Advanced Statistical Mechanics

Advanced Statistical Mechanics 171.704

Course information.
Syllabus.
Literature.
Lecture notes.
Homework.
Course project.
Miscellania.

Course information:

Lectures:
Thursday 9-10:30, 178 Bloomberg.
Friday 3:15-4:45, 361 Bloomberg.
Prof. Oleg Tchernyshyov.
Office 323 Bloomberg.
Office hours: Thursday 10:30-12 or by appointment.
Tel. (410)516-8586.
Email olegt@jhu.edu.

Syllabus:

Exactly solvable systems in 1 and 2 dimensions. Critical exponents. Universality classes.
Numerical simulations in statistical mechanics. Monte Carlo methods: algorithms of Metropolis and Wolff.
Landau's theory of phase transitions. Spontaneously broken symmetry. Continuous and discontinuous transitions. Multicritrical points. Critical fluctuations and the breakdown of the Landau theory.
Renormalization group. Stable and unstable fixed points. Universal scaling functions. Scaling variables and critical exponents. RG flows and phase diagrams. RG methods: real-space, perturbative renormalization in 4−ε dimensions. Finite-size scaling. Quantum critical behavior.
Random systems. Harris criterion. Random fixed point. Percolation.
Dynamical critical behavior.

Literature:

Required texts (available in the bookstore):

P.M. Chaikin and T.C. Lubensky, Principles of Condensed Matter Physics (Cambridge University Press, New York, 1995).
J. Cardy, Scaling and Renormalization in Statistical Physics (Cambridge University Press, New York, 1996).

Auxiliary texts (in the library):

J.J. Binney et al., The theory of critical phenomena. Contains a detailed description of the Monte Carlo algorithms you will need for the course project.
C. Domb, The Critical Point. An introduction into scaling and renormalization written from a historical perspective.

Reviews:

M.E. Fisher, Renormalization group theory: Its basis and formulation in statistical physics, Rev. Mod. Phys. 70, 653 (1998).
Phase transitions and critical phenomena, eds. C. Domb and M.S. Green (later C. Domb and J.L. Lebowitz). Continuously published reference volumes.
K. Binder, Applications of Monte Carlo methods to statistical physics, Rep. Prog. Phys. 60, 487 (1997).
F.Y. Wu, The Potts model, Rev. Mod. Phys. 54, 235 (1982).
S.L. Sondhi et al., Continuous quantum phase transitions, Rev. Mod. Phys. 69, 315 (1997).

Lecture notes:

Overview of the key concepts. Spontaneous symmetry breaking. Scale invariance at a critical point. Universality of critical behavior.
Exactly solvable model: 1d Ising ferromagnet. Thermodynamics from transfer matrix. Demise of long-range order in one dimension.
Exactly solvable model: 2d Ising ferromagnet. Survival of order at low temperatures. Kramers-Wannier duality. Critical temperature.
Numerical simulations. Monte Carlo methods. Critical slowdown. Finding the critical point in a finite system.
Mean-field theory. Landau's expansion of free energy. Continuous and discontinuous phase transitions. Ising and 3-state Potts ferromagnets.
Landau theory I. Tricritical point. Ising antiferromagnet in magnetic field.
Landau theory II. Bi- and tetracritical points.
Landau theory III. Continuous symmetry. Landau functional.
Ginzburg-Landau theory of superconductivity.

Homework:

Homework is given out every Friday. It is due the next Friday in class.

Assignment 1. Due February 10.
Assignment 2. Due February 17.
Assignment 3. Due February 24.
Assignment 4. Due March 3.
Assignment 5. Due March 10.
Assignment 6. Due March 31.
Assignment 7. Due April 14.
Assignment 8. Due April 21.
Assignment 9: Prob. 6.1 in Cardy. Due May 5.

Course project:

Write your own Monte Carlo simulation of a classical statistical model and study its critical behavior. In particular, measure the critical exponents. You will get the most bang for the buck from a two-dimensional system (not much happens in 1 dimension; 3-dimensional systems require a lot of CPU time). The most obvious choices are the Ising model or the Potts model with, say, q = 3 states. The ferromagnetic Ising model is the most elementary. If you get bored, try the Potts antiferromagnet.

Understand what you are doing. Before you start growing your virtual crystals and building imaginary magnetometers, play with the simulator that I used in the first lecture. You will find pertinent information below in Miscellania.

Start early in the course, preferably as soon as we have discussed the subject of numerical simulations in class. See Binney's book for a description of Monte Carlo algorithms. The local Metropolis scheme is the most straightforward but becomes very slow in the critical region. Cluster algorithms (Swendsen-Wang or Wolff) will do a much better job. Sample C++ code implementing the Metropolis algorithm for the q=3 Potts ferromagnet can be found here.

These notes describe the extraction of critical indices in the following systems:

3-state Potts ferromagnet,
Ising antiferromagnet.
Frustrated Ising antiferromagnet (non-universal critical exponents).

Observation of critical behavior and especially extraction of critical exponents require systems of large enough sizes. As you will need a fast running code, Matlab, Mathematica, and Maple are most definitely out. Use a compiled programming language of your choice. Compilers of C, C++, and FORTRAN are available on any UNIX workstation or Linux PC. If you don't know any of these, C is your best bet. It is a small language: you can learn its syntax and start writing a code in a week. The C Programming Language by B.W. Kernighan and D.M. Ritchie is the one and only reference you'll ever need.

Miscellania:

In my lecture demonstration I used xpotts, a simulation program written by Michael Creutz of the Brookhaven National Lab. You can download the code freely from his web site, along with quite a few other xtoys including forest fires, sand piles, and cellular automata. The programs compile and run on pretty much any UNIX or Linux machine with X Windows.
Because xpotts is a microcanonical simulator, its controls are a little idiosyncratic. Instead of temperature and magnetic field, your knobs are energy and magnetic moment. However, both T and h are measured and shown. The microcanonical approach has, in fact, two advantages: it is simple to implement and allows us to observe phase coexistence (try that in a nonzero magnetic field).