Theory of multiple-phase competition in pyrochlore magnets with anisotropic exchange with application to Yb 2 Ti 2 O 7 , Er 2 Ti 2 O 7 , and Er 2 Sn 2 O 7

The family of magnetic rare-earth pyrochlore oxides R2M2O7 plays host to a diverse array of exotic phenomena, driven by the interplay between geometrical frustration and spin-orbit interaction, which leads to anisotropy in both magnetic moments and their interactions. In this article we establish a general, symmetry-based theory of pyrochlore magnets with anisotropic exchange interactions. Starting from a very general model of nearest-neighbor exchange between Kramers ions, we find four distinct classical ordered states, all with q=0, competing with a variety of spin liquids and unconventional forms of magnetic order. The finite-temperature phase diagram of this model is determined by Monte Carlo simulation, supported by classical spin-wave calculations. We pay particular attention to the region of parameter space relevant to the widely studied materials Er2Ti2O7, Yb2Ti2O7, and Er2Sn2O7. We find that many of the most interesting properties of these materials can be traced back to the “accidental” degeneracies where phases with different symmetries meet. These include the ordered ground-state selection by fluctuations in Er2Ti2O7, the dimensional reduction observed in Yb2Ti2O7, and the lack of reported magnetic order in Er2Sn2O7. We also discuss the application of this theory to other pyrochlore oxides.

[1]  M. Lees,et al.  Static magnetic moments revealed by muon spin relaxation and thermodynamic measurements in the quantum spin ice Yb2Ti2O7 , 2013, 1311.5526.

[2]  L. Jaubert,et al.  Living on the edge : ground-state selection in quantum spin-ice pyrochlores , 2013, 1311.3501.

[3]  M. Gingras,et al.  Thermodynamic and single-ion properties of Tb3+ within the collective paramagnetic-spin liquid state of the frustrated pyrochlore antiferromagnet Tb2Ti2O7 , 2000 .

[4]  Antiferromagnetic order in the pyrochlores R$_{2}$Ge$_{2}$O$_{7}$ (R = Er, Yb) , 2015 .

[5]  J. Alam,et al.  Magnetic ground-state of strongly frustrated pyrochlore anti-ferromagnet Er2Sn2O7 , 2014 .

[6]  T. Kimura,et al.  Experimental realization of a quantum breathing pyrochlore antiferromagnet , 2014, 1404.6439.

[7]  P. Bonville,et al.  Low temperature magnetic properties of geometrically frustrated Gd2Sn2O7 and Gd2Ti2O7 , 2003 .

[8]  M. Gingras,et al.  Thermodynamic properties of Yb 2 Ti 2 O 7 pyrochlore as a function of temperature and magnetic field: Validation of a quantum spin ice exchange Hamiltonian , 2012, 1211.5934.

[9]  Yong Baek Kim,et al.  Unusual liquid state of hard-core bosons on the pyrochlore lattice. , 2007, Physical review letters.

[10]  S. Petit,et al.  Spin dynamics in the presence of competing ferromagnetic and antiferromagnetic correlations inYb2Ti2O7 , 2015, 1506.01729.

[11]  H. Zhou,et al.  Fragile singlet ground-state magnetism in the pyrochlore osmates R2Os2O7 ( R=Y and Ho) , 2016, 1602.05121.

[12]  L. Corruccini,et al.  Evidence for gapped spin-wave excitations in the frustrated Gd2Sn2O7 pyrochlore antiferromagnet from low-temperature specific heat measurements. , 2007, Physical review letters.

[13]  A. Amato,et al.  Nd2Sn2O7: an all-in-all-out pyrochlore magnet with no divergence-free field and anomalously slow paramagnetic spin dynamics , 2015, 1511.00987.

[14]  E. Lhotel,et al.  Candidate quantum spin ice in the pyrochlore Pr 2 Hf 2 O 7 , 2016, 1601.05071.

[15]  J. Copley,et al.  Order by disorder spin wave gap in the XY pyrochlore magnet Er2Ti2O7. , 2014, Physical review letters.

[16]  P. McClarty,et al.  Order induced by dilution in pyrochlore XY antiferromagnets , 2014, 1408.7119.

[17]  S. H. Cumoe.  Quantum spin configurations in Tb2Ti2O7 , 2007 .

[18]  Chandra,et al.  Possible spin-liquid state at large S for the frustrated square Heisenberg lattice. , 1988, Physical review. B, Condensed matter.

[19]  M. Gingras,et al.  Quantum spin ice: a search for gapless quantum spin liquids in pyrochlore magnets , 2013, Reports on progress in physics. Physical Society.

[20]  A. Kasahara,et al.  Quadrupole Order in the Frustrated Pyrochlore Tb , 2016 .

[21]  G. Marsaglia.  Choosing a Point from the Surface of a Sphere , 1972 .

[22]  M. Lees,et al.  First-order magnetic transition in Yb2Ti2O7 , 2014 .

[23]  L. Pauling.  The Structure and Entropy of Ice and of Other Crystals with Some Randomness of Atomic Arrangement , 1935 .

[24]  R. Moessner,et al.  Spin Ice, Fractionalization, and Topological Order , 2011, 1112.3793.

[25]  J. Copley,et al.  Antiferromagnetic spin ice correlations at ((1)/(2),(1)/(2),(1)/(2)) in the ground state of the pyrochlore magnet Tb 2 Ti 2 O 7 , 2012, 1210.1242.

[26]  J. Bouchaud,et al.  Transitions and Spin Dynamics at Very Low Temperature in the Pyrochlores Yb2Ti2O7 and Gd2Sn2O7 , 2003, cond-mat/0306470.

[27]  M. Gingras,et al.  Rods of neutron scattering intensity in Yb2Ti2O7: compelling evidence for significant anisotropic exchange in a magnetic pyrochlore oxide. , 2010, Physical review letters.

[28]  N. Shannon,et al.  Ground State Selection and Spin-Liquid Behaviour in the Classical Heisenberg Model on the Breathing Pyrochlore Lattice , 2015, 1506.01822.

[29]  M. Gingras,et al.  Magnetic Pyrochlore Oxides , 2009, 0906.3661.

[30]  R. Moessner.  Relief and generation of frustration in pyrochlore magnets by single-ion anisotropy , 1998 .

[31]  J. Attfield,et al.  Long-range antiferromagnetic order in the frustrated XY pyrochlore antiferromagnet Er2Ge2O7 , 2014 .

[32]  A. S. Wills,et al.  Magnetic ordering in Gd2Sn2O7: the archetypal Heisenberg pyrochlore antiferromagnet , 2006, cond-mat/0602138.

[33]  T. Rojo,et al.  CdEr2Se4: a new erbium spin ice system in a spinel structure. , 2010, Physical review letters.

[34]  M. Gingras,et al.  Dynamically induced frustration as a route to a quantum spin ice state in Tb2Ti2O7 via virtual crystal field excitations and quantum many-body effects. , 2006, Physical review letters.

[35]  C. Lacroix,et al.  Ising-like order by disorder in the pyrochlore antiferromagnet with Dzyaloshinskii-Moriya interactions , 2008, 0807.0352.

[36]  L. Balents,et al.  Coulombic quantum liquids in spin-1/2 pyrochlores. , 2011, Physical review letters.

[37]  T. Ziman,et al.  Magnetic Order and Disorder in the Frustrated Quantum Heisenberg Antiferromagnet in Two Dimensions , 1994, cond-mat/9402061.

[38]  R. Moessner,et al.  Acoustic signatures of the phases and phase transitions in Yb2Ti2O7 , 2015, 1508.00925.

[39]  B. Schmidt,et al.  Finite temperature properties and frustrated ferromagnetism in a square lattice Heisenberg model , 2003, cond-mat/0312160.

[40]  Andrew S. Wills,et al.  Magnetic ordering in the XY pyrochlore antiferromagnet Er2Ti2O7: a spherical neutron polarimetry study , 2007 .

[41]  G. Granroth,et al.  Neutron spectroscopic study of crystal field excitations in Tb2Ti2O7 and Tb2Sn2O7 , 2013, 1310.3264.

[42]  Y. Kao,et al.  Higgs transition from a magnetic Coulomb liquid to a ferromagnet in Yb 2 Ti 2 o 7 , 2012 .

[43]  K. Kanki,et al.  Efficiency of the microcanonical over-relaxation algorithm for vector spins analyzing first and second order transitions , 2005 .

[44]  Gang Chen,et al.  Symmetry enriched U(1) topological orders for dipole-octupole doublets on a pyrochlore lattice , 2016, 1607.02287.

[45]  Masafumi Ito,et al.  Ferromagnetic transition of pyrochlore compound Yb2Ti2O7 , 2003 .

[46]  M. Gingras,et al.  Bosonic many-body theory of quantum spin ice , 2014, 1409.5127.

[47]  J. Gaudet,et al.  XY antiferromagnetic ground state in the effective S = 1 2 pyrochlore Yb 2 Ge 2 O 7 , 2016, 1602.03833.

[48]  A. Hallas,et al.  Frustration under pressure: Exotic magnetism in new pyrochlore oxides , 2015 .

[49]  S. Petit,et al.  Order by virtual crystal field fluctuations in pyrochlore XY antiferromagnets , 2015, 1510.04292.

[50]  O. Tchernyshyov,et al.  Low-energy electrodynamics of novel spin excitations in the quantum spin ice Yb2Ti2O7 , 2014, Nature Communications.

[51]  M. Fisher,et al.  Pyrochlore photons: The U ( 1 ) spin liquid in a S = 1 2 three-dimensional frustrated magnet , 2003, cond-mat/0305401.

[52]  J. Oitmaa,et al.  Are Multiphase Competition and Order by Disorder the Keys to Understanding Yb(2)Ti(2)O(7)? , 2015, Physical review letters.

[53]  J. Attfield,et al.  Chemical pressure effects on magnetism in the quantum spin liquid candidates Yb2X2O7 (X = Sn, Ti, Ge) , 2014 .

[54]  G. Chern.  Novel Magnetic Orders and Ice Phases in Frustrated Kondo-Lattice Models , 2015 .

[55]  谢华,et al.  Er 2 Ti 2 O 7 烧绿石基玻璃陶瓷固化体的制备工艺研究 , 2019 .

[56]  L. Balents,et al.  Quantum Excitations in Quantum Spin Ice , 2011, 1107.0761.

[57]  E. Ressouche,et al.  Magnetic order, magnetic correlations, and spin dynamics in the pyrochlore antiferromagnet Er2Ti2O7 , 2012, 1209.6153.

[58]  M. Green,et al.  Order in the Heisenberg pyrochlore: The magnetic structure of Gd2Ti2O7 , 2001, cond-mat/0105043.

[59]  P. Sindzingre,et al.  Novel phases in a square-lattice frustrated ferromagnet : 1/3-magnetisation plateau, helicoidal spin-liquid and vortex crystal , 2015, 1509.03850.

[60]  B. Roessli,et al.  A novel type of splayed ferromagnetic order observed in Yb2Ti2O7 , 2016, Journal of physics. Condensed matter : an Institute of Physics journal.

[61]  G. Ehlers,et al.  Gapless quantum excitations from an icelike splayed ferromagnetic ground state in stoichiometric Yb 2 Ti 2 O 7 , 2016, 1601.05139.

[62]  P. Fulde,et al.  Quantum ice: a quantum Monte Carlo study. , 2011, Physical review letters.

[63]  S. Petit,et al.  Magnetization process in Er2Ti2O7 at very low temperature , 2013, Journal of physics. Condensed matter : an Institute of Physics journal.

[64]  L. Balents.  Spin liquids in frustrated magnets , 2010, Nature.

[65]  J. Rodríguez-Carvajal,et al.  Ordered Spin Ice State and Magnetic Fluctuations in Tb~2Sn~2O~7 , 2005 .

[66]  R. Moessner,et al.  Nature of finite-temperature transition in anisotropic pyrochlore Er 2 Ti 2 O 7 , 2014, 1402.4689.

[67]  M. Gingras,et al.  Ground state phase diagram of generic XY pyrochlore magnets with quantum fluctuations , 2013, 1311.3299.

[68]  P. Bonville,et al.  The crystal field and exchange interactions in Yb2Ti2O7 , 2001 .

[69]  T. Sakakibara,et al.  Low-temperature magnetic properties of pyrochlore stannates , 2002 .

[70]  R. Cava,et al.  Static and dynamic X Y -like short-range order in a frustrated magnet with exchange disorder , 2015, 1503.04494.

[71]  I. Swainson,et al.  Cooperative Paramagnetism in the Geometrically Frustrated Pyrochlore Antiferromagnet Tb 2 Ti 2 O 7 , 1999 .

[72]  M. Rams,et al.  First-order transition in the spin dynamics of geometrically frustrated Yb2Ti2O7. , 2002, Physical review letters.

[73]  M. Gingras,et al.  Order-by-disorder in the XY pyrochlore antiferromagnet , 2011, 1108.6053.

[74]  M. Tachibana,et al.  Relief of frustration in the Heisenberg pyrochlore antiferromagnet Gd 2 Pt 2 O 7 , 2016, 1607.08657.

[75]  M. Lees,et al.  Structural and magnetic investigations of single-crystalline neodymium zirconate pyrochlore Nd2Zr2O7 , 2015 .

[76]  M. Lumsden,et al.  Anisotropic Exchange within Decoupled Tetrahedra in the Quantum Breathing Pyrochlore Ba_{3}Yb_{2}Zn_{5}O_{11}. , 2016, Physical review letters.

[77]  R. Moessner,et al.  Quantum order by disorder and accidental soft mode in Er2Ti2O7. , 2012, Physical review letters.

[78]  V. Pomjakushin,et al.  Candidate Quantum Spin Liquid in the Ce3+ Pyrochlore Stannate Ce2Sn2O7. , 2015, Physical review letters.

[79]  M. Gingras,et al.  Energetic selection of ordered states in a model of the Er2Ti2O7 frustrated pyrochlore XY antiferromagnet , 2008, 0810.2483.

[80]  Koutarou D. Kimura,et al.  Quantum fluctuations in spin-ice-like Pr2Zr2O7 , 2013, Nature Communications.

[81]  J. Attfield,et al.  Magnetic frustration in lead pyrochlores , 2015, 1502.00721.

[82]  S. Bramwell,et al.  GEOMETRICAL FRUSTRATION IN THE FERROMAGNETIC PYROCHLORE HO2TI2O7 , 1997 .

[83]  J. Delrieu,et al.  Magnetism in solidHe3 , 1983 .

[84]  M. Gingras,et al.  Vindication of Yb2Ti2O7 as a model exchange quantum spin ice. , 2012, Physical review letters.

[85]  P. Sindzingre,et al.  Nematic order in square lattice frustrated ferromagnets. , 2005, Physical review letters.

[86]  Ludovic D. C. Jaubert,et al.  Magnetic-Moment Fragmentation and Monopole Crystallization , 2013 .

[87]  L. Jaubert,et al.  A spin-liquid with pinch-line singularities on the pyrochlore lattice , 2015, Nature Communications.

[88]  J. R. Stewart,et al.  LETTER TO THE EDITOR: Phase transitions, partial disorder and multi-k structures in Gd2Ti2O7 , 2004 .

[89]  T. Sakakibara,et al.  Possible observation of highly itinerant quantum magnetic monopoles in the frustrated pyrochlore Yb2Ti2O7 , 2016, Nature Communications.

[90]  L. Balents,et al.  Generic quantum spin ice , 2012, 1204.2262.

[91]  J. Gardner,et al.  Two-dimensional kagome correlations and field induced order in the ferromagnetic XY pyrochlore Yb2Ti2O7. , 2009, Physical review letters.

[92]  J. Attfield,et al.  Breathing pyrochlore lattice realized in A-site ordered spinel oxides LiGaCr4O8 and LiInCr4O8. , 2013, Physical review letters.

[93]  Gang Chen,et al.  Quantum spin ices and topological phases from dipolar-octupolar doublets on the pyrochlore lattice. , 2013, Physical review letters.

[94]  J. Rau,et al.  Magnitude of quantum effects in classical spin ices , 2015 .

[95]  L. Balents,et al.  Order by Quantum Disorder in Er _ { 2 } Ti _ { 2 } O _ { 7 } , 2014 .

[96]  S. Petit,et al.  Spin liquid correlations, anisotropic exchange, and symmetry breaking in Tb2Ti2O7 , 2012, 1210.1428.

[97]  A. Amato,et al.  Dynamical Splayed Ferromagnetic Ground State in the Quantum Spin Ice Yb2Sn2O7 , 2013 .

[98]  Wang,et al.  Replica Monte Carlo simulation of spin glasses. , 1986, Physical review letters.

[99]  P. Bonville,et al.  Ising versus XY anisotropy in frustrated R(2)Ti(2)O(7) compounds as "Seen" by Polarized Neutrons. , 2009, Physical review letters.

[100]  J. Gaudet,et al.  Universal dynamic magnetism in Yb pyrochlores with disparate ground states , 2016, 1602.09011.

[101]  S. Petit,et al.  Spin dynamics in the ordered spin ice Tb$_2$Sn$_2$O$_7$ , 2011, 1109.1960.

[102]  S. Onoda,et al.  Quantum fluctuations in the effective pseudospin- 1 2 model for magnetic pyrochlore oxides , 2010, 1011.4981.

[103]  N. Shannon,et al.  Seeing the light: Experimental signatures of emergent electromagnetism in a quantum spin ice , 2012, 1204.1325.

[104]  L. Balents,et al.  Spin liquid regimes at nonzero temperature in quantum spin ice , 2013, 1301.5643.

[105]  S. Petit,et al.  Observation of magnetic fragmentation in spin ice , 2016, Nature Physics.

[106]  R. Moessner,et al.  LOW-TEMPERATURE PROPERTIES OF CLASSICAL GEOMETRICALLY FRUSTRATED ANTIFERROMAGNETS , 1998, cond-mat/9807384.

[107]  M. Gingras,et al.  Fluctuation-driven selection at criticality in a frustrated magnetic system: the case of multiple-k partial order on the pyrochlore lattice. , 2013, Physical review letters.

[108]  J. Attfield,et al.  Incipient ferromagnetism in Tb2Ge2O7: application of chemical pressure to the enigmatic spin-liquid compound Tb2Ti2O7. , 2014, Physical review letters.