Spin-1 pyrochlore antiferromagnets: Theory, model, and materials’ survey

Abstract

Pyrochlore magnets can be a unique platform to demonstrate numerous important concepts and applications of frustrated magnetic physics in modern condensed matter physics. Most works on pyrochlore magnets deal with the interacting spin-1/2 local moments, while much less works have studied the spin-1 systems. We here review the physics with interacting spin-1 local moments on the pyrochlore lattice to illustrate the potentially interesting physics associated with spin-1 magnets. The generic pyrochlore spin-1 model includes the antiferromagnetic Heisenberg interaction, the Dzyaloshinskii– Moriya interaction and the single-ion spin anisotropy. The global phase diagram of this generic spin model is reviewed, and the relation between different quantum phases in the phase diagram is clarified. The critical properties of the transition from the parent quantum paramagnet to the proximate orders are discussed. The presence of quantum order by disorder in the parts of the ordered phases is analyzed. The elementary excitations with respect to the ground states are further reviewed, and the topological natures of these excitations are carefully addressed. The materials’ relevance of the spin-1 pyrochlore magnets are finally reviewed. This review may provide insights about the interesting spin-1 local moments on frustrated systems.

Keywords

topological magnon / quantum order by disorder / Dzyaloshinskii–Moriya interaction / flavor wave theory

References

[1] F. D. M. Haldane, Nonlinear field theory of large-spin Heisenberg antiferromagnets: Semiclassically quantized solitons of the one-dimensional easy-axis Néel state, Phys. Rev. Lett. 50(15), 1153 (1983)

[2] F. Haldane, Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the O(3) nonlinear sigma model, Phys. Lett. A 93(9), 464 (1983)

[3] I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Rigorous results on valence-bond ground states in antiferromagnets, Phys. Rev. Lett. 59(7), 799 (1987)

[4] X. Chen, Z. C. Gu, and X. G. Wen, Classification of gapped symmetric phases in one-dimensional spin systems, Phys. Rev. B 83(3), 035107 (2011)

[5] C. Wang, A. Nahum, and T. Senthil, Topological paramagnetism in frustrated spin-1 Mott insulators, Phys. Rev. B 91(19), 195131 (2015)

[6] G. Chen, Quantum paramagnet and frustrated quantum criticality in a spin-one diamond lattice antiferromagnet, Phys. Rev. B 96(2), 020412 (2017)

[7] L. Savary, Quantum loop states in spin–orbital models on the honeycomb lattice, arXiv: 1511.01505 (2015)

[8] Z. Wang, A. E. Feiguin, W. Zhu, O. A. Starykh, A. V. Chubukov, and C. D. Batista, Chiral liquid phase of simple quantum magnets, Phys. Rev. B 96(18), 184409 (2017)

[9] F. L. Buessen, M. Hering, J. Reuther, and S. Trebst, Quantum spin liquids in frustrated spin-1 diamond antiferromagnets, arXiv: 1706.06299 (2017)

[10] J. R. Chamorro, L. Ge, J. Flynn, M. A. Subramanian, M. Mourigal, and T. M. McQueen, Frustrated spin one on a diamond lattice in NiRh₂O₄, Phys. Rev. Mater. 2(3), 034404 (2018)

[11] J. G. Cheng, G. Li, L. Balicas, J. S. Zhou, J. B. Goodenough, C. Xu, and H. D. Zhou, High-pressure sequence of Ba₃NiSb₂O₉ structural phases: New S= 1 quantum spin liquids based on Ni²⁺, Phys. Rev. Lett. 107(19), 197204 (2011)

[12] M. Serbyn, T. Senthil, and P. A. Lee, Exotic S= 1 spinliquid state with fermionic excitations on the triangular lattice, Phys. Rev. B 84(18), 180403 (2011)

[13] S. Bieri, M. Serbyn, T. Senthil, and P. A. Lee, Paired chiral spin liquid with a Fermi surface in S= 1 model on the triangular lattice, Phys. Rev. B 86(22), 224409 (2012)

[14] C. Xu, F. Wang, Y. Qi, L. Balents, and M. P. A. Fisher, Spin liquid phases for spin-1 systems on the triangular lattice, Phys. Rev. Lett. 108(8), 087204 (2012)

[15] G. Chen, M. Hermele, and L. Radzihovsky, Frustrated quantum critical theory of putative spin-liquid phenomenology in 6H-B-Ba₃NiSb₂O₉, Phys. Rev. Lett. 109(1), 016402 (2012)

[16] K. Hwang, T. Dodds, S. Bhattacharjee, and Y. B. Kim, Three-dimensional nematic spin liquid in a stacked triangular lattice 6H-B structure, Phys. Rev. B 87(23), 235103 (2013)

[17] J. A. Quilliam, F. Bert, A. Manseau, C. Darie, C. Guillot-Deudon, C. Payen, C. Baines, A. Amato, and P. Mendels, Gapless quantum spin liquid ground state in the spin-1 antiferromagnet 6HB-Ba₃NiSb₂O₉, Phys. Rev. B 93(21), 214432 (2016)

[18] J. S. Gardner, M. J. P. Gingras, and J. E. Greedan, Magnetic pyrochlore oxides, Rev. Mod. Phys. 82(1), 53 (2010)

[19] S. T. Bramwell and M. J. P. Gingras, Spin ice state in frustrated magnetic pyrochlore materials, Science 294(5546), 1495 (2001)

[20] R. G. Melko, B. C. den Hertog, and M. J. P. Gingras, Long-range order at low temperatures in dipolar spin ice, Phys. Rev. Lett. 87(6), 067203 (2001)

[21] C. Castelnovo, R. Moessner, and S. L. Sondhi, Magnetic monopoles in spin ice, Nature 451(7174), 42 (2008)

[22] H. R. Molavian, M. J. P. Gingras, and B. Canals, Dynamically induced frustration as a route to a quantum spin ice state in Tb₂Ti₂O₇ via virtual crystal field excitations and quantum many-body effects, Phys. Rev. Lett. 98(15), 157204 (2007)

[23] 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(5), 056501 (2014)

[24] L. Savary and L. Balents, Quantum spin liquids: A review, Rep. Prog. Phys. 80(1), 016502 (2017)

[25] S. Onoda and Y. Tanaka, Quantum melting of spin ice: Emergent cooperative quadrupole and chirality, Phys. Rev. Lett. 105(4), 047201 (2010)

[26] L. Savary and L. Balents, Coulombic quantum liquids in spin-1/2 pyrochlores, Phys. Rev. Lett. 108(3), 037202 (2012)

[27] S. Lee, S. Onoda, and L. Balents, Generic quantum spin ice, Phys. Rev. B 86(10), 104412 (2012)

[28] L. Savary and L. Balents, Spin liquid regimes at nonzero temperature in quantum spin ice, Phys. Rev. B 87(20), 205130 (2013)

[29] H. Fukazawa, R. G. Melko, R. Higashinaka, Y. Maeno, and M. J. P. Gingras, Magnetic anisotropy of the spin-ice compound Dy₂Ti₂O₇, Phys. Rev. B 65(5), 054410 (2002)

[30] S. T. Bramwell, M. J. Harris, B. C. den Hertog, M. J. P. Gingras, J. S. Gardner, D. F. McMorrow, A. R. Wildes, A. L. Cornelius, J. D. M. Champion, R. G. Melko, and T. Fennell, Spin correlations in Ho₂Ti₂O₇: A dipolar spin ice system, Phys. Rev. Lett. 87(4), 047205 (2001)

[31] K. A. Ross, J. P. C. Ruff, C. P. Adams, J. S. Gardner, H. A. Dabkowska, Y. Qiu, J. R. D. Copley, and B. D. Gaulin, Two-dimensional Kagome correlations and field induced order in the ferromagnetic XY pyrochlore Yb₂Ti₂O₇, Phys. Rev. Lett. 103(22), 227202 (2009)

[32] Y. P. Huang, G. Chen, and M. Hermele, Quantum spin ices and topological phases from dipolar–octupolar doublets on the pyrochlore lattice, Phys. Rev. Lett. 112(16), 167203 (2014)

[33] G. Chen, “Magnetic monopole” condensation of the pyrochlore ice U(1) quantum spin liquid: Application to Pr₂Ir₂O₇ and Yb₂Ti₂O₇, Phys. Rev. B 94(20), 205107 (2016)

[34] Y. Wan and O. Tchernyshyov, Quantum strings in quantum spin ice, Phys. Rev. Lett. 108(24), 247210 (2012)

[35] Y. D. Li and G. Chen, Symmetry enriched U(1) topological orders for dipole–octupole doublets on a pyrochlore lattice, Phys. Rev. B 95(4), 041106 (2017)

[36] H. Yan, O. Benton, L. Jaubert, and N. Shannon, Theory of multiple-phase competition in pyrochlore magnets with anisotropic exchange with application to Yb₂Ti₂O₇, Er₂Ti₂O₇, and Er₂Sn₂O₇, Phys. Rev. B 95(9), 094422 (2017)

[37] L. Savary, X. Wang, H. Y. Kee, Y. B. Kim, Y. Yu, and G. Chen, Quantum spin ice on the breathing pyrochlore lattice, Phys. Rev. B 94(7), 075146 (2016)

[38] T. Fennell, M. Kenzelmann, B. Roessli, M. K. Haas, and R. J. Cava, Power-law spin correlations in the pyrochlore antiferromagnet Tb₂Ti₂O₇, Phys. Rev. Lett. 109(1), 017201 (2012)

[39] Y. Yasui, M. Kanada, M. Ito, H. Harashina, M. Sato, H. Okumura, K. Kakurai, and H. Kadowaki, Static correlation and dynamical properties of Tb³⁺-moments in Tb₂Ti₂O₇ – neutron scattering study, J. Phys. Soc. Jpn. 71(2), 599 (2002)

[40] J. S. Gardner, B. D. Gaulin, A. J. Berlinsky, P. Waldron, S. R. Dunsiger, N. P. Raju, and J. E. Greedan, Neutron scattering studies of the cooperative paramagnet pyrochlore Tb₂Ti₂O₇, Phys. Rev. B 64(22), 224416 (2001)

[41] Z. Hao, A. G. R. Day, and M. J. P. Gingras, Bosonic many-body theory of quantum spin ice, Phys. Rev. B 90(21), 214430 (2014)

[42] L. J. Chang, S. Onoda, Y. Su, Y. J. Kao, K. D. Tsuei, Y. Yasui, K. Kakurai, and M. R. Lees, Higgs transition from a magnetic Coulomb liquid to a ferromagnet in Yb₂Ti₂O₇, Nat. Commun. 3(1), 992 (2012)

[43] K. Kimura, K. Nakatsuji, J. J. Wen, C. Broholm, M. Stone, E. Nishibori, and H. Sawa, Quantum fluctuations in spin-ice-like Pr₂Zr₂O₇, Nat. Commun. 4(1), 1934 (2013)

[44] E. Lhotel, S. R. Giblin, M. R. Lees, G. Balakrishnan, L. J. Chang, and Y. Yasui, First-order magnetic transition in Yb₂Ti₂O₇, Phys. Rev. B 89(22), 224419 (2014)

[45] L.J. Chang, M. R. Lees, I. Watanabe, A. D. Hillier, Y. Yasui, and S. Onoda, Static magnetic moments revealed by muon spin relaxation and thermodynamic measurements in the quantum spin ice Yb₂Ti₂O₇, Phys. Rev. B 89(18), 184416 (2014)

[46] Y. Yasui, M. Soda, S. Iikubo, M. Ito, M. Sato, N. Hamaguchi, T. Matsushita, N. Wada, T. Takeuchi, N. Aso, and K. Kakurai, Ferromagnetic transition of pyrochlore compound Yb₂Ti₂O₇, J. Phys. Soc. Jpn. 72(11), 3014 (2003)

[47] K. Ross, L. Savary, B. Gaulin, and L. Balents, Quantum excitations in quantum spin ice, Phys. Rev. X 1(2), 021002 (2011)

[48] N. Shannon, O. Sikora, F. Pollmann, K. Penc, and P. Fulde, Quantum ice: A quantum Monte Carlo study, Phys. Rev. Lett. 108(6), 067204 (2012)

[49] P. Goswami, B. Roy, and S. Das Sarma, Itinerant spin ice order, Weyl metal, and anomalous Hall effect in Pr₂Ir₂O₇, arXiv: 1603.02273 (2016)

[50] K. E. Arpino, B. A. Trump, A. O. Scheie, T. M. Mc-Queen, and S. M. Koohpayeh, Impact of stoichiometry of Yb₂Ti₂O₇ on its physical properties, Phys. Rev. B 95(9), 094407 (2017)

[51] J. J. Wen, S. M. Koohpayeh, K. A. Ross, B. A. Trump, T. M. McQueen, K. Kimura, S. Nakatsuji, Y. Qiu, D. M. Pajerowski, J. R. D. Copley, and C. L. Broholm, Disordered route to the Coulomb quantum spin liquid: Random transverse fields on spin ice in Pr₂Zr₂O₇, Phys. Rev. Lett. 118(10), 107206 (2017)

[52] D. E. MacLaughlin, O. O. Bernal, L. Shu, J. Ishikawa, Y. Matsumoto, J. J. Wen, M. Mourigal, C. Stock, G. Ehlers, C. L. Broholm, Y. Machida, K. Kimura, S. Nakatsuji, Y. Shimura, and T. Sakakibara, Unstable spin-ice order in the stuffed metallic pyrochlore Pr_2+xIr_2−xO_7−δ, Phys. Rev. B 92(5), 054432 (2015)

[53] G. Chen, H. Y. Kee, and Y. B. Kim, Fractionalized charge excitations in a spin liquid on partially filled pyrochlore lattices, Phys. Rev. Lett. 113(19), 197202 (2014)

[54] J. Fu, J. G. Rau, M. J. Gingras, and N. B. Perkins, Fingerprints of quantum spin ice in Raman scattering, arXiv: 1703.03836 (2017)

[55] O. Benton, O. Sikora, and N. Shannon, Seeing the light: Experimental signatures of emergent electromagnetism in a quantum spin ice, Phys. Rev. B 86(7), 075154 (2012)

[56] L. D. C. Jaubert, O. Benton, J. G. Rau, J. Oitmaa, R. R. P. Singh, N. Shannon, and M. J. P. Gingras, Are multiphase competition and order by disorder the keys to understanding Yb₂Ti₂O₇? Phys. Rev. Lett. 115(26), 267208 (2015)

[57] R. Applegate, N. R. Hayre, R. R. P. Singh, T. Lin, A. G. R. Day, and M. J. P. Gingras, Vindication of Yb₂Ti₂O₇ as a model exchange quantum spin ice, Phys. Rev. Lett. 109(9), 097205 (2012)

[58] S. R. Dunsiger, A. A. Aczel, C. Arguello, H. Dabkowska, A. Dabkowski, M. H. Du, T. Goko, B. Javanparast, T. Lin, F. L. Ning, H. M. L. Noad, D. J. Singh, T. J. Williams, Y. J. Uemura, M. J. P. Gingras, and G. M. Luke, Spin ice: Magnetic excitations without monopole signatures using muon spin rotation, Phys. Rev. Lett. 107(20), 207207 (2011)

[59] 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(9), 097202 (2015)

[60] M. Taillefumier, O. Benton, H. Yan, L. D. C. Jaubert, and N. Shannon, Competing spin liquids and hidden spinnematic order in spin ice with frustrated transverse exchange, Phys. Rev. X 7(4), 041057 (2017)

[61] G. Chen, Spectral periodicity of the spinon continuum in quantum spin ice, Phys. Rev. B 96(8), 085136 (2017)

[62] L. Savary and L. Balents, Disorder-induced quantum spin liquid in spin ice pyrochlores, Phys. Rev. Lett. 118(8), 087203 (2017)

[63] G. Chen, Dirac’s “magnetic monopoles” in pyrochlore ice U(1) spin liquids: Spectrum and classification, Phys. Rev. B 96(19), 195127 (2017)

[64] S. H. Curnoe, Structural distortion and the spin liquid state in Tb₂Ti₂O₇, Phys. Rev. B 78(9), 094418 (2008)

[65] S. Onoda, Effective quantum pseudospin-1/2 model for Yb pyrochlore oxides, J. Phys.: Conf. Ser. 320, 012065 (2011)

[66] J. W. Krizan and R. J. Cava, NaCaNi2F7: A single-crystal high-temperature pyrochlore antiferromagnet with S= 1 Ni²⁺, Phys. Rev. B 92, 014406 (2015)

[67] J. W. Krizan and R. J. Cava, NaCaCo₂F₇: A singlecrystal high-temperature pyrochlore antiferromagnet, Phys. Rev. B 89(21), 214401 (2014)

[68] K. A. Ross, J. M. Brown, R. J. Cava, J. W. Krizan, S. E. Nagler, J. A. Rodriguez-Rivera, and M. B. Stone, Singleion properties of the S_eff= 1/2 XY antiferromagnetic pyrochlores NaA′Co₂F₇ (A′= Ca²⁺,Sr²⁺), Phys. Rev. B 95(14), 144414 (2017)

[69] M. B. Sanders, J. W. Krizan, K. W. Plumb, T. M. Mc-Queen, and R. J. Cava, NaSrMn₂F₇, NaCaFe₂F₇, and NaSrFe₂F₇: Novel single crystal pyrochlore antiferromagnets, J. Phys.: Condens. Matter 29(4), 045801 (2017)

[70] W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Balents, Correlated quantum phenomena in the strong spin–orbit regime, Annu. Rev. Condens. Matter Phys. 5(1), 57 (2014)

[71] F. Y. Li and G. Chen, Competing phases and topological excitations of spin-1 pyrochlore antiferromagnets, Phys. Rev. B 98(4), 045109 (2018)

[72] M. Elhajal, B. Canals, R. Sunyer, and C. Lacroix, Ordering in the pyrochlore antiferromagnet due to Dzyaloshinsky–Moriya interactions, Phys. Rev. B 71(9), 094420 (2005)

[73] S. Maekawa, T. Tohyama, S. Barnes, S. Ishihara, W. Koshibae, and G. Khaliullin, Physics of Transition Metal Oxides, Springer, 2004

[74] A. Joshi, M. Ma, F. Mila, D. N. Shi, and F. C. Zhang, Elementary excitations in magnetically ordered systems with orbital degeneracy, Phys. Rev. B 60(9), 6584 (1999)

[75] Y. Q. Li, M. Ma, D. N. Shi, and F. C. Zhang, SU(4) theory for spin systems with orbital degeneracy, Phys. Rev. Lett. 81(16), 3527 (1998)

[76] S. E. Palmer and J. T. Chalker, Order induced by dipolar interactions in a geometrically frustrated antiferromagnet, Phys. Rev. B 62(1), 488 (2000)

[77] A. Poole, A. S. Wills, and E. Lelièvre-Berna, Magnetic ordering in the XY pyrochlore antiferromagnet Er₂Ti₂O₇: A spherical neutron polarimetry study, J. Phys.: Condens. Matter 19(45), 452201 (2007)

[78] L. Savary, K. A. Ross, B. D. Gaulin, J. P. C. Ruff, and L. Balents, Order by quantum disorder in Er₂Ti₂O₇, Phys. Rev. Lett. 109(16), 167201 (2012)

[79] M. E. Zhitomirsky, M. V. Gvozdikova, P. C. W. Holdsworth, and R. Moessner, Quantum order by disorder and accidental soft mode in Er₂Ti₂O₇, Phys. Rev. Lett. 109(7), 077204 (2012)

[80] M. E. Zhitomirsky, P. C. W. Holdsworth, and R. Moessner, Nature of finite-temperature transition in anisotropic pyrochlore Er₂Ti₂O₇, Phys. Rev. B 89(14), 140403 (2014)

[81] F. Y. Li, Y. D. Li, Y. B. Kim, L. Balents, Y. Yu, and G. Chen, Weyl magnons in breathing pyrochlore antiferromagnets, Nat. Commun. 7(1), 12691 (2016)

[82] A. Mook, J. Henk, and I. Mertig, Tunable magnon Weyl points in ferromagnetic pyrochlores, Phys. Rev. Lett. 117(15), 157204 (2016)

[83] F. Y. Li, Y. D. Li, Y. Yu, A. Paramekanti, and G. Chen, Kitaev materials beyond iridates: Order by quantum disorder and Weyl magnons in rare-earth double perovskites, Phys. Rev. B 95(8), 085132 (2017)

[84] S. A. Owerre, Magnonic analogs of topological Dirac semimetals, J. Phys. Commun. 1(2), 025007 (2017)

[85] S. A. Owerre, Topological magnetic excitations on the distorted Kagomé antiferromagnets: Applications to volborthite, vesignieite, and edwardsite, EPL 117(3), 37006 (2017)

[86] J. Fransson, A. M. Black-Schaffer, and A. V. Balatsky, Magnon Dirac materials, Phys. Rev. B 94(7), 075401 (2016)

[87] K. Li, C. Li, J. Hu, Y. Li, and C. Fang, Dirac and nodal line magnons in three-dimensional antiferromagnets, Phys. Rev. Lett. 119(24), 247202 (2017)

[88] X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates, Phys. Rev. B 83(20), 205101 (2011)

[89] A. A. Burkov, M. D. Hook, and L. Balents, Topological nodal semimetals, Phys. Rev. B 84(23), 235126 (2011)

[90] R. Moessner and J. T. Chalker, Low-temperature properties of classical geometrically frustrated antiferromagnets, Phys. Rev. B 58(18), 12049 (1998)

[91] R. Moessner and J. T. Chalker, Properties of a classical spin liquid: The Heisenberg pyrochlore antiferromagnet, Phys. Rev. Lett. 80(13), 2929 (1998)

[92] R. Kmieć, Z. Świątkowska, J. Gurgul, M. Rams, A. Zarzycki, and K. Tomala, Investigation of the magnetic properties of Y₂Ru₂O₇ by ⁹⁹Ru Mössbauer spectroscopy, Phys. Rev. B 74(10), 104425 (2006)

[93] S. Lee, J. G. Park, D. T. Adroja, D. Khomskii, S. Streltsov, K. A. McEwen, H. Sakai, K. Yoshimura, V. I. Anisimov, D. Mori, R. Kanno, and R. Ibberson, Spin gap in Tl₂Ru₂O₇ and the possible formation of Haldane chains in three-dimensional crystals,Nat. Mater. 5(6), 471 (2006)

[94] S. M. Perez1, R. Cobas, J. M. Cadogan, J. A. Aguiar, C. Frontera, T. Puig, G. Long, M. DeMarco, D. Coffey, and X. Obradors, Anomalous electronic and magnetic properties of the Eu₂Ru₂O₇ pyrochlore, J. Appl. Phys. 113, 17E102 (2013)

[95] M. Tachibanaa, Heat capacity of pyrochlore Pr₂Ru₂O₇, J. Appl. Phys. 101, 09D502 (2007)

[96] S. Zouari, R. Ballou, A. Cheikhrouhou, and P. Strobel, Structural and magnetic properties of the (Bi_2−xPr_x)Ru₂O₇ pyrochlore solid solution (0≤x≤2), J. Alloys Compd. 476(1–2), 43 (2009)

[97] M. W. Gaultois, P. T. Barton, C. S. Birkel, L. M. Misch, E. E. Rodriguez, G. D. Stucky, and R. Seshadri, Structural disorder, magnetism, and electrical and thermoelectric properties of pyrochlore Nd₂Ru₂O₇, J. Phys.: Condens. Matter 25(18), 186004 (2013)

[98] J. Gurgul, M. Rams, Z. Świątkowska, R. Kmieć, and K. Tomala, Bulk magnetic measurements and ⁹⁹Ru and ¹⁵⁵Gd Mössbauer spectroscopies of Gd₂Ru₂O₇, Phys. Rev. B 75(6), 064426 (2007)

[99] L. J. Chang, M. Prager, J. Perβon, J. Walter, E. Jansen, Y. Y. Chen, and J. S. Gardner, Magnetic order in the double pyrochlore Tb₂Ru₂O₇, J. Phys.: Condens. Matter 22(7), 076003 (2010)

[100] Z. C. Xu, M. F. Liu, L. Lin, H. Liu, Z. B. Yan, and J. M. Liu, Experimental observations of ferroelectricity in double pyrochlore Dy₂Ru₂O₇, Front. Phys. 9(1), 82 (2014)

[101] C. R. Wiebe, J. S. Gardner, S. J. Kim, G. M. Luke, A. S. Wills, B. D. Gaulin, J. E. Greedan, I. Swainson, Y. Qiu, and C. Y. Jones, Magnetic ordering in the spinice candidate Ho₂Ru₂O₇, Phys. Rev. Lett. 93(7), 076403 (2004)

[102] N. Taira, M. Wakeshima, and Y. Hinatsu, Magnetic susceptibility and specific heat studies on heavy rare earth ruthenate pyrochlores R₂Ru₂O₇ (R= Gd–Yb), J. Mater. Chem. 12(5), 1475 (2002)

[103] J. S. Gardner and G. Ehlers, Magnetic order and crystal field excitations in Er₂Ru₂O₇: A neutron scattering study, J. Phys.: Condens. Matter 21(43), 436004 (2009)

[104] N. Taira, M. Wakeshima, Y. Hinatsu, A. Tobo, and K. Ohoyama, Magnetic structure of pyrochlore-type Er₂Ru₂O₇, J. Solid State Chem. 176(1), 165 (2003)

[105] A. Keren and J. S. Gardner, Frustration driven lattice distortion: An NMR investigation of Y₂Mo₂O₇, Phys. Rev. Lett. 87(17), 177201 (2001)

[106] P. M. M. Thygesen, J. A. M. Paddison, R. Zhang, K. A. Beyer, K. W. Chapman, H. Y. Playford, M. G. Tucker, D. A. Keen, M. A. Hayward, and A. L. Goodwin, Orbital dimer model for the spin-glass state in Y₂Mo₂O₇, Phys. Rev. Lett. 118(6), 067201 (2017)

[107] H. J. Silverstein, K. Fritsch, F. Flicker, A. M. Hallas, J. S. Gardner, Y. Qiu, G. Ehlers, A. T. Savici, Z. Yamani, K. A. Ross, B. D. Gaulin, M. J. P. Gingras, J. A. M. Paddison, K. Foyevtsova, R. Valenti, F. Hawthorne, C. R. Wiebe, and H. D. Zhou, Liquidlike correlations in single-crystalline Y₂Mo₂O₇: An unconventional spin glass,Phys. Rev. B 89(5), 054433 (2014)

[108] S. R. Dunsiger, R. F. Kiefl, K. H. Chow, B. D. Gaulin, M. J. P. Gingras, J. E. Greedan, A. Keren, K. Kojima, G. M. Luke, W. A. MacFarlane, N. P. Raju, J. E. Sonier, Y. J. Uemura, and W. D. Wu, Muon spin relaxation investigation of the spin dynamics of geometrically frustrated antiferromagnets Y₂Mo₂O₇ and Tb₂Mo₂O₇, Phys. Rev. B 54(13), 9019 (1996)

[109] L. Clark, G. J. Nilsen, E. Kermarrec, G. Ehlers, K. S. Knight, A. Harrison, J. P. Attfield, and B. D. Gaulin, From spin glass to quantum spin liquid ground states in molybdate pyrochlores, Phys. Rev. Lett. 113(11), 117201 (2014)

[110] Y. Jiang, A. Huq, C. H. Booth, G. Ehlers, J. E. Greedan, and J. S. Gardner, Order and disorder in the local and long-range structure of the spin-glass pyrochlore, Tb₂Mo₂O₇, J. Phys.: Condens. Matter 23(16), 164214 (2011)

[111] G. Ehlers, J. E. Greedan, J. R. Stewart, K. C. Rule, P. Fouquet, A. L. Cornelius, C. Adriano, P. G. Pagliuso, Y. Qiu, and J. S. Gardner, High-resolution neutron scattering study of Tb₂Mo₂O₇: A geometrically frustrated spin glass, Phys. Rev. B 81(22), 224405 (2010)

[112] D. K. Singh, J. S. Helton, S. Chu, T. H. Han, C. J. Bonnoit, S. Chang, H. J. Kang, J. W. Lynn, and Y. S. Lee, Spin correlations in the geometrically frustrated pyrochlore Tb₂Mo₂O₇, Phys. Rev. B78(22), 220405 (2008)

[113] K. W. Plumb, H. J. Changlani, A. Scheie, S. Zhang, J. W. Krizan, J. A. Rodriguez-Rivera, Y. Qiu, B. Winn, R. J. Cava, and C. L. Broholm, Continuum of quantum fluctuations in a three-dimensional s= 1 heisenberg magnet, arXiv: 1711.07509 (2017)

[114] T. Moriya, Anisotropic superexchange interaction and weak ferromagnetism, Phys. Rev. 120(1), 91 (1960)

[115] D. L. Bergman, R. Shindou, G. A. Fiete, and L. Balents, Models of degeneracy breaking in pyrochlore antiferromagnets, Phys. Rev. B 74(13), 134409 (2006)

[116] K. Penc, N. Shannon, and H. Shiba, Half-magnetization plateau stabilized by structural distortion in the antiferromagnetic Heisenberg model on a pyrochlore lattice, Phys. Rev. Lett. 93(19), 197203 (2004)

[117] G. Chen and L. Balents, Spin–orbit coupling in d² ordered double perovskites, Phys. Rev. B 84(9), 094420 (2011)

[118] Z. Y. Zhao, S. Calder, A. A. Aczel, M. A. McGuire, B. C. Sales, D. G. Mandrus, G. Chen, N. Trivedi, H. D. Zhou, and J. Q. Yan, Fragile singlet ground-state magnetism in the pyrochlore osmates R₂Os₂O₇ (R= Y and Ho), Phys. Rev. B 93(13), 134426 (2016)

[119] G. Khaliullin, Excitonic magnetism in Van Vleck–type d⁴ Mott insulators, Phys. Rev. Lett. 111(19), 197201 (2013)

[120] A. Georges, L. Medici, and J. Mravlje, Strong correlations from Hund’s coupling, Annu. Rev. Condens. Matter Phys. 4(1), 137 (2013)

[121] K. I. Kugel and D. I. Khomskii, The Jahn–Teller effect and magnetism: Transition metal compounds, Sov. Phys. Usp. 25(4), 231 (1982)

[122] S. H. Lee, D. Louca, H. Ueda, S. Park, T. J. Sato, M. Isobe, Y. Ueda, S. Rosenkranz, P. Zschack, J. Íñiguez, Y. Qiu, and R. Osborn, Orbital and spin chains in ZnV₂O₄, Phys. Rev. Lett. 93(15), 156407 (2004)

[123] T. Maitra and R. Valentí, Orbital order in ZnV₂O₄, Phys. Rev. Lett. 99(12), 126401 (2007)

[124] G. Giovannetti, A. Stroppa, S. Picozzi, D. Baldomir, V. Pardo, S. Blanco-Canosa, F. Rivadulla, S. Jodlauk, D. Niermann, J. Rohrkamp, T. Lorenz, S. Streltsov, D. I. Khomskii, and J. Hemberger, Dielectric properties and magnetostriction of the collinear multiferroic spinel CdV₂O₄, Phys. Rev. B 83(6), 060402 (2011)

[125] D. I. Khomskii and T. Mizokawa, Orbitally induced peierls state in spinels, Phys. Rev. Lett. 94(15), 156402 (2005)

[126] S. Niitaka, H. Ohsumi, K. Sugimoto, S. Lee, Y. Oshima, K. Kato, D. Hashizume, T. Arima, M. Takata, and H. Takagi, A-type antiferro-orbital ordering with I4₁/asymmetry and geometrical frustration in the spinel vanadate MgV₂O₄, Phys. Rev. Lett. 111(26), 267201 (2013)

[127] E. M. Wheeler, B. Lake, A. T. M. N. Islam, M. Reehuis, P. Steffens, T. Guidi, and A. H. Hill, Spin and orbital order in the vanadium spinel MgV₂O₄, Phys. Rev. B 82(14), 140406 (2010)

RIGHTS & PERMISSIONS

2020 Higher Education Press