Course Syllabus Syllabus

Lecture notes

For Lecture 1 to 4, read Herbut book Chapter 1, and 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 : Onsager solution

Lecture 5 Ginzburg Landau mean-field theory

For Lecture 5 to 10, read Goldenfeld book, Chapter 5, 6, 7, 9

Lecture 6 Gaussian model and Ginzburg criterion

Lecture 7 Scaling hypothesis

Lecture 8 Dimension and Anomalous dimension

Lectures 9 and 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 and 17 K-T transition of XY model

More is different The famous paper by P. W. Anderson.

Howework assignment

HW1: Goldenfeld book "Lectures on phase transitions and RG", Chapter 3, Excersie 3.1, 3.2, 3.3, due time April 18, on class Solutions posted Apirl 22 .

HW2: Goldenfeld book "Lectures on phase transitions and RG" Chapter 5, Excercise 5.1, 6.3; Chapter Excercise 7.1, due time May 14 on class. Solutions posted on May 22 .

HW3: Goldenfeld book "Lectures on phase transitions and RG" Chapter 9, Excercise 9.1, 9.3; Chapter 12 Excercise 12-3 due time May 28 on class
Solutions posted on Jun. 11 .


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.