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Lecture Notes on Many-Body Theory

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Lecture Notes on Many-Body Theory Lecture notes on many-body theory Michele Fabrizio updated to 2013, but still work in progress!! All comments are most welcome! Contents 1 Landau-Fermi-liquid theory: Phenomenology 5 1.1 The Landau energy functional and the concept of quasiparticles . .5 1.2 Quasiparticle thermodynamics . .9 1.2.1 Specific heat . 13 1.2.2 Compressibility and magnetic susceptibility . 13 1.3 Quasiparticle transport equation . 15 1.3.1 Quasiparticle current density . 18 1.4 Collective excitations in the collisionless regime . 20 1.4.1 Zero sound . 22 1.5 Collective excitations in the collision regime: first sound . 25 1.6 Linear response functions . 27 1.6.1 Formal solution . 28 1.6.2 Low frequency limit of the response functions . 31 1.6.3 High frequency limit of the response functions . 32 1.7 Charged Fermi-liquids: the Landau-Silin theory . 34 1.7.1 Formal solution of the transport equation . 37 1.8 Dirty quasiparticle gas . 39 1.8.1 Conductivity . 41 1.8.2 Diffusive behavior . 43 2 Second Quantization 48 2.1 Fock states and space . 48 2.2 Fermionic operators . 50 2.2.1 Second quantization of multifermion-operators . 52 2.2.2 Fermi fields . 55 2.3 Bosonic operators . 57 2.3.1 Bose fields and multiparticle operators . 57 2.4 Canonical transformations . 59 2.4.1 More general canonical transformations . 60 1 2.5 Examples and Exercises . 62 2.6 Application: fermionic lattice models and the emergence of magnetism . 65 2.6.1 Hubbard models . 69 2.6.2 The Mott insulator within the Hubbard model . 71 2.7 Spin wave theory in the Heisenberg model . 73 2.7.1 More rigorous derivation: the Holstein-Primakoff transformation . 81 3 Linear Response Theory 83 3.1 Linear Response Functions . 83 3.2 Kramers-Kronig relations . 86 3.2.1 Symmetries . 87 3.3 Fluctuation-Dissipation Theorem . 88 3.4 Spectral Representation . 90 3.5 Power dissipation . 91 3.5.1 Absorption/Emission Processes . 93 3.5.2 Thermodynamic Susceptibilities . 94 4 Hartree-Fock Approximation 98 4.1 Hartree-Fock Approximation for Fermions at Zero Temperature . 98 4.1.1 Alternative approach . 102 4.2 Hartree-Fock approximation for fermions at finite temperature . 106 4.2.1 Preliminaries . 106 4.2.2 Variational approach at T 6=0 ........................ 110 4.3 Mean-Field approximation for bosons and superfluidity . 114 4.3.1 Superfluid properties of the gauge symmetry breaking wavefunction . 119 4.4 Time-dependent Hartree-Fock approximation for fermions . 124 4.4.1 Bosonic representation of the low-energy excitations . 126 4.5 Application: antiferromagnetism in the half-filled Hubbard model . 130 4.5.1 Spin-wave spectrum by time dependent Hartree-Fock . 134 4.6 Linear response of an electron gas . 139 4.6.1 Response to an external charge . 142 4.6.2 Response to a transverse field . 147 4.6.3 Limiting values of the response functions . 148 4.6.4 Power dissipated by the electromagnetic field . 156 4.6.5 Reflectivity . 158 4.7 Random Phase Approximation for the electron gas . 160 5 Feynman diagram technique 163 5.1 Preliminaries . 163 5.1.1 Imaginary-time ordered products . 163 2 5.1.2 Matsubara frequencies . 164 5.1.3 Single-particle Green's functions . 168 5.2 Perturbation expansion in imaginary time . 172 5.2.1 Wick's theorem . 173 5.3 Perturbation theory for the single-particle Green's function and Feynman diagrams174 5.3.1 Diagram technique in momentum and frequency space . 177 5.3.2 The Dyson equation . 180 5.4 Other kinds of perturbations . 185 5.4.1 Scalar potential . 185 5.4.2 Coupling to bosonic modes . 186 5.5 Two-particle Green's functions and correlation functions . 189 5.5.1 Diagrammatic representation of the two-particle Green's function . 190 5.5.2 Correlation functions . 191 5.6 Coulomb interaction and proper and improper response functions . 194 5.6.1 Screened interaction and corresponding Dyson equation . 195 5.7 Irreducible vertices and the Bethe-Salpeter equations . 197 5.7.1 Bethe-Salpeter equation for the vertex functions . 198 5.8 The Ward identities . 200 5.9 Consistent approximation schemes . 204 5.9.1 Example: the Hartree-Fock approximation . 206 5.10 Some additional properties and useful results . 208 5.10.1 The occupation number and the Luttinger-Ward functional . 208 5.10.2 The thermodynamic potential . 211 5.10.3 The Luttinger theorem . 214 6 Landau-Fermi liquid theory: a microscopic justification 217 6.1 Preliminaries . 217 6.1.1 Vertex and Ward identities . 220 6.2 Correlation functions . 223 6.2.1 Conserved quantities . 225 6.3 Coulomb interaction . 226 7 Kondo effect and the physics of the Anderson impurity model 228 7.1 Brief introduction to scattering theory . 229 7.1.1 General analysis of the phase-shifts . 234 7.2 The Anderson Impurity Model . 235 7.2.1 Variation of the electron number . 239 7.2.2 Energy variation . 240 7.2.3 Mean-field analysis of the interaction . 242 7.3 From the Anderson to the Kondo model . 244 3 7.3.1 The emergence of logarithmic singularities and the Kondo temperature . 245 7.3.2 Anderson's scaling theory . 248 8 Introduction to abelian Bosonization 252 8.1 Interacting spinless fermions . 252 8.1.1 Spin wave theory . 255 8.1.2 Construction of the effective Hamiltonian . 256 8.2 Interactions . 263 8.2.1 Umklapp processes . 266 8.3 Bosonization of the Heisenberg model . 268 8.4 The Hubbard model . 272 4 Chapter 1 Landau-Fermi-liquid theory: Phenomenology One of the first thing that one learns in a Solid State Physics course is that the thermodynamic and transport properties of metals are well described in terms of non-interacting electrons; the Drude-Sommerfeld-Boltzmann theory of metals. Yet, this evidence is somehow surprising in view of the fact that actual electrons interact mutually via Coulomb repulsion, which is not weak at all. This puzzle was solved brilliantly in the end of the 50's by Landau, as we are going to discuss in what follows. We will start by analyzing the case of a neutral Fermi system, like 3He. Later we shall discuss charged Fermi systems, relevant to metals. 1.1 The Landau energy functional and the concept of quasipar- ticles Let us start from a non-interacting Fermi gas at very low temperature T . Right at T = 0 we know that the ground state is the Fermi sea that is obtained by filling with two opposite-spin electrons all momentum states with energy smaller than the chemical potential µ (the Fermi energy). In other words, if nkσ is the occupation number of each momentum state { nkσ being zero or one because of Pauli principle { then the Fermi-sea occupation numbers are ( (0) (0) 1 if k ≤ µ nkσ = (0) : 0 if k > µ At finite temperature, the equilibrium values of the occupation numbers are given by the Fermi- Dirac distribution −1 (0) (0) (0) β (k −µ) nkσ = f k − µ = 1 + e ; 5 with β = 1=KBT the inverse temperature. Any excited state can be uniquely identified by the variation of the occupation numbers with respect to equilibrium, namely through (0) δnkσ = nkσ − nkσ ; (1.1) and costs an energy (0) X (0) δE [fδnpαg] = k − µ δnkσ; (1.2) kσ that is a simple functional of the δnkσ's. Now suppose that we switch on smoothly the interaction. Each non-interacting excited state will evolve smoothly into a fully-interacting one. The Landau hypothesis was that the non-interacting and the fully interacting states are adiabatically connected; in other words they are in one-to-one correspondence. This implies in particular that each fully-interacting excited state can be uniquely identified, like its non-interacting partner, by the deviation with respect to equilibrium of the occupation numbers δnkσ. Consequently, its excitation energy δE must be a functional of the δnkσ's, i.e. δE [fδnpαg] : (1.3) ||| Remark This hypothesis may look simple but it is actually deeply counter-intuitive. Suppose we have a \non-interacting".
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