Fakultät > Vorlesungen > Sommersemester 2020 > Condensed Matter Theory 2

Condensed Matter Theory 2

Sommersemester 2020
Dozent: Prof. Dr. Tilman Enss
Link zum LSF
17 Teilnehmer/innen

Curriculum / Timeline

The lectures are available as a movie for download (possibly stream) every Monday and Wednesday at 11:00am; see the highlighted section in the time line below for the current lecture. The accompanying lecture notes can be downloaded as a PDF file; they evolve from week to week. You may ask questions about the lecture at the CMT2 chat; I will collect your questions every Monday and Wednesday at 11:00am and answer them subsequently.

Every Wednesday there will be new set of homework problems in the exercise section below. When you have solved them, please write your solution clearly and succinctly on paper+scan or tablet and have them ready to present for the tutorial.

The tutorial is held Tuesdays at 9:15-11:00am as a live zoom video call. The tutorial is hosted by our tutor Dr. Yi Lu. During the tutorial you are encouraged to present your solutions to the class by sharing your screen. When someone else is presenting, please takes notes for yourselves; there will be no transcript of the video call.

The zoom meeting takes place in Zoom Room Physics 5 that you can find at the following address:

Link: https://zoom.us/j/3141177797?pwd=MEhNWDJ6M3hQMEErT0xRMWNMdTlwQT09

Alternatively, you can also join the meeting using the following credentials:
Meeting ID: 314-117-7797
Passwort: 334833

The address of the zoom room will be valid for the whole semester. It will be used by other tutorials at other times of the week, so you can leave the zoom room after our tutorial is finished. If you haven't used zoom before, please follow the link above to find the room, install the zoom client, and familiarize yourself with muting/unmuting your mic and sharing your screen, so you are ready to use it before the tutorial starts.

Enjoy the course!

 

  1. Introduction
    • Lecture 01 (2020-04-20): More is different
      even from a simple Hamiltonian of electron and ions, interesting symmetry broken states like magnets and superconductors can emerge at low energy
      a. watch the introduction (stream(experimental) / download video / slides)
      b. reading assignment: lecture notes ch. 1 (pp. 5-6) on the Hamiltonian, energy scales, and "more is different"
       
  2. Green functions and perturbation theory
    • Lecture 01 (2020-04-20): Green functions
      how to describe the properties of a single particle in a many-body environment
      c. watch the lecture (stream(experimental) / download video)
      accompanying lecture notes ch. 2.1 (pp. 7-11): time-ordered Green functions at zero temperature
      further literature: Negele/Orland ch. 5.1, Fetter/Walecka sec. 7
       
    • Lecture 02 (2020-04-22): Finite temperature Green functions
      which states are populated at finite temperature?
      a. watch the lecture (stream(experimental) / download video)
      work through the lecture notes 2.1.1-2.2 (pp. 11-15): advanced and retarded Green functions, finite temperature, spectral decomposition
      b. get the first exercise sheet (below) and solve it for the tutorial next Tuesday
      further literature: Negele/Orland ch. 5.1, Fetter/Walecka sec. 23
       
    • Lecture 03 (2020-04-27): Imaginary time Green functions
      clever computations in thermal equilibrium
      a. watch the lecture (download video)
      work through the lecture notes 2.2-2.2.1 (pp. 15-20): imaginary time and Matsubara frequency, analytical properties of Green functions
       
    • Lecture 04 (2020-04-29): Coherent states (bosonic)
      eigenstates of the ladder operators
      a. watch the lecture (download video)
      work through the lecture notes 2.2.1-2.3.1 (pp. 20-23): Matsubara sums, properties of coherent states for bosons, closure relation
      b. get the second exercise sheet (below) and solve it for the tutorial next Tuesday
      further literature: Altland/Simons p. 170 (Matsubara sums); Negele/Orland ch. 1.5, Altland/Simons 4.1 (coherent states)
       
    • Lecture 05 (2020-05-04): Coherent states (fermionic)
      can one represent anticommutation relations with complex numbers?
      a. watch the lecture (download video)
      work through the lecture notes 2.3.1-2.3.5 (pp. 23-28): Grassmann algebra, fermionic coherent states
       
    • Lecture 06 (2020-05-06): Path integral
      how to sum over all possible paths a quantum particle can take
      a. watch the lecture (download video)
      work through the lecture notes 2.4-2.5.1 (pp. 28-31): path integral, partition function, Wick's theorem
      b. get the exercise sheet (below) and solve it for the tutorial next Tuesday
      further literature: Negele/Orland ch. 2.2, Altland/Simons
       
    • Lecture 07 (2020-05-11): Perturbation theory
      how does a single particle interact with surrounding medium?
      a. watch the lecture (download video)
      work through the lecture notes 2.5.1-2.5.3 (pp. 32-35): perturbation theory, self-energy, Dyson equation
       
  3. Metals
    • Lecture 08 (2020-05-13): Metals: Hartree-Fock
      how to include Coulomb interaction between electrons
      a. watch the lecture (download video)
      work through the lecture notes 3.1-3.2 (pp. 37-41): Hartree-Fock, Jellium model
      b. get the exercise sheet (below) and solve it for the tutorial next Tuesday
      further literature: Fetter/Walecka sec. 10, sec. 3
       
    • Lecture 09 (2020-05-18): Jellium model
      can Hartree-Fock explain the stability of matter?
      a. watch the lecture (download video)
      work through the lecture notes 3.2-3.3 (pp. 41-44): Fock term in Jellium model, ground-state energy and stability of matter
      further literature: Fetter/Walecka sec. 3
       
    • Lecture 10 (2020-05-20): Screening
      how does a photon travel through an electron gas?
      a. watch the lecture (download video)
      work through the lecture notes 3.3-3.3.2 (pp. 45-48): Charge excitations, static and dynamic screening, Thomas-Fermi and random-phase approximation
      b. get the exercise sheet (below) and solve it for the tutorial next Tuesday
      further literature: Fetter/Walecka sec. 12
       
    • Lecture 11 (2020-05-25): Structure factor
      which density excitations are possible for an interacting electron gas?
      a. watch the lecture (download video)
      work through the lecture notes 3.3.2-3.3.3 (pp. 48-52): Static screening at finite momentum, dynamical screening, structure factor of interacting electron gas
      further literature: Fetter/Walecka sec. 12, 14, 15
       
    • Lecture 12 (2020-05-27): Phonons
      how can one describe lattice vibrations?
      a. watch the lecture (download video)
      work through the lecture notes 3.4-3.4.2 (pp. 53-56): Normal modes of lattice vibrations, acoustic phonons, phonon density of states and specific heat
      b. get the exercise sheet (below) and solve it for the tutorial next Tuesday
      further literature: Ashcroft/Mermin 22, 23, 26
       
    • Lecture 13 (2020-06-03): Electron-phonon interaction
      how do phonons affect the electron gas?
      a. watch the lecture (download video)
      work through the lecture notes 3.4.3-3.4.5 (pp. 57-60): Electon-phonon interaction, phonon induced electron interaction, phonons in metals
      b. get the exercise sheet (below) and solve it for the tutorial next Tuesday
       
  4. Superconductivity
    • Lecture 14 (2020-06-08): Superconductivity
      what is the phenomenon of superconductivity?
      a. watch the lecture (download video)
      work through the lecture notes 3.4.5-4.2 (pp. 61-64): Peierls instability, phonon dielectric function; superconductors: phenomena, phonon induced attraction, Cooper bound state
      further literature: Tinkham ch. 3; Altland/Simons ch. 6.4; Fetter/Walecka sec. 51
       
    • Lecture 15 (2020-06-10): BCS theory
      how does superconductivity arise microscopically?
      a. watch the lecture (download video)
      work through the lecture notes 4.2-4.3 (pp. 65-68): Cooper pairs, BCS theory, quasiparticle dispersion, gap equation
      b. get the exercise sheet (below) and solve it for the tutorial next Tuesday
       
    • Lecture 16 (2020-06-15): Gap equation and pair propagation
      what determines the value of the gap?
      a. watch the lecture (download video)
      work through the lecture notes 4.3-4.5 (pp. 69-72): gap equation, temperature dependence of the gap, propagation of pairs, Hubbard-Stratonovich transformation
      further literature: Altland/Simons ch. 6.4; Fetter/Walecka sec. 51
       
    • Lecture 17 (2020-06-17): Landau-Ginzburg action, Hubbard model
      which action governs the order parameter fluctuations?
      a. watch the lecture (download video)
      work through the lecture notes 4.5-4.6 (pp. 73-76): Landau-Ginzburg-Wilson action, Hubbard model, high-temperature superconductors, antiferromagnetic Mott insulator
      b. get the exercise sheet (below) and solve it for the tutorial next Tuesday
       
  5. Magnetism
    • Lecture 18 (2020-06-22): d-wave superconductors
      how can repulsively interacting electrons form pairs?
      a. watch the lecture (download video)
      work through the lecture notes 4.6-5.1 (pp. 77-80): AFM Mott insulator, d-wave superconductivity; classical magnets and critical behavior
      further literature: Altland/Simons ch. 6.4; Di Castro/Raimondi ch. 8
       
    • Lecture 19 (2020-06-24): Stoner ferromagnetism
      how can a Fermi gas become magnetic?
      a. watch the lecture (download video)
      work through the lecture notes 5.2 (pp. 81-84): magnetism in solids, Stoner ferromagnetism of itinerant electrons, spin susceptibility
      b. get the exercise sheet (below) and solve it for the tutorial next Tuesday
       
    • Lecture 20 (2020-06-29): Transverse field Ising model
      what effect can quantum tunneling have on a magnetic state?
      a. watch the lecture (download video)
      work through the lecture notes 5.2-5.3 (pp. 85-88): Spin excitations in a ferromagnet, transverse field Ising model, phase diagram, excitation spectrum in FM and PM states
      further literature: Sachdev ch. 1, 5
       
    • Lecture 21 (2020-07-01): Quantum phase transition
      what distinguishes a quantum from a thermal phase transition?
      a. watch the lecture (download video)
      work through the lecture notes 5.3-5.4 (pp. 89-92): mean-field solution, exact solution in one dimension, quantum phase transitions at zero temperature and quantum critical region
      b. get the exercise sheet (below) and solve it for the tutorial next Tuesday
       
  6. Quantum liquids
    • Lecture 22 (2020-07-06): Fermi liquid theory
      how can one describe the low-temperature properties of a Fermi liquid exactly?
      a. watch the lecture (download video)
      work through the lecture notes 6.1 (pp. 93-96): Mapping of free fermions to quasiparticles, quasiparticle excitations near the Fermi surface, lifetime of quasiparticles
      further literature: Di Castro/Raimondi ch. 12
       
    • Lecture 23 (2020-07-08): Effective mass and interaction in FLT
      how are mass and interaction linked by Galilei invariance?
      a. watch the lecture (download video)
      work through the lecture notes 6.1.2-6.1.5 (pp. 97-100): quasiparticle energy, quasiparticle interaction, specific heat, effective mass and interaction
      further literature: Di Castro/Raimondi ch. 12
       
    • Lecture 24 (2020-07-13): Microscopic foundation of FLT
      how do we find the interaction between quasiparticles from a microscopic Hamiltonian?
      a. watch the lecture (download video)
      work through the lecture notes 6.1.6-6.3 (pp. 101-104): Compressibility and spin susceptibility, stability of a Fermi liquid; microscopic derivation of quasiparticle interaction; failure of FLT in one dimension and Tomonaga-Luttinger liquids
      further literature: Negele/Orland ch. 6.3, Di Castro/Raimondi ch. 19 (FLT); Di Castro/Raimondi ch. 20, Altland/Simons ch. 2.2 (TLL).
       
    • Lecture 25 (2020-07-15): Tomonaga-Luttinger liquid
      can one excite one fermion in an interacting 1D Fermi gas?
      a. watch the lecture (download video)
      work through the lecture notes 6.3 (pp. 105-107): decomposition into left- and right-movers, effective bosonic low-energy Hamiltonian, spin-charge separation, power-law corrections and single-particle Green function
      further literature: Schönhammer (see copy in Materials section); Di Castro/Raimondi ch. 20

The exam takes place on 27 July 2020. If you signed up I already sent you an email with your exam time; if you haven't received it please contact me.

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Condensed Matter Theory 2
Sommersemester 2020
T. Enss
Link zum LSF
17 Teilnehmer/innen
Termine