Physics 217 -- Phase transitions and RG

Course Syllabus Syllabus

Lecture notes
  • More is different The famous paper by P. W. Anderson.
  • For Lecture 1 to 4, Goldenfeld book Chapter 2 and 3
  • Lecture 1 One dimensional Ising model
  • Lecture 2 Two dimensional Ising model (I)
  • Lecture 3 Two dimensional Ising model (II)
  • Lecture 4 Two dimensional Ising model and 1D quantum Ising model (III)
  • Supplemental Material Correlation function of 2D Ising model
  • Supplemental Material Onsager solution to 2D Ising model
  • Lecture 5 Ginzburg Landau mean-field theory
  • Lecture 6 Gaussian model and Ginzburg criterion
  • Lecture 7 Scaling hypothesis
  • Lecture 8 Dimension and Anomalous dimension
  • Lecture 9-10 Real Space Renormalization Group I and II
  • Lecture 11 4-epsilon (I): Gaussian model and scaling
  • Lecture 12 4-epsilon (II): phi-4 theory and RG equations
  • Lecture 13 4-epsilon (III): calculation of critical exponents
  • Lecture 14 4-epsilon (IV): Integration of RG equations, crossover
  • Lecture 15 Non-liner sigma model, asymptotic freedom
  • Lecture 16-17 K-T transition of XY model

    Howework assignment
  • HW1: Goldenfeld book "Lectures on phase transitions and RG", Chapter 3, Excersie 3.1, 3.2, 3.3, due time Oct 16, on class Solutions posted on Nov 3 .
  • HW 2: Do the Onsager's solution to the 2D Ising model with two different couplings Jx and Jy along the x and y-directions, respectively. Figure the critical temperature, Free energy, internal energy, and specific heat due time Nov 1 class.
  • HW 3: Do a real space RG for the Ising model for a 2D lattice except the triangular lattice example taught in class. Due time Nov 13 class.
  • HW 4: 1) Goldenfeld book "Lectures on phase transitions and RG", Chapter 12 Excercise 12-3
    2) Derive the RG equation for the O(n) phi-4 model, and analyze the fixed points, and critical exponents nu and eta. due time Nov 27 on class

    Final projects (to be added more) .
  • Exact solution to 2D Ising model. Ref. Schultz, Lieb, and Mattis, Rev. Mod. Phys. 36, 856 (1964).
  • Properties of quantum Ising model, Chapter 4 of the book "Quantum Phase transition" by S. Sachdev.
  • Fluctuation induced first-order phase transition (Weinberg-Coleman mechanism) (I) Ref: B. I. Halperin, T. C. Lubensky and Shang-keng Ma, PRL 47, 1469 (1974);
  • Fluctuation induced first-order phase transition (Weinberg-Coleman mechanism (II) Ref: "Radiative Corrections as the Origin of Spontaneous Symmetry Breaking", Weinberg-Coleman, Phys. Rev. D 7, 1888 (1973), or Peskin book P469.
  • Quantum critical behavior of Heisenberg model in 2D Ref S. Charkaravarty, B. I. Halperin, and D. R. Nelson, PRB 39, 2344 (1989)
  • Mermin-Wagner theorem and related things Ref: Auberbach's book: Interacting electrons and quantum magnetism, Chapter 6.
  • Application of non-linear sigma model to quantum spin chain Ref: Auberbach's book: Interacting electrons and quantum magnetism Chapter 12 and 14
  • RG for dynamic systems. Ref: Goldenfel's text book
  • RG in the field theory method: Callan-Symmanzik equaiton. Ref: Peskin's textbook. Chapter 12 and 13
  • Quantum phase transition of itinerant electrons. Hertz-Millis Ref: Ben Simons' texbbook "Condensed matter field theory", Chapter 8, problem 8.8.2.
  • The density-matrix renormalization group: Rev. Mod. Phys. 77, 259 (2005) U. Schollwock.
  • An introduction to lattice gauge theory and spin systems Rev. Mod. Phys. 51, 659 (1979), John B. Kogut.
  • Quantum phase transtion, Rev. Mod. Phys. 69, 315 (1997) S. L. Sondhi, S. M. Girvin, J. P. Carini, and D. Shahar.
  • Criticality on fractals, arXiv:1404.6311, "Quantum criticality from Ising model on fractal lattices" Beni Yoshida, Aleksander Kubica.

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    Last modified: Jan 7, 2010.