Renormalization group flow in the (r,w) plane of the q = 1 Potts model in 6.2 dimensions (left) and 5.8 dimensions (right). Above six dimensions there is only the Gaussian fixed point (r,w) = (0,0), with one stable and one unstable axis. Another fixed point appears below six dimensions, the Wilson-Fisher fixed point, also with stable and unstable axes. Below six dimensions both axes of the Gaussian fixed point (open circle) are unstable.

Announcements

• Welcome! The course roster is not in error: lectures are in 200 Savage.
• Watch this video to see a macroscopic manifestation of quantum mechanics! 
• (9/3) Good news -- the course will continue despite the small enrollment! The only change is that the lecturer will also be the grader.
• (9/11) The function urand( ) in rowrot.c returns uniform random numbers between 0 and 1. The global variable rub counts sign changes in the matrix. In the hard matrix model these are the "rubicons" that enable free diffusion.
• (10/1) Read this Science article by Donald Knuth, paying special attention to the section "Counting the Paths on a Grid". HW4 will include a problem on this topic.
• (10/21) I've had to schedule an urgent eye exam for 8am Tuesday and, depending on the status of the waiting room and luck in hiring an Uber ride, may be slightly late for the lecture.
• (10/21) Regarding HW3, I thought it was clear that the constants e0, e1 and c2 are explicit numbers to be worked out analytically. Also, the energy and heat capacity outputs of hmm.c are already per-matrix-element (no need to divide by n^2). Please redo these problems if these remarks apply to your work.
• (12/16) q = 1 Potts model RG notes
• (12/16) Hint for Lionel Levine's hat puzzle

Requirements and prerequisites

• No auditors: if you attend you also must do the work.
• You must already have taken 6562 (or an equivalent course).
• Resources and ability to run a computer program written in a compute-efficient language like C (not python).

Course Description

If you have not been scared away by the information above, here is what we plan to do this semester.

• A class research project that may turn into a publication. The nature of glasses, specifically continuum models without quenched disorder, may be one of the greatest outstanding mysteries of statistical mechanics. In Fall 2019 we studied a new system, the hard matrix model, to gain insights. Here is a summary of what we learned. In this semester we pursue in greater detail some of the questions raised by that first study. Could it be that the glass transition goes away when one is careful about taking the thermodynamic limit? Is the energy landscape picture fundamentally misleading when saddle points abound? And how should we think about first order phase transitions when there is no obvious way to have phase coexistence? Here is a list of possible projects you could work on individually or as a team.
• The percolation limit of the Potts model as a case study of critical behavior. We start with the tractable Erdos-Renyi random graph model and end up with a continuum field theory. This part of the course is all about the strange transformations and limits that have become standard tools for researchers in this field.
• Renormalization group analysis of the Potts field theory. You will get hands-on experience with Ken Wilson's famous technique by calculating the exponents of percolation just below six dimensions. In the last weeks of the course, when you calculate a single fraction, you will get a harder (analytical) workout than in the quantum field theory course!

Homework

• HW1  rowrot.c
• HW2  hmm.c
• HW3
• HW4  self-avoiding paths and dual variables
• HW5  hmm.c (new version)

Course grade

• Based on five homework assignments which may include computer simulations of the hard matrix model. There are no exams.

Text

• Because this is the advanced course we assume you are familiar with the basic concepts and will not follow any textbook. While there may be some review at times, the approach will be hands-on calculations with no omitted details. Bring pen and paper to lecture and be prepared to do short calculations.

 Office hours

• Veit Elser: Tuesdays, 3-4pm, 426 PSB