii-Moriya interaction, arms of the triple-qstructure might become helical and thus give rise to a skyrmion lattice.
For the spin-Peierls system MgCr2O4, one big challenge is to find a unified description of the spin wave and resonance-like excitations. Further neutron inelastic scattering experi-ments with better-resolution will be beneficial because the interference between the spin wave and resonance-like excitations is still unclear at the moment. If the spin wave modes crossed the resonance-like modes without much interference, then these two modes might be independent and thus a non-trivial ground state with co-existence of long-range ordered and short-range fluctuating spins could be the solution [235]. On the other hand, if the spin wave and resonance-like modes interfere strongly and it is more favorable to calculate all the excitations from a long-range ordered ground state, then the spin-lattice coupling, or more specifically the magnon-phonon coupling [236], might be a necessary ingredient of the spin Hamiltonian.
Bibliography
[1] Fazekas, P. & Anderson, P. W. On the ground state properties of the anisotropic triangular antiferromagnet. Philos. Mag.30, 423–440 (1974).
[2] Coleman, P. Phil Anderson’s magnetic ideas in science.ArXiv e-prints1605.06993 (2016).
[3] Gottfried, K. & Yan, T.-M.Quantum mechanics: fundamentals(Springer-Verlag, Berlin, 2004).
[4] Fazekas, P.Lecture notes on electron correlation and magnetism(World Scientific Pub-lishing, Singapore, 2003).
[5] Sachdev, S.Quantum phase transitions(Cambridge University Press, Cambridge, 2013).
[6] Normand, B. Frontiers in frustrated magnetism. Contemp. Phys.50, 533–552 (2009).
[7] Villain, J., Bidaux, R., Carton, J.-P. & Conte, R. Order as an effect of disorder.J. de Phys.
41, 1263 (1980).
[8] Henley, C. L. Ordering due to disorder in a frustrated vector antiferromagnet.Phys. Rev.
Lett.62, 2056–2059 (1989).
[9] Cheong, S.-W. & Mostovoy, M. Multiferroics: a magnetic twist for ferroelectricity.Nat.
Mater.6, 13–20 (2007).
[10] Fiebig, M., Lottermoser, T., Meier, D. & Trassin, M. The evolution of multiferroics. Nat.
Rev. Mater.1, 16046 (2016).
[11] Katsura, H., Nagaosa, N. & Balatsky, A. V. Spin current and magnetoelectric effect in noncollinear magnets. Phys. Rev. Lett.95, 057205 (2005).
[12] Mostovoy, M. Ferroelectricity in spiral magnets.Phys. Rev. Lett.96, 067601 (2006).
[13] Kimura, T.et al.Magnetic control of ferroelectric polarization.Nature426, 55–58 (2003).
[14] Choi, Y. J., Zhang, C. L., Lee, N. & Cheong, S.-W. Cross-control of magnetization and polarization by electric and magnetic fields with competing multiferroic and weak-ferromagnetic phases.Phys. Rev. Lett.105, 097201 (2010).
[15] Tsui, D. C., Stormer, H. L. & Gossard, A. C. Two-dimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett.48, 1559–1562 (1982).
[16] Laughlin, R. B. Anomalous quantum hall effect: An incompressible quantum fluid with fractionally charged excitations. Phys. Rev. Lett.50, 1395–1398 (1983).
[17] Moore, G. & Read, N. Nonabelions in the fractional quantum hall effect. Nuc. Phys. B 360, 362 – 396 (1991).
[18] Kalmeyer, V. & Laughlin, R. B. Equivalence of the resonating-valence-bond and fractional quantum hall states. Phys. Rev. Lett.59, 2095–2098 (1987).
[19] Kitaev, A. Fault-tolerant quantum computation by anyons.Ann. Phys.303, 2 – 30 (2003).
[20] Levin, M. A. & Wen, X.-G. String-net condensation: A physical mechanism for topologi-cal phases.Phys. Rev. B71, 045110 (2005).
[21] Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Das Sarma, S. Non-Abelian anyons and topological quantum computation.Rev. Mod. Phys.80, 1083–1159 (2008).
[22] Mermin, N. D. & Wagner, H. Absence of ferromagnetism or antiferromagnetism in one-or two-dimensional isotropic heisenberg models. Phys. Rev. Lett.17, 1133–1136 (1966).
[23] Giamarchi, T. Quantum physics in one dimension(Oxford Science Publications, Oxford, 2003).
[24] Wen, X.-G. Quantum field theory of many-body systems: from the origin of sound to an origin of light and electrons(Oxford University Press on Demand, 2004).
[25] Haldane, F. Continuum dynamics of the 1-d heisenberg antiferromagnet: Identification with the O(3) nonlinear sigma model.Phys. Lett. A93, 464 – 468 (1983).
[26] Zhou, Y., Kanoda, K. & Ng, T.-K. Quantum spin liquid states. Rev. Mod. Phys.89, 025003 (2017).
[27] Bethe, H. Zur theorie der metalle.Z. Phys.71, 205–226 (1931).
[28] Karabach, M., Müller, G., Gould, H. & Tobochnik, J. Introduction to the Bethe Ansatz I.
Comp. in Phys.11, 36–43 (1997).
[29] Faddeev, L. & Takhtajan, L. What is the spin of a spin wave? Phys. Lett. A85, 375–377 (1981).
[30] Lake, B., Tennant, D. A., Frost, C. D. & Nagler, S. E. Quantum criticality and universal scaling of a quantum antiferromagnet. Nat. Mater.4, 329–334 (2005).
[31] Mourigal, M.et al. Fractional spinon excitations in the quantum Heisenberg antiferro-magnetic chain. Nat. Phys.9, 435–441 (2013).
Bibliography
[32] Sierra, G. On the application of the Non-linear sigma model to spin chains and spin ladders. In Sierra, G. & Martín-Delgado, M. A. (eds.)Lecture Notes in Physics, Berlin Springer Verlag, vol. 478 (1997). cond-mat/9610057.
[33] Affleck, I., Kennedy, T., Lieb, E. H. & Tasaki, H. Rigorous results on valence-bond ground states in antiferromagnets. Phys. Rev. Lett.59, 799–802 (1987).
[34] Buyers, W. J. L.et al. Experimental evidence for the haldane gap in a spin-1 nearly isotropic, antiferromagnetic chain. Phys. Rev. Lett.56, 371–374 (1986).
[35] Kenzelmann, M.et al.Properties of haldane excitations and multiparticle states in the antiferromagnetic spin-1 chain compound CsNiCl3.Phys. Rev. B66, 024407 (2002).
[36] Matan, K. et al. Pinwheel valence-bond solid and triplet excitations in the two-dimensional deformed kagome lattice.Nat. Phys.6, 865–869 (2010).
[37] Hwang, K., Huh, Y. & Kim, Y. B. Z2gauge theory for valence bond solids on the kagome lattice.Phys. Rev. B92, 205131 (2015).
[38] Anderson, P. W. The resonating valence bond state in La2CuO4and superconductivity.
Science235, 1196–1198 (1987).
[39] Savary, L. & Balents, L. Quantum spin liquids: a review. Rep. Prog. Phys.80, 016502 (2017).
[40] Coleman, P. New approach to the mixed-valence problem.Phys. Rev. B29, 3035–3044 (1984).
[41] Arovas, D. P. & Auerbach, A. Functional integral theories of low-dimensional quantum heisenberg models. Phys. Rev. B38, 316–332 (1988).
[42] Shen, Y.et al.Evidence for a spinon Fermi surface in a triangular-lattice quantum-spin-liquid candidate.Nature540, 559–562 (2016).
[43] Paddison, J. A. M.et al.Continuous excitations of the triangular-lattice quantum spin liquid YbMgGaO4.Nat. Phys.13, 117–122 (2017).
[44] Wen, X.-G. Quantum orders and symmetric spin liquids.Phys. Rev. B65, 165113 (2002).
[45] Wen, X. G. Vacuum degeneracy of chiral spin states in compactified space. Phys. Rev. B 40, 7387–7390 (1989).
[46] Levin, M. & Wen, X.-G. Detecting topological order in a ground state wave function.
Phys. Rev. Lett.96, 110405 (2006).
[47] Wen, X.-G. Zoo of quantum-topological phases of matter. ArXiv e-prints1610.03911 (2016).
[48] Kitaev, A. Anyons in an exactly solved model and beyond. Ann. of Phys.321, 2–111 (2006).
[49] Tchernyshyov, O., Yao, H. & Moessner, R. Valence-bond crystal in a {111} slice of the pyrochlore antiferromagnet. Phys. Rev. B69, 212402 (2004).
[50] Radaelli, P. G.et al.Formation of isomorphic Ir3+and Ir4+octamers and spin dimeriza-tion in the spinel CuIr2S4.Nature416, 155–158 (2002).
[51] Khomskii, D. I. & Mizokawa, T. Orbitally induced Peierls state in spinels.Phys. Rev. Lett.
94, 156402 (2005).
[52] Okamoto, Y., Nohara, M., Aruga-Katori, H. & Takagi, H. Spin-liquid state in the S=1/2 hyperkagome antiferromagnet Na4Ir3O8.Phys. Rev. Lett.99, 137207 (2007).
[53] Chen, G. & Balents, L. Spin-orbit effects in Na4Ir3O8: A hyper-kagome lattice antiferro-magnet.Phys. Rev. B78, 094403 (2008).
[54] Takayama, T.et al. Hyperhoneycomb iridateβ-Li2IrO3as a platform for Kitaev mag-netism. Phys. Rev. Lett.114, 077202 (2015).
[55] Biffin, A.et al.Unconventional magnetic order on the hyperhoneycomb kitaev lattice in β-Li2IrO3: Full solution via magnetic resonant X-ray diffraction.Phys. Rev. B90, 205116 (2014).
[56] Lee, E. K.-H., Schaffer, R., Bhattacharjee, S. & Kim, Y. B. Heisenberg-Kitaev model on the hyperhoneycomb lattice. Phys. Rev. B89, 045117 (2014).
[57] Bi, Z., You, Y.-Z. & Xu, C. Anyon and loop braiding statistics in field theories with a topologicalΘterm.Phys. Rev. B90, 081110 (2014).
[58] Wang, C. & Levin, M. Braiding statistics of loop excitations in three dimensions.Phys.
Rev. Lett.113, 080403 (2014).
[59] Wang, J. C. & Wen, X.-G. Non-abelian string and particle braiding in topological order:
Modular SL(3,Z) representation and (3+1)-dimensional twisted gauge theory. Phys.
Rev. B91, 035134 (2015).
[60] Hermele, M., Fisher, M. P. A. & Balents, L. Pyrochlore photons: The U(1) spin liquid in a S=1/2 three-dimensional frustrated magnet. Phys. Rev. B69, 64404 (2004).
[61] Bramwell, S. T. & Gingras, M. J. P. Spin ice state in frustrated magnetic pyrochlore materia.Science294, 1495–1501 (2001).
[62] Castelnovo, C., Moessner, R. & Sondhi, S. L. Magnetic monopoles in spin ice. Nature 451, 42–45 (2008).
[63] Gingras, M. J. P. & McClarty, P. A. Quantum spin ice: A search for gapless quantum spin liquids in pyrochlore magnets. Rep. Prog. Phys.77, 056501 (2014).
Bibliography
[64] Squires, G. L. Introduction to the theory of thermal neutron scattering (Cambridge University Press, Cambridge, 2012).
[65] Furrer, A., Mesot, J. & Strässle, T.Neutron scattering in condensed matter physics(World Scientific Publishing, Singapore, 2009).
[66] Shirane, G., Shapiro, S. M. & Tranquada, J. M. Neutron scattering with a triple-axis spectrometer(Cambridge University Press, Cambridge, 2004).
[67] Willis, B. T. M. & Carlile, C. J.Experimental neutron scattering(Oxford University Press, Oxford, 2009).
[68] Ament, L. J. P., van Veenendaal, M., Devereaux, T. P., Hill, J. P. & van den Brink, J. Resonant inelastic x-ray scattering studies of elementary excitations.Rev. Mod. Phys.83, 705–767 (2011).
[69] Shannon, R. D. Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides. Acta Cryst. A32, 751–767 (1976).
[70] Bauer, B.et al.The ALPS project release 2.0: open source software for strongly correlated systems.J. Stat. Mech.: Theo. and Exp.2011, P05001 (2011).
[71] Chernyshev, A. L. & Zhitomirsky, M. E. Spin waves in a triangular lattice antiferromagnet:
Decays, spectrum renormalization, and singularities.Phys. Rev. B79, 144416 (2009).
[72] Toth, S. & Lake, B. Linear spin wave theory for single-q incommensurate magnetic structures. J. Phys. Condens. Matt.27, 166002 (2015).
[73] Jensen, J. & Mackintosh, A. R.Rare Earth Magnetism(Clarendon Press, Oxford, 1991).
[74] McPhase User Manual, chap. Appendix G. URL http://www2.cpfs.mpg.de/~rotter/
homepage_mcphase/manual/node132.html.
[75] Bertin A. and Chapuis Y. and Dalmas de Réotier P. and Yaouanc A. Crystal electric field in the R2Ti2O7pyrochlore compounds.J. Phys. Condens. Matt.24, 256003 (2012).
[76] 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).
[77] Ramirez, A. P., Hayashi, A., Cava, R. J., Siddharthan, R. & Shastry, B. S. Zero-point entropy in ‘spin ice’. Nature399, 333–335 (1999).
[78] Matsuhira K. and Hinatsu Y. and Tenya K. and Sakakibara T. Low temperature magnetic properties of frustrated pyrochlore ferromagnets Ho2Sn2O7and Ho2Ti2O7. J. Phys.
Condens. Matt.12, L649 (2000).
[79] Kadowaki, H., Ishii, Y., Matsuhira, K. & Hinatsu, Y. Neutron scattering study of dipolar spin ice Ho2Sn2O7: frustrated pyrochlore magnet.Phys. Rev. B65, 144421 (2002).
[80] Matsuhira, K., Hinatsu, Y., Tenya, K., Amitsuka, H. & Sakakibara, T. Low-temperature magnetic properties of pyrochlore stannates.J. Phys. Soc. Jpn.71, 1576–1582 (2002).
[81] Zhou, H. D.et al. Chemical pressure effects on pyrochlore spin ice.Phys. Rev. Lett.108, 207206 (2012).
[82] Zhou, H. D.et al.High pressure route to generate magnetic monopole dimers in spin ice. Nat. Commun.2, 478 (2011).
[83] 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).
[84] Castelnovo, C., Moessner, R. & Sondhi, S. Spin ice, fractionalization, and topological order.Ann. Rev. Condens. Matt. Phys.3, 35–55 (2012).
[85] Paulsen, C. et al. Experimental signature of the attractive Coulomb force between positive and negative magnetic monopoles in spin ice.Nat. Phys.12, 661–666 (2016).
[86] Ryzhkin, I. A. Magnetic relaxation in rare-earth oxide pyrochlores.J. Exp. Theo. Phys.
101, 481–486 (2005).
[87] Castelnovo, C., Moessner, R. & Sondhi, S. L. Debye-Hückel theory for spin ice at low temperature.Phys. Rev. B84, 144435 (2011).
[88] Morris, D. J. P.et al. Dirac strings and magnetic monopoles in the spin ice Dy2Ti2O7. Science326, 411–414 (2009).
[89] Bramwell, S. T.et al.Measurement of the charge and current of magnetic monopoles in spin ice. Nature461, 956–959 (2009).
[90] Kaiser, V., Bramwell, S. T., Holdsworth, P. C. W. & Moessner, R. Onsager’s Wien effect on a lattice.Nat. Mater.12, 1033–1037 (2013).
[91] Paulsen, C.et al.Far-from-equilibrium monopole dynamics in spin ice. Nat. Phys.10, 135–139 (2014).
[92] Kaiser, V., Bramwell, S., Holdsworth, P. & Moessner, R. ac Wien effect in spin ice, manifest in nonlinear, nonequilibrium susceptibility.Phys. Rev. Lett.115, 037201 (2015).
[93] Henley, C. L. Power-law spin correlations in pyrochlore antiferromagnets.Phys. Rev. B 71, 14424 (2005).
[94] Henley, C. L. The “Coulomb phase” in frustrated systems.Ann. Rev. Condens. Matt. Phys.
1, 179–210 (2010).
[95] Fennell, T., Bramwell, S. T., McMorrow, D. F., Manuel, P. & Wildes, A. R. Pinch points and Kasteleyn transitions in kagome ice.Nat. Phys.3, 566–572 (2007).
Bibliography
[96] Fennell, T.et al.Magnetic coulomb phase in the spin ice Ho2Ti2O7.Science326, 415–417 (2009).
[97] Savary, L. & Balents, L. Coulombic quantum liquids in spin-1/2 pyrochlores.Phys. Rev.
Lett.108, 037202 (2012).
[98] Savary, L. & Balents, L. Spin liquid regimes at nonzero temperature in quantum spin ice.
Phys. Rev. B87, 205130 (2013).
[99] Lee, S., Onoda, S. & Balents, L. Generic quantum spin ice.Phys. Rev. B86, 104412 (2012).
[100] McClarty, P. A.et al.Chain-based order and quantum spin liquids in dipolar spin ice.
Phys. Rev. B92, 094418 (2015).
[101] Benton, O., Sikora, O. & Shannon, N. Seeing the light: Experimental signatures of emergent electromagnetism in a quantum spin ice.Phys. Rev. B86, 075154 (2012).
[102] Rau, J. G. & Gingras, M. J. P. Magnitude of quantum effects in classical spin ices.Phys.
Rev. B92, 144417 (2015).
[103] Ruminy, M.et al.Crystal-field parameters of the rare-earth pyrochlores R2Ti2O7(R=Tb, Dy, and Ho).Phys. Rev. B94, 024430 (2016).
[104] Fennell, T., Kenzelmann, M., Roessli, B., Haas, M. K. & Cava, R. J. Power-law spin correlations in the pyrochlore antiferromagnet Tb2Ti2O7. Phys. Rev. Lett.109, 17201 (2012).
[105] Zhou, H. D.et al.Dynamic spin ice: Pr2Sn2O7.Phys. Rev. Lett.101, 227204 (2008).
[106] Princep, A. J., Prabhakaran, D., Boothroyd, A. T. & Adroja, D. T. Crystal-field states of Pr3+in the candidate quantum spin ice Pr2Sn2O7.Phys. Rev. B88, 104421 (2013).
[107] Kimura, K.et al.Quantum fluctuations in spin-ice-like Pr2Zr2O7.Nat. Commun.4, 1934 (2013).
[108] Sibille, R.et al.Candidate quantum spin ice in the pyrochlore Pr2Hf2O7.Phys. Rev. B94, 024436 (2016).
[109] Petit, S.et al.Observation of magnetic fragmentation in spin ice. Nat. Phys.12, 746–750 (2016).
[110] Xu, J.et al.Magnetic structure and crystal-field states of the pyrochlore antiferromagnet Nd2Zr2O7.Phys. Rev. B92, 224430 (2015).
[111] Sibille, R. et al. Candidate quantum spin liquid in the Ce3+ pyrochlore stannate Ce2Sn2O7.Phys. Rev. Lett.115, 097202 (2015).
[112] Lau, G. C., Freitas, R. S., Ueland, B. G., Schiffer, P. & Cava, R. J. Geometrical magnetic frustration in rare-earth chalcogenide spinels. Phys. Rev. B72, 54411 (2005).
[113] Lago, J.et al.CdEr2Se4: A new erbium spin ice system in a spinel structure.Phys. Rev.
Lett.104, 247203 (2010).
[114] Legros, A., Ryan, D. H., Dalmas de Réotier, P., Yaouanc, A. & Marin, C.166Er Mössbauer spectroscopy study of magnetic ordering in a spinel-based potential spin-ice system:
CdEr2S4.J. App. Phys.117, 17C701 (2015).
[115] Rodríguez-Carvajal, J. Recent advances in magnetic structure determination by neutron powder diffraction. Phys. B: Condens. Matt.192, 55–69 (1993).
[116] Prather, J. L.Atomic energy levels in crystals(National Bureau of Standards, 1961).
[117] Rotter, M. Using mcphase to calculate magnetic phase diagrams of rare earth com-pounds.J. Mag. Magn. Mater.272–276, E481–E482 (2004).
[118] Dasgupta, P., Jana, Y. & Ghosh, D. Crystal field effect and geometric frustration in Er2Ti2O7–an XY antiferromagnetic pyrochlore.Sol. Stat. Commun.139, 424–429 (2006).
[119] Cao, H.et al.Ising versusX Y anisotropy in frustrated R2Ti2O4compounds as "seen" by polarized neutrons.Phys. Rev. Lett.103, 056402 (2009).
[120] Huang, Y.-P., Chen, G. & Hermele, M. Quantum spin ices and topological phases from dipolar-octupolar doublets on the pyrochlore lattice.Phys. Rev. Lett.112, 167203 (2014).
[121] Li, Y.-D., Wang, X. & Chen, G. Hidden multipolar orders of dipole-octupole doublets on a triangular lattice.Phys. Rev. B94, 201114 (2016).
[122] Li, Y.-D. & Chen, G. Symmetry enriched U(1) topological orders for dipole-octupole doublets on a pyrochlore lattice. Phys. Rev. B95, 041106 (2017).
[123] Le, D. SAFiCF. URL http://yocto.me/work/saficf/index.html.
[124] Kanada, M.et al.Neutron scattering study of the spin correlation in the spin ice system Ho2Ti2O7.J. Phys. Soc. Jpn.71, 313–318 (2002).
[125] Fennell, T.et al.Neutron scattering investigation of the spin ice state in Dy2Ti2O7.Phys.
Rev. B70, 134408 (2004).
[126] Yavors’kii, T., Fennell, T., Gingras, M. J. P. & Bramwell, S. T. Dy2Ti2O7spin ice: A test case for emergent clusters in a frustrated magnet.Phys. Rev. Lett.101, 37204 (2008).
[127] Chang, L.-J.et al.Higgs transition from a magnetic Coulomb liquid to a ferromagnet in Yb2Ti2O7.Nat. Commun.3, 992 (2012).
[128] Stewart, J. R.et al.Disordered materials studied using neutron polarization analysis on the multi-detector spectrometer, D7. J. App. Cryst.42, 69–84 (2009).
[129] Ehlers, G., Stewart, J. R., Wildes, A. R., Deen, P. P. & Andersen, K. H. Generalization of the classical xyz-polarization analysis technique to out-of-plane and inelastic scattering.
[129] Ehlers, G., Stewart, J. R., Wildes, A. R., Deen, P. P. & Andersen, K. H. Generalization of the classical xyz-polarization analysis technique to out-of-plane and inelastic scattering.