UNIVERSITEIT VAN AMSTERDAM Instituut voor Theoretische Fysica

Statistical Physics and Condensed Matter Theory II , second semester 2008/2009

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Course description, Literature, Teachers, Schedule, Week-by-week, Assignments and exam.

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Course description

SPCMT II is an optional course in the Master Program Theoretical Physics. At the same time it is open to all interested. We follow up on the introductory course SPCMT I, giving center stage to the Renormalization Group as a conceptual framework and theoretical tool for the analysis of many-body systems and quantum field theories. The second half of the course will mainly deal with the one-dimensional electron gas and the quantum Hall effect.

Literature

We use the book

`Condensed Matter Field Theory' by Alexander Altland and Ben Simons, Cambridge University Press 2006

A further source are the 1994 Les Houches lecture notes by Heinz Schulz on `Fermi-liquids and non-Fermi liquids'

• 1994 Les Houches lecture notes by H. Schulz,

• 1998 Les Houches lecture notes by S. Girvin,

Teachers

Prof. J. de Boer

Instituut voor Theoretische Fysica
Office 3.57
Tel: 020 - 525 5769/5773

Home page Jan de Boer

Email: J.deBoer@...

Prof A. Pruisken

Instituut voor Theoretische Fysica
Office 2.63
Tel: 020 - 525 5746/5773

Email: A.M.M.Pruisken@...

Schedule

Integrated lectures and practice sessions: Thursday morning 10.00-13.00, in room J/K 3.85. First session February 5, last session May 14; holidays March 26 and April 30. Exam May 28. Take home tests will be handed out on March 19 and April 23.

Week-by-week

• Week 6, February 5.
  Presentation grand plan; Review of SPCM I.
• Week 7, February 12.
  1D Ising model: scaling and block spin RG [AS sect 8.1]; problems: naive mean field approach to d-dimensional Ising model; exercise page 424; exercise page 432.
• Week 8, February 19.
  Dissipative quantum tunneling [AS sect 8.2] and beginning of general RG [AS sect 8.3]; problems: (i) problem 8.1 of the book and (ii) work out the RG transformations for the coupling constants of a d-dimensional field theory with arbitrary potential when we simply put the fast degrees of freedom equal to zero.
• Week 9, February 26.
  General RG [AS sect 8.3]; problems: (i) review the definition of the cricital exponents in the info block in sect 8.3 and (ii) show that there are only two independent critical exponents and express the others in these two (exercise at the end of sect 8.3).
• Week 10, March 5.
  Ferromagnetic transition [AS sect 8.4]; problems: (i) Write the scaling relation for the reduced free energy near the non-trivial fixed point of (8.30); keep all three parameters r,h,lambda. (ii) Consider a generalization of the Ising model where the spins take values in (-1,0,1). Keep the usual nearest neighbor coupling and magnetic field coupling, but also add a coupling with a parameter h' which multiplies an operator which counts the number of spins with value zero. Perform the Hubbard-Stratanovich transformation to write this as a field theory. Show that for generic values of the couplings the effective field theory is similar to that of the Ising model, but that one can tune paramters in such a way that the phi^4 term in the action is absent, and there only is a phi^6 term. Find the mean field critical exponents for this special case.
• Week 11, March 12.
  BKT transition [AS sect 8.6]; problems: (i) exercise page 482. (ii) Redo the RG analysis of the BKT transition in 2+ε dimensions. The RG eqns in (8.47) are slightly modified, the first eqn has an extra term -ε/J on the right hand side, the second equation remains unchanged. Find the fixed points, linearize the RG flow near them, find an expression for the correlation length close to the fixed points, and determine the critical exponent ν. Take ε=1 and compare with the results in e.g. cond-mat/0605083 (optional).
• Week 22, May 28.
  Written exam.

Assignments and exam

Partial credit given for take-home problem sets. Final grade is MAX(E,(T1+T2+4E)/6) with E=exam and T1,T2=take home 1,2.

The final exam will take place Thursday May 28, 10.00-13.00, room J/K 3.85.