Topological Constraints and Defects in Spin Ice Ludovic D.C. Jaubert To cite this version: Ludovic D.C. Jaubert. Topological Constraints and Defects in Spin Ice. Condensed Matter [cond-mat]. Ecole normale supérieure de lyon - ENS LYON, 2009. English. tel-00462970 HAL Id: tel-00462970 https://tel.archives-ouvertes.fr/tel-00462970 Submitted on 10 Mar 2010 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License N◦ d’ordre : 528 N◦ attribu´epar la biblioth`eque : ENSL528 Th`ese en vue d’obtenir le grade de Docteur de l’Universit´ede Lyon - Ecole Normale Sup´erieure de Lyon Laboratoire de Physique, ENS Lyon Ecole Doctorale de Physique et d’Astrophysique de Lyon pr´esent´ee et soutenue publiquement le 23 / IX / 2009 par Ludovic D. C . Jaubert Topological Constraints and Defects in Spin Ice r´ealis´ee sous la direction de Peter C . W . Holdsworth Devant la commission d’examen form´ee de : M. Fabien Alet Membre/Rapporteur M. Leon Balents Membre M. Benjamin Canals Membre/Rapporteur M. Peter Holdsworth Directeur de th`ese M. Philippe Mendels Pr´esident M. Fr´ed´eric Mila Membre i Table of Contents Acknowledgements 1 I Introduction 5 I.1 MagneticMaterials ............................... 5 I.1.a Origin of magnetism in condensed matter . ... 6 I.1.b Phasetransitions ............................ 11 I.1.c 2-dimensional XY -model........................ 15 I.2 FrustrationinPhysics ............................ 18 I.2.a Randomness............................... 19 I.2.b Geometricalfrustration. 19 I.3 SpinIce ..................................... 23 I.3.a Origin .................................. 23 I.3.b DipolarInteractions . .. .. 28 I.3.c Magnetic Monopoles in Spin ice . 33 I.3.d Magneticfield.............................. 39 I.3.e Dynamics ................................ 46 I.3.f Spinicecandidates ........................... 48 I.4 Plan ....................................... 49 II Methods 51 II.1 Presentationofthemodel . .. 51 II.2 MonteCarlo................................... 53 II.3 Wormalgorithm................................. 57 II.4 EwaldMethod.................................. 62 II.4.a Generalpresentation . 62 II.4.b Analytical calculation of the series . ..... 64 II.4.c Numerical application to spin ice . ... 67 II.5 Transfermatrix ................................. 69 II.5.a One dimensional Ising model . 69 II.5.b 6-vertexmodel ............................. 71 II.6 Husimitree ................................... 74 II.6.a Betheapproximation . 74 II.6.b Bethelattice............................... 75 II.6.c Husimitree ............................... 77 ii TABLE OF CONTENTS II.6.d Recursion on the Husimi tree for spin ice . .... 78 III Topological phase transitions 81 III.1ConstrainedManifold. ... 81 III.1.a Natureofthetransitions . .. 81 III.1.b Topologicalconstraints . ... 84 III.2 Kasteleyn transition: δ =0........................... 88 III.2.aHusimiTree ............................... 89 III.2.bWormalgorithm ............................ 92 III.2.c Upper-critical dimension . ... 94 III.2.dWorldlinesforbosons . 95 III.2.e Experiments............................... 96 III.3 A tuneable Kasteleyn transition: h> 0, δ> 0................ 98 III.4 KDP transition: h =0 .............................101 III.4.a Apparent 1st ordertransition . .101 III.4.b Multicritical Point of -order.....................106 ∞ III.4.c Rigourous proof: Transfer Matrix method . .114 III.4.d Mapping onto a quantum phase transition . .122 III.4.e Correlations ...............................125 III.4.f Experiments...............................126 IV Constrained Monopoles Dynamics 133 IV.1Presentationoftheproblem . 133 IV.1.a Magneticmonopoles . .133 IV.1.bDynamicsinspinice . .136 IV.2 Deconfined quasi-particles . .137 IV.2.a Quantumtunnelling . .138 IV.2.b Arrheniusargument . .139 IV.3Spinfreezing................................... 142 IV.3.a Dipolarspinice .............................142 IV.3.b Magnetic monopoles and Dirac strings . 143 IV.3.c Comparisontoexperiments . 148 IV.4Monopolesinafield ..............................153 V Zero Field Susceptibility in Ho2Ti2O7 155 V.1 An appealing experimental result . .155 V.2 Husimitree ...................................156 V.3 Q-dependance of the susceptibility . 160 VI Conclusion 165 A Generalisation of Pauling’s argument 169 TABLE OF CONTENTS iii B Demagnetisation Effect 171 C Neutron Scattering & Pinch Points 173 iv List of Figures I.1 Hydrogenmolecule .............................. 8 I.2 GibbsenergyforLandautheory. 13 I.3 Signature of phase transitions in the susceptibility . ........... 16 I.4 ExcitationsoutoftheXYgroundstate. ... 17 I.5 Frustration .................................. 18 I.6 Frustratedlattices .............................. 20 I.7 Signature of frustration in the susceptibility . ......... 21 I.8 Pyrochlorelattice............................... 24 I.9 Models equivalent to spin ice . 26 I.10 Degeneracyinspinice ............................ 27 I.11 Zero-point entropy in Dy2Ti2O7 ....................... 29 I.12 Hyperfine coupling in Ho2Ti2O7 ....................... 30 I.13 Scattering function in spin ice . .... 31 I.14 Phase transition in dipolar spin ice . ..... 33 I.15 Ground state of the dipolar spin ice model . ..... 34 I.16 Coulombpotential .............................. 38 I.17 Dumbbellmodel ............................... 38 I.18 Groundstateina110field. .. .. 41 I.19 Phase diagram of dipolar spin ice in a 110 field . ..... 42 I.20 GroundstateofKagomeIce . 43 I.21 Phase Diagram and Kagome Ice plateau in a 111 field . ..... 44 I.22 Decoupled spin in a 112-tilted field . .... 45 I.23 Spinicedynamics............................... 46 II.1 Presentationofthemodel . 52 II.2 Degeneracy lift of the 2 in - 2 out tetrahedra in a 001 magnetic field . 52 II.3 Degeneracy lift of the 2 in - 2 out tetrahedra with bond distortion. 53 II.4 LoopsfromtheWormalgorithm . 58 II.5 Mapping from a spin configuration to an ensemble of strings ....... 59 II.6 IllustrationoftheEwaldsum . .. 63 II.7 Periodic boundary conditions in the Ewald method . ....... 69 II.8 Isingchain................................... 70 II.9 6-vertexmodel ................................ 72 II.10 Configurationsofthe6-vertexmodel . ..... 73 LIST OF FIGURES v II.11 Betheapproximation . 75 II.12 Bethelattice ................................. 76 II.13 Husimitree .................................. 78 III.1 Constructionofastring . .. 83 III.2 Gibbspotential ................................ 85 III.3 Smallclosedloop ............................... 86 III.4 Kasteleyn transition for biomembranes . ....... 89 III.5 3d Kasteleyntransitioninspinice. 92 III.6 Finitesizeeffects .............................. 94 III.7 Logarithmic corrections at the upper-critical dimension.......... 95 III.8 Correlationsina001field . .. 96 III.9 Magnetisation in a 001 field for Dy2Ti2O7 and Ho2Ti2O7 ......... 98 III.10 Tuneable Kasteleyn transition . ...... 99 III.11 Logarithmic corrections at the upper-critical dimension . .100 III.12 Entropyontheconstrainedmanifold . ......105 III.13 KDPtransitionwithdefects . .106 III.14 Gibbs energy for Landau theory (multi-critical point)...........107 III.15 Equiprobability of the sectors . ......112 III.16 Energy E vs magnetisation M ........................113 III.17 Probability distribution of the energy . .........114 III.18 Densityoftetrahedra. 115 III.19 3d TransferMatrix ..............................116 III.20 2d projectionofthepyrochlorelattice. 118 III.21 MeanFieldcriticalcorrelations . .......126 III.22 Anisotropiccorrelations . .127 III.23 Finite size effects in the correlation . ........128 III.24 Uniaxialpressure . .. .. .. 130 III.25 Dy2Ti2O7 underuniaxialpressure . 131 IV.1 Coulombpotential ..............................135 IV.2 Emergenceofmonopoles . 136 IV.3 Experimentalrelaxationtime . 137 IV.4 Deconfinedmonopoles ............................140 IV.5 Densityofmonopoles............................. 144 IV.6 Creation/annihilation of a pair of quasi-particles . ............146 IV.7 Spin freezing in Dy2Ti2O7 ..........................150 IV.8 Influence of a magnetic field on the dynamics . .151 IV.9 Influence of dilution on the dynamics . .152 IV.10 Transientmonopolecurrent: . .154 IV.11 Monopoledensityprofile . 154 V.1 Experimental susceptibility . 156 V.2 Susceptibility crossover . 159 V.3 Scatteringfunction .............................. 162 vi LIST OF FIGURES V.4 Susceptibility χ(q,T ).............................163 vii List of Tables I.1 Groundstateelectronicconfiguration . ..... 25 I.2 Crystalfieldlevel............................... 25 I.3 Nearest neighbour interactions . .... 32 II.1 ProbabilitiesoftheWormalgorithm . .... 61 IV.1 Mapping from the dipolar spin ice to the dumbbell model . .......133 IV.2 Energy scales resulting from a single spin flip . .......141 Acknowledgements Ces trois derni`eres ann´ees ayant ´et´eassez pleines (voire intenses), cela risque de me pren- dre pas mal de temps pour remercier ou saluer tout ceux qui m’ont aid´eou que j’ai rencontr´eau fil de ma