Metallic magnets without inversion symmetry and antiferromagnetic quantum critical points [Elektronische Ressource] / vorgelegt von Inga Anita Fischer

Sujets

• Physik

Informations

• Publié par: universitat_zu_koln
• Publié le: 01 janvier 2006
• Nombre de lectures: 11
• Langue: English
• Poids de l'ouvrage: 2 Mo

Extrait

Metallic Magnets without
Inversion Symmetry
and
Antiferromagnetic
Quantum Critical Points
Inaugural-Dissertation
zur
Erlangung des Doktorgrades
der Mathematisch-Naturwissenschaftlichen Fakult¨at
der Universit¨at zu K¨oln
vorgelegt von
Inga Anita Fischer
aus G¨ottingen
K¨oln 2006Berichterstatter: Prof. Dr. A. Rosch
Prof. Dr. E. Mu¨ller-Hartmann
Tag der mu¨ndlichen Pru¨fung: 30. Juni 2006Contents
0. Introduction 1
I. Metallic Magnets without Inversion Symmetry: MnSi 3
1. Introduction 5
1.1. Experimental observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2. Electrons in the Ordered Phase 9
2.1. Band structure from coupling of the electrons to the helix . . . . . . . . . . . 9
2.2. Form of the spin-orbit coupling term in the bandstructure . . . . . . . . . . . 12
2.3. Electron band structure and the Fermi surface . . . . . . . . . . . . . . . . . 12
2.3.1. Band structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.2. Shape of the Fermi surface . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4. Fermi surface: Experimental consequences . . . . . . . . . . . . . . . . . . . . 17
2.4.1. De-Haas–van-Alphen Effect . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.2. Impurity scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4.3. Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4.4. Anomalous Skin effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5. Minibands and Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.6. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3. New Phases from a Ginzburg-Landau theory 29
3.1. Ginzburg-Landau theory and the helix . . . . . . . . . . . . . . . . . . . . . . 29
3.2. Stability of the helix solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2.1. Quantum phase transitions of magnetic rotons . . . . . . . . . . . . . 31
3.2.2. A new phase for magnetic rotons? . . . . . . . . . . . . . . . . . . . . 32
3.2.3. Stability analysis of the Ginzburg-Landau expansion . . . . . . . . . . 32
3.3. Blue Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3.1. Blue Phases in cholesteric liquid crystals . . . . . . . . . . . . . . . . . 34
3.3.2. Chiral Ferromagnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3.3. Single Double-Twist Cylinders . . . . . . . . . . . . . . . . . . . . . . 42
3.3.4. Crystal of Double Twist Cylinders I: Square Lattice . . . . . . . . . . 44
3.3.5. Crystal of Double Twist Cylinders II: Cubic Lattice . . . . . . . . . . 46
3.3.6. Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3.7. Blue Phases and MnSi . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3.8. Other propositions for helical spin crystals . . . . . . . . . . . . . . . . 53
3.3.9. Summary and further directions . . . . . . . . . . . . . . . . . . . . . 55
iContents
A. Minibands 59
A.1. Dzyaloshinsky-Moriya interaction . . . . . . . . . . . . . . . . . . . . . . . . . 59
A.2. The space group P2 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601
A.3. Representations and multiplication tables for T . . . . . . . . . . . . . . . . . 60
A.4. Polarization bubble: Calculations . . . . . . . . . . . . . . . . . . . . . . . . . 61
B. New Phases from a Ginzburg-Landau Theory 65
B.1. Square lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
B.2. Cubic lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
II. Quantum Phase Transitions in Antiferromagnetic Metals 73
4. Introduction 75
4.1. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.1.1. Hertz-Millis theory of Quantum Critical Points in Metals . . . . . . . 77
4.1.2. Quantum critical points in itinerant ferromagnets . . . . . . . . . . . . 79
4.1.3. Hertz-Millis theory and antiferromagnets . . . . . . . . . . . . . . . . 80
4.1.4. Other theories for heavy fermion criticality . . . . . . . . . . . . . . . 83
4.2. Antiferromagnetic QCPs in Experiments . . . . . . . . . . . . . . . . . . . . . 85
5. Field-tuned Quantum Phase Transitions 89
5.1. Model and Effective action . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.1.1. Magnetic insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2. Renormalization group equations and correlation length . . . . . . . . . . . . 94
5.3. Thermodynamic quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.3.1. Specific heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3.2. Magnetization, magnetocaloric effect and Gru¨neisen parameter . . . . 98
5.3.3. Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.4. Scattering rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6. Beyond Hertz-Millis Theory 107
6.1. Functional Renormalization Group . . . . . . . . . . . . . . . . . . . . . . . . 107
6.2. Model and fRG-equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.2.1. Definitions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.2.2. fRG Equation for the Effective Action Γ . . . . . . . . . . . . . . . . . 111
6.2.3. Choice of Cutoff Functions . . . . . . . . . . . . . . . . . . . . . . . . 113
6.3. Flow Equations for the 1-PI Vertex Functions . . . . . . . . . . . . . . . . . . 116
6.3.1. Approximations in the Vertex Functions . . . . . . . . . . . . . . . . . 117
6.3.2. Flow for a constant Fermion-Boson Vertex . . . . . . . . . . . . . . . . 118
6.3.3. Flow Equations for the Vertex Functions. . . . . . . . . . . . . . . . . 120
6.4. Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
A. Field-tuned Quantum Phase Transitions 131
A.1. Cubic terms in the effective action . . . . . . . . . . . . . . . . . . . . . . . . 131
A.2. Frustration in BEC of Magnons . . . . . . . . . . . . . . . . . . . . . . . . . . 131
A.3. Derivation of RG-equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
iiContents
B. Beyond Hertz-Millis Theory 139
B.1. Bosonic self-energy Σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139B
B.2. Fermionic self-energy Σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140F
B.3. Fermion-boson vertex V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
B.3.1. Fermion-boson vertex V at zero external frequencies . . . . . . . . . . 142
B.3.2. Fermion-boson vertex V at s . . . . . . . . . . . . . . . . . . . . . . 143F
B.3.3. Fully frequency-dependentV . . . . . . . . . . . . . . . . . . . . . . . 144
B.4. Four-boson vertex Γ 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145φ
B.5. Continuous Hubbard–Stratonovich transformation . . . . . . . . . . . . . . . 148
B.5.1. Field redefinition invariance of the partition sum . . . . . . . . . . . . 149
B.5.2. Method 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
B.5.3. Method 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
B.5.4. Method 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Bibliography 155
Danksagung 161
Anh¨ange gem¨aß Pru¨fungsordnung 163
Kurze Zusammenfassung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Erkl¨arung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Teilpublikationen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
iii0. Introduction
The Fermi liquid description of metals is of paramount importance to the quantum theory
of solids. First formulated by Landau in the 1950’s, the theory predicts that at very low
temperaturesasystemofstronglyinteractingelectronscanbedescribedbyweaklyinteracting
electron-like “quasiparticles”. Although these quasiparticles are in fact complex many-body
approximations, they have many properties in common with electrons: the quasiparticles
have the same quantum numbers as electrons and the quasiparticle energy spectrum is in
one-to-one correspondence with the energy spectrum of a free Fermi gas.
Starting in the early eighties, more and more materials have emerged whose experimentally
observedbehaviourisinconsistentwithFermiliquidphenomenology. Mostnotably,exponents
characterizing the temperature-dependence of thermodynamic and transport quantities can
deviate from those predicted by Fermi liquid theory. This “Non-Fermi liquid behaviour”
encompasses a wide range of physical phenomena, many of which are subject of current
theoretical and experimental interest. For example, collective excitations of the electronic
quasiparticlescandominatethelow-energybehaviourofasysteminthevicinityofaquantum
critical point. In this case, the Fermi liquid paradigm in fact does not fail, since electronic
quasiparticlesarestil