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Lecture Notes for Quantum Matter

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Lecture Notes for Quantum Matter Lecture Notes for Quantum Matter MMathPhys c Professor Steven H. Simon Oxford University July 24, 2019 Contents 1 What we will study 1 1.1 Bose Superfluids (BECs, Superfluid He, Superconductors) . .1 1.2 Theory of Fermi Liquids . .2 1.3 BCS theory of superconductivity . .2 1.4 Special topics . .2 2 Introduction to Superfluids 3 2.1 Some History and Basics of Superfluid Phenomena . .3 2.2 Landau and the Two Fluid Model . .6 2.2.1 More History and a bit of Physics . .6 2.2.2 Landau's Two Fluid Model . .7 2.2.3 More Physical Effects and Their Two Fluid Pictures . .9 2.2.4 Second Sound . 12 2.2.5 Big Questions Remaining . 13 2.3 Curl Free Constraint: Introducing the Superfluid Order Parameter . 14 2.3.1 Vorticity Quantization . 15 2.4 Landau Criterion for Superflow . 17 2.5 Superfluid Density . 20 2.5.1 The Andronikoshvili Experiment . 20 2.5.2 Landau's Calculation of Superfluid Density . 22 3 Charged Superfluid ≈ Superconductor 25 3.1 London Theory . 25 3.1.1 Meissner-Ochsenfeld Effect . 27 3 3.1.2 Quantum Input and Superfluid Order Parameter . 29 3.1.3 Superconducting Vortices . 30 3.1.4 Type I and Type II superconductors . 32 3.1.5 How big is Hc ............................... 33 4 Microscopic Theory of Bosons 37 4.1 Mathematical Preliminaries . 37 4.1.1 Second quantization . 37 4.1.2 Coherent States . 38 4.1.3 Multiple orbitals . 40 4.2 BECs and the Gross-Pitaevskii Equation . 41 4.2.1 Noninteracting BECs as Coherent States . 41 4.3 Interacting Bosons and the Gross-Pitaevskii Equation . 42 4.3.1 Rederivation without Coherent States . 44 4.4 Order Parameter and Off-Diagonal Long Ranged Order . 44 4.4.1 non-interacting bosons at T > 0 .................... 47 4.5 Bogoliubov Theory for the Weakly Interacting Bose Gas . 48 4.5.1 Bogoliubov Transform . 50 5 Feynman Theory of Helium-4 55 5.1 Ground State and Low Energy Excitations . 55 5.2 Single Mode Excitation Spectrum . 62 6 Ginzburg-Landau Theory 69 6.1 Neutral Superfluids . 69 6.1.1 Spatially Uniform Solution . 70 6.1.2 Spatial Dependence: Ginzburg Landau Theory . 71 6.2 Charged Superfluids (i.e., Superconductors) . 74 6.2.1 Anderson-Higgs Mechanism . 74 6.2.2 Equations of Motion . 77 6.2.3 Energetics from Ginzburg-Landau theory: Type I and Type II su- perconductors revisited . 78 7 Interacting Fermions 83 7.1 Why study fermions . 83 7.2 Some Mathematical Preliminaries Regarding Fermions . 84 7.2.1 Second Quantization . 85 7.2.2 Change of Basis . 87 7.2.3 Writing our Hamiltonian in Second Quantized Form . 88 7.3 First Order Perturbation Theory . 91 7.3.1 Hartree and Fock Terms . 92 7.4 Hartree-Fock Approximation . 94 7.4.1 Hartree Fock as Optimal Slater Determinant . 97 7.5 Application of Hartree-Fock to Translationally Invariant Fermions . 100 7.5.1 Hartree Fock Effective Mass . 105 7.6 Appendix: Showing the Log Divergence at The Fermi Surface . 106 8 Screening and Linear Response 109 8.1 Thomas-Fermi Screening . 109 8.2 Response More Generally . 112 8.3 Lindhard Response Function (Response of Noninteracting Electrons) . 117 8.3.1 Interpretations . 119 8.3.2 High Frequency ! EF and small k .................. 120 8.4 Response of Interacting Electrons: RPA . 123 8.4.1 Relation to Dielectric Constant . 124 8.4.2 High Frequency, Low k response: Plasmons . 125 8.5 Chapter Appendix: Deriving Lindhard Screening expression . 127 9 Landau Fermi Liquid Theory 129 9.1 Background . 129 9.2 Basics Idea of Fermi Liquid Theory . 130 9.2.1 Landau's Conjecture . 130 9.3 Properties of the Dressed Fermi Liquid . 133 9.4 Landau Free Energy Functional . 135 9.5 Results from Fermi Liquid Theory . 139 9.5.1 Compressibility . 139 9.5.2 Spin Susceptibility . 140 9.5.3 Mass Renormalization . 141 9.5.4 More? . 142 9.6 Further extensions of Fermi Liquid Theory . 142 9.6.1 Local Dynamical Properties and Boltzmann Transport . 142 9.6.2 Landau-Silin Theory: Long Range Coulomb Interactions . 143 10 Leading up to BCS Theory: Electron Binding 145 10.1 Mechanism of Attraction . 146 10.1.1 Canonical Transformation . 146 10.1.2 RPA Screening and Ion Gas Model . 149 10.2 Bound States and the Cooper Problem . 151 10.2.1 Non-Existence of Weakly Bound States in 3D . 151 10.2.2 The Cooper Problem: Bound States in The Presence of a Fermi Sea 153 11 BCS Theory 157 11.1 The BCS wavefunction . 158 11.1.1 Fixed N wavefunction . 158 11.1.2 Coherent State Wavefunction . 159 11.2 BCS Hamiltonian, Energy Minimization, and Gap Equation . 161 11.3 More Exotic Pairing . 167 11.3.1 Anisotropic Interactions and Anisotropic Gaps: The Example of High Tc .................................. 167 11.3.2 Triplet Superconductors: Helium-3 . 168 11.4 BCS Excitation Spectrum . 169 11.5 Finite Temperature Gap Equation . 172 11.6 Inhomogeneous Superconductors . 174 11.7 Ginzburg-Landau, Number-Phase, and Josephson . 175 11.7.1 Josephson Effect . 176 i Preface This course covers interesting quantum states of matter: Superfluids, Superconduc- tors, and Fermi Liquids. This is where condensed matter really starts to get bizarre | and therefore even more fun (at least in my opinion). I realize that possibly not everyone will have actually attended the official prerequi- site courses (or possibly did attend, but didn't pay attention), so I will try to make this course as self-contained as possible, while still making contact with what has been taught in prior courses. Particularly for those who have read the Oxford Solid-State Basics, I will warn you right from the front that this book is higher level and is likely to be more difficult going. Whereas that book was written as a third year undergrad course, this book is written at a masters level for students who have a good amount of theoretical background already. Lets get started! ii At early points in one's career there is an inevitable tension between learning physics and learning mathematical formalism. While learning to manipulate commutation rela- tions, and even learning the rules for green's function expansions can be fun | and indeed, by doing this one gets a feeling that one is actually learning something | the real hard part is learning to develop intuition for physics. Developing intuition is extremely hard, and it is something that professional physicists struggle with throughout their lives. This course needs to strike a balance between the two tasks. For some we will not learn enough formalism. For others we will not learn enough physics. I hope we will be able to satisfy both factions, and I hope everyone will learn at least a bit of both. Chapter 1 What we will study 1.1 Bose Superfluids (BECs, Superfluid He, Superconduc- tors) Much of the term will be focused on discussion of superfluids and superconductors. We will discuss both Bose Einstein Condensates (BECs) and superfluids | the main example being superfluid Helium 4. The difference here is that a BEC is by definition a gas of noninteracting bosons, whereas Helium 4 is strongly interacting. In the modern era one calls a weakly interacting bose gas a BEC also { since it can be treated as a weak per- turbation of the noninteracting case. Helium 4, however, is very far from noninteracting. The tension between the physics of the interacting and noninteracting cases will be a theme. We will then turn to study of superconductors. First we will view these as simply being a superfluid of charged bosons. Of course superconductors are things like aluminum1 are regular metals, where the charged objects are electrons, which are fermions of charge −e. However, as we will see later, these charged objects can pair into bosons of charge −2e. Why are such pairs bosons? Recall that fermions accumulate a minus sign when they are exchanged. Exchanging a pair of fermions with another pair of fermions accumulates an even number of minus signs, hence a plus sign, so the pair of fermions is a boson. This picture of electrons pairing into bosons is not entirely accurate and there are good reasons to believe that even using this as a cartoon picture is quite dangerous. However, in some other sense, this picture does make sense. In order to figure out how it makes sense we must first take a detour into a study of Fermions. 1Those using the british spelling and pronounciation "aluminium" will be ridiculed. 1 2 CHAPTER 1. WHAT WE WILL STUDY 1.2 Theory of Fermi Liquids Fermi liquids, metals and other systems of interacting fermions have some unusual prop- erties | some of which you have probably studied in your introductory solid state or condensed matter courses2. However, in most of these treatments, the interaction be- tween fermions is completely neglected. This is a terrible thing to do being that the interaction energy scale is usually just as a big (and sometimes even bigger) than the fermi energy | and both are immensely bigger than the temperature for usual metals at room tempertature. We will need to understand some of the properties of these fermi systems before we can go on to understand the more exotic physics of electron pairing. 1.3 BCS theory of superconductivity This is not examinable 1.4 Special topics also not examined. Subjects we may cover include: Integer quantum Hall effect. Topological Insulators. Majoranas 2If you have not studied introductory solid state physics, you should do so. I can recommend a good book.
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