Fakultät > Vorlesungen > Wintersemester 2020/2021 > Advanced Statistical Physics

Advanced Statistical Physics

Wintersemester 2020/2021
Dozent: Prof. Tilman Enss
Link zum LSF
ws20-ast
37 Teilnehmer/innen

This advanced theory course builds on the statistical physics course (MKTP1) and introduces paradigmatic models of statistical physics and their critical properties near phase transitions.  In particular, we shall discuss the Heisenberg and O(N) vector models, the nonlinear sigma model, the XY model, the Sine-Gordon model, and the spherical model, as time permits.  By computing their critical behavior, one can understand the phase transitions in many different systems in statistical physics, condensed matter physics and beyond, which belong to the same universality classes.  We will use field theoretic methods and introduce renormalization, epsilon and 1/N expansions, and duality transformation.

This two-hour lecture-only course is planned to take place on campus. Due to the pandemic, in January/February no students can attend the lecture in the auditorium. Lecture notes are available online. All registered participants can take part in the final exam, which will be held online.

Curriculum / Timeline

The lectures are made available as a movie for download every Thursday at lecture time; see the highlighted section in the time line below for the current lecture. The accompanying lecture notes can be downloaded as a PDF file. You may discuss with your fellow students at the AST chat.

Enjoy the course!
 

  1. Landau Theory and O(N) vector model
     
    • Lecture 01 (2020-11-05): Ising model
      phase transitions exhibit universal critical properties, as exemplified in the Ising model
      watch the lecture (download video)
      lecture notes ch. 1.1-1.2 (pp. 1-4) on the Ising model, universality and the mean-field ansatz
       
    • Lecture 02 (2020-11-12): Fluctuations
      fluctuations around the mean-field solution become important in low dimensions
      watch the lecture (download video)
      lecture notes ch. 1.3 (pp. 5-8) on fluctuations around mean field, loop expansion, upper critical dimension
       
    • Lecture 03 (2020-11-19): N-vector model
      how to treat magnets of different spin symmetry in a unified description
      watch the lecture (download video)
      lecture notes ch. 1.4 (pp. 9-10) on the N-vector model, and a summary of critical exponents
       
  2. Renormalization group and universality
     
    • Lecture 03 continued (2020-11-19): Renormalization
      renormalization fixes divergences and corrects the value of critical exponents
      continue watching the lecture (download video)
      lecture notes ch. 2.1 (pp. 11-12) on scaling dimensions and the idea of renormalization
       
    • Lecture 04 (2020-11-26): Renormalization group equations
      where do the RG equations come from?
      a. do the online quiz (11:15-11:20h)
      b. watch the lecture (download video)
      lecture notes ch. 2.2 (pp. 12-16) on deriving the RG equations, epsilon expansion, fixed points, and corrections to the gamma exponent
       
    • Lecture 05 (2020-12-03): Universality
      why can different systems show similar behavior?
      a. study the lecture notes ch. 2.3+ (pp. 17-20)
      b. do the online quiz (Thursday 11:15-11:20h)
      c. live discussion (Thursday 11:20 via zoom)
      lecture notes ch. 2.3-4 (pp. 17-20) on universality and multiple fixed points
       
    • Lecture 06 (2020-12-10): Crossover
      if there is more than one fixed point, which one matters?
      a. do the online quiz (11:15-11:20h)
      b. watch the lecture (download video)
      lecture notes ch. 2.4 (pp. 21-22) on crossovers between multiple fixed points and the Ginzburg criterion
       
  3. Nonlinear sigma model and epsilon expansion
     
    • Lecture 06 continued (2020-12-10): Goldstone modes
      are there low-energy excitations in the symmetry-broken state?
      c. continue watching the lecture (download video)
      lecture notes       ch. 3.1 (pp. 23-25) on longitudinal and transverse excitations and the appearance of Goldstone modes
       
    • Lecture 07 (2020-12-17): Nonlinear sigma model
      what is the effective action for Goldstone modes?
      a. do the online quiz (Thursday 11:15-11:20h)
      b.       watch the lecture (download video      )
      c. live discussion session (Thursday 12:40 via zoom)
      lecture notes ch. 3.2-3 (pp. 26-29) on deriving the nonlinear sigma model and its renormalization
       
    • Lecture 08 (2021-01-14): Critical exponents in the nonlinear sigma model
      how strongly does the correlation length diverge in different dimensions?
      a. do the online quiz (Thursday 11:15-11:20h)
      b.       watch the lecture (download video      )
      lecture notes ch. 3.3 (pp. 30) on critical exponents in the nonlinear sigma model
       
  4. Topological excitations in the XY and Sine-Gordon models and the Kosterlitz-Thouless transition
     
    • Lecture 08 continued (2021-01-14): XY model and spin waves
      why is the transition of planar rotors such an important universality class?
      c. continue watching the lecture (download video)
      lecture notes       ch. 4-4.1 (pp. 31-34) on XY magnets, order parameter, spin correlations and the effect of spin-wave excitations
       
    • Lecture 09 (2021-01-21): Vortices and Coulomb gas
      what are topological excitations?
      a. do the online quiz (Thursday 11:15-11:20h)
      b.       watch the lecture (download video      )
      c. live question & answer session (Thursday 12:35 via zoom)
      lecture notes       ch. 4.2-4.3 (pp. 35-38) on vortices, winding number, Coulomb gas description of vortices and partition function

    • Lecture 10 (2021-01-28): Sine-Gordon model
      how are domain walls in a magnet related to vortices?
      a. do the online quiz (Thursday 11:15-11:20h)
      b.       watch the lecture (download video      )
      lecture notes       ch. 4.4 (pp. 39-41) on the Sine-Gordon model, domain walls, mapping to the Coulomb gas, and the relevance of vortex excitations
       
    • Lecture 11 (2021-02-04): Berezinski-Kosterlitz-Thouless transition
      how do vortex pairs screen the Coulomb interaction?
      a. do the online quiz (Thursday 11:15-11:20h)
      b.       watch the lecture (download video      )
      lecture notes       ch. 4.5 (pp. 42-43) on the screening of Coulomb interaction and the universal features of the BKT phase transition
       
  5. Spherical model and quantum phase transitions
     
    • Lecture 11 continued (2021-02-04): Spherical Model
      do quantum fluctuations affect a phase transition?
      c. continue watching the lecture (download video)
      d. live question & answer session (Thursday 12:30 via zoom)
      lecture notes       ch. 5-5.1 (pp. 44-45) on thermal vs quantum fluctuations, quantum phase transitions, and the spherical model (classical and quantum)
       
    • Lecture 12 (2021-02-11): Critical Properties of the Spherical Model
      is a phase transition at zero temperature different?
      a. do the online quiz (Thursday 11:15-11:20h)
      b.       watch the lecture (download video      )
      c. live question & answer session (Thursday 12:30 via zoom)
      lecture notes             ch. 5.1-5.4 (pp. 46-49) on critical behavior at nonzero and zero temperature, and the quantum critical regime; Vojta article on the spherical model.

Literature

As an introduction, the lecture notes by Mudry are recommended; Mudry chapter 1 introduces the field theoretical language.
For starters:

  • Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press (1996)
  • Mudry, Lecture Notes on Field Theory in Condensed Matter Physics, World Scientific (2014)

Further reading:

  • Altland and Simons, Condensed Matter Field Theory, Cambridge University Press (2010)
  • Negele and Orland, Quantum Many-Particle Systems, Addison-Wesley (1988)
  • Sachdev, Quantum Phase Transitions, Cambridge University Press, 2nd edition (2011)
  • Zinn-Justin, Phase Transitions and Renormalization Group, Oxford University Press (2007)

Übungsgruppen

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Advanced Statistical Physics
Wintersemester 2020/2021
T. Enss
Link zum LSF
37 Teilnehmer/innen
Termine