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.

Back to home

---

Last modified: Jan 7, 2010.