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New Route to Frustration by Quantum Many-Body Effects in the Spin

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New Route to Frustration by Quantum Many-Body Effects in the Spin New Route to Frustration by Quantum Many-Body Effects in the Spin Liquid Pyrochlore Tb2Ti2O7 by Hamid Reza Molavian A thesis presented to the University of Waterloo in fulfilment of the thesis requirement for the degree of Doctor of Philosophy in Physics Waterloo, Ontario, Canada, 2007 c Hamid Reza Molavian 2007 AUTHOR'S DECLARATION FOR ELECTRONIC SUBMISSION OF A THESIS I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract In this thesis we investigate the frustrated spin liquid Tb2Ti2O7 theoretically. The low-energy effective Hamiltonian of this compound is derived by integrating out the excited crystal field states. It is shown that the pairwise interaction in the effective Hamiltonian is renormalized by all the other Tb3+ ions in the sys- tem and dynamically generate frustration. The phase diagram of Tb2Ti2O7 in the single tetrahedron approximation is calculated. It is shown that Tb2Ti2O7 is in a singlet state which is a linear combination of all frustrated two-in/two-out states. Motivated by experimental results, the diffuse neutron scattering of Tb2Ti2O7 is obtained within the single tetrahedron approximation. This diffuse neutron scat- tering captures semi-quantitatively most of the experimental neutron scattering features. The magnetization of Tb2Ti2O7 in the single tetrahedron approximation is calculated. Two experiments based on diffuse neutron scattering and magneti- zation in high symmetry directions are proposed to verify the spin ice like ground state of Tb2Ti2O7. iii Acknowledgements I would like to thank my supervisor Michel Gingras for his enthusiasm, thought- ful suggestions and continual support of this project. I would like to acknowledge my Ph.D. advisory committee Tom Devereaux, Jan Kycia, Bill Power, Robert Mann and Achim Kempf for their useful suggestions and comments during my study at Waterloo and also I would like to thank Prof. Oleg Tchernyshyov who accepted to be my Ph.D external examinar and for his useful suggestions on the thesis. During this project I also benefitted from many useful discussions with Bruce Gaulin and Jason Gardner and here I would like to thank them. I wish to acknowledge Prof. Yoshiteru Maeno and Dr. Ryuji Higashinaka for the stimulating collaboration from which the experimental data presented in Appendix A originate. I would like to give my special thanks to Samad Bazargan, Azad Qazi-zade and Sattar Taheri-Araghi for their help in the final revision of the thesis. My special thank to my colleagues Matthew Enjalran, Adrian Del Maestro , Jean-Yves Delannoy, Jacob Ruff, Taras Yavors'kii, Jeffrey Quilliam, Tom Fennell, Ali Tabei, Francois Vernay, Pawel Stasiak, Mahmoud Ghaznavi, Roger Melko, Mar- tin Weigel, Mohammad Kohandel, Farhad Shahbazi, Koroush Afrousheh, and Jacob Cosman. iv Dedicated to My Parents, v Contents 1 Introduction 1 1.1 Frustration . 1 1.2 Spin Ice Systems Ho2Ti2O7 and Dy2Ti2O7 . 8 1.3 Spin Liquid Tb2Ti2O7 . 14 1.4 Structure of the Thesis . 25 2 Crystal Field Effect in Tb2Ti2O7 28 2.1 Point Charge Approximation . 29 2.2 Crystal Field Effects in Tb2Ti2O7 . 33 2.3 Summary . 39 3 Effective Hamiltonian Method 41 3.1 Brillouin-Wigner Perturbation Theory . 42 3.2 Rayleigh-Schr´'odinger Perturbation Theory . 45 3.3 Effective Hamiltonian . 48 4 Effective Hamiltonian of Tb2Ti2O7 52 4.1 Total Hamiltonian of Tb2Ti2O7 . 53 4.2 The Classical Term of the Effective Hamiltonian P V P . 56 vi 4.3 Quantum Term PVRVP . 57 4.3.1 Single Ion Excitation . 58 4.4 Two Ion Excitations . 70 4.4.1 Two Ion Excitations, Spin Flip Mechanism and Transverse Magnetic Field . 70 4.5 Summary . 75 5 Single Tetrahedron Approximation 76 5.1 Single Tetrahedron Approximation . 77 5.2 Phase Diagram of Tb2Ti2O7 in the Single Tetrahedron Approximation 78 5.3 Phase Diagram of the Effective Hamiltonian in the Single Tetrahe- dron Approximation . 87 5.4 Summary . 91 6 Neutron Scattering of Tb2Ti2O7 93 6.1 Diffuse Neutron Scattering . 94 6.2 Diffuse Neutron Scattering of Tb2Ti2O7 . 96 6.3 Possible Experimental Realization of Quantum Spin Ice . 102 6.4 Summary . 105 7 The Ground State of Tb2Ti2O7 in the Classical Limit 107 7.1 Ewald Method for E-D Interaction . 108 7.2 D-D Interaction in a Simulation Box . 109 7.3 Classical Ground State of Tb2Ti2O7 . 113 7.4 Summary . 116 vii 8 Tb2Ti2O7 in a Magnetic Field 117 8.1 Crystal Field Levels of Tb2Ti2O7 in a Magnetic Field . 118 8.2 Non-Interacting Tb3+ Ions in a Magnetic field . 121 8.3 Interacting Tb3+ Ions in a Magnetic Field . 126 8.4 Probing the Ground State of Tb2Ti2O7 . 131 8.5 Summary . 137 9 Summary and Future Work 138 9.1 Effective Hamiltonian of Tb2Ti2O7 . 138 9.2 Tb2Ti2O7 in the Single Tetrahedron Approximation . 139 9.3 Beyond the Single Tetrahedron Approximation . 140 9.4 Future Work . 140 A Exchange Couplings of Ho2Ti2O7 and Dy2Ti2O7 142 A.1 Effective Hamiltonian of Spin Ice Systems . 143 A.2 High Temperature Expansion of DC Susceptibility for Spin Ice Systems146 A.3 Exchange Couplings of Ho2Ti2O7 and Dy2Ti2O7 . 149 A.4 Summary . 158 B Stevens Operator of Tb2Ti2O2 159 C Local Basis Vector 160 viii List of Figures 1.1 Frustration on a Square . 2 1.2 Arrangement of Hydrogens around Oxygen in Ice IH Phase . 2 1.3 Geometrically Frustration with Three Antiferromagnetic Ising Spins on a Triangle . 4 1.4 Geometrically Frustration with Antiferromagnetic Heisenberg Spins on a Tetrahedron . 4 1.5 Pyrochlore Lattice . 5 3+ 1.6 The Distorted Cubic Environment of A in A2Ti2O7 . 8 1.7 Magnetization of Dy2Ti2O7 . 9 1.8 Mapping the Two-in/Two-out State on a Tetrahedron to the Two- Close/Two-Far State in Ice IH . 11 1.9 Neutron Scattering of Ho2Ti2O7 . 12 1.10 Phase Diagram of 111 Ising Spins on the Pyrochlore Lattice . 13 h i 1.11 Ordered Phase of Spin Ice System . 14 1.12 All-in/All-out State . 16 1.13 Muon Spin Relaxation of Tb2Ti2O7 and (TbpY1 p)2Ti2O7 . 17 − 1.14 Diffuse Neutron Scattering of Tb2Ti2O7 . 18 1.15 Specific Heat of Tb2Ti2O7 . 21 1.16 Experimental Magnetization of Tb2Ti2O7 . 22 ix 1.17 Diffuse Neutron Scattering of Tb2Ti2O7 under a Magnetic Field . 23 2.1 The Crystal Environment of an Ion . 30 2.2 The Crystal Environment of Tb3+ Ion . 34 2.3 The Evolution of the Crystal Field States of Tb2Ti2O7 . 36 4.1 Single Ion Process with Two Ions . 59 4.2 Single Ion Process with Three Ions . 60 4.3 Generated First, Second and Third Nearest Neighbor Interactions . 65 4.4 First, Second, Third NN Interactions on a Pyrochlore Lattice . 66 4.5 Log-Log Plot of E-D Interaction . 67 4.6 A Procedure for Calculating the D-D Interaction . 69 4.7 Log-Log Plot of the D-D Coupling . 69 4.8 J Matrix Elements between the Ground State and First Excited Crystal Field State . 71 4.9 Spin Flip Mechanism via the Crystal Field Excited States . 72 5.1 Classical Phase Diagram of a Local 111 Ising Spins . 79 h i 5.2 J Matrix Elements between the Crystal Field Ground State and First Excited State . 80 5.3 Phase Diagram of Tb2Ti2O7 with Four Lowest Crystal Field States and Zero off-Diagonal J z . 82 3+ 5.4 Phase Diagram of Tb2Ti2O7 with Four Tb Lowest Crystal Field States . 84 5.5 Phase Diagram of Tb2Ti2O7 as a Function of Exchange Interaction 85 5.6 Phase Diagram of Tb2Ti2O7 as a Function of Dipole-Dipole . 86 z 5.7 Phase Diagram of the Effective Hamiltonian of Tb2Ti2O7 with J =0 in the Single Tetrahedron Approximation . 89 x 5.8 Phase Diagram of Effective Hamiltonian in the Single Tetrahedron Approximation . 90 6.1 Experimental Diffuse Neutron Scattering of Tb2Ti2O7 . 97 6.2 Calculated Diffuse Neutron Scattering with Two States of the Crys- tal Field Ground State . 98 6.3 Diffuse Neutron Scattering with Four Lowest States of the Crystal z Field States of Tb2Ti2O7 (J =0) . 100 6.4 Calculated Diffuse Neutron Scattering with Four Lowest States of the Crystal Field States of Tb2Ti2O7 . 101 6.5 Calculated Line Scan of Scattering Function in the Highly Symmetry Directions . 103 6.6 Calculated Line Scan of Diffuse Scattering Function . 104 6.7 Quantum Spin Ice Test Based on Neutron Scattering . 106 7.1 E-D and D-D Couplings as a Function of Distance 1-10 Unit Cell . 111 7.2 E-D and D-D Couplings as a Function of Distance 10-50 Unit Cell . 112 7.3 Competing Classical Phases of Tb2Ti2O7 in the Single Tetrahedron Approximation . 114 7.4 Competing Classical Phases of Tb2Ti2O7 in a Single Unit Cell . 115 8.1 The Azimuthal and Polar Angles of a Magnetic Field in a Global Coordinate of Frame . 118 8.2 Four Lowest Levels of Crystal Field and Zeeman Hamiltonians as a Function of Polar Angle . 120 8.3 Energy Gap as a Function of Magnetic Field along the 110¯ Direction 121 8.4 Four Lowest Energy Levels of Tb3+as a Function of Polar Angle . 122 8.5 Energy Gap as a Function of Magnetic Field along the 112¯ Direction 123 xi 8.6 Four Lowest Levels of Tb3+ as a Function of Magnetic Field in the Highly Symmetry Directions . 124 8.7 Magnetization of Non-Interacting Tb3+ Ions in the Highly Symmetry Direction . 125 8.8 Magnetization of Non-Interacting Tb3+ Ions in Highly Symmetry Directions and at Different Temperatures .
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