Renormalization group flows, above (left) and below (right) the critical dimension D = 6, for the Potts model in the limit of q = 1. The initial w parameter (of the original model) is large and positive, and flows to the pure quadratic model (w = 0) on long length scales when D > 6. Critical behavior is then determined by the "Gaussian" fixed point (solid red). When D < 6 the downward flow is intercepted by the new "Wilson-Fisher" fixed point (solid red). Notice the Wilson-Fisher fixed point has both stable and unstable axes, just as the Gaussian fixed point for D > 6. The Gaussian fixed point still exists for D < 6 (empty red) but only has unstable axes.

Potts field theory RG-calculation summary

Announcements

•  (9/11) Please have your homework ready by the due date in lecture.
•  (9/20) The second problem of assignment 2 turned out to be more work than what was intended, and so the two pair distributions have been posted below (f₁(u,v) for u and v on the same row/column, f₂(u,v) when they appear on distinct rows and columns). These formulas include corrections up to O(1/n²), which is what you will need for problem 3. You are only required to derive f₁, the easier one because it involves just a single vector on the hyper-sphere.

Requirements and prerequisites

• No auditors: if you attend you also must do the work.
• You must already have taken 6562 (or an equivalent course).
• Some knowledge or concurrent study of a low-level programming language (C or 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 topic is the Hadamard phase of orthogonal matrices. This is one of the simplest systems that exhibits a first-order phase transition. It was chosen for the research project because not much is known about it and even less (zero) has been published. We will develop basic analysis skills in a setting where the answers are not yet known, and where coming up with good questions and hypotheses is just as important as answers. The prospect of novel insights (first order transitions, Hadamard matrices) is high, especially for engaged students.
• 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 any you will encounter over your entire education!

Homework

•  assignment 1 (due 9/12) solution
•  assignment 2 (due 9/26) updated 9/13, pairDistributions.nb solution
•  assignment 3 (due 10/10) solution
•  assignment 4 (due 10/31)
•  assignment 5 (due 12/10) updated 11/4

Course grade

• Based on five homework assignments which may include computer simulations of the Hadamard 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: W 12-3pm, PSB 426
• Jaron Kent-Dobias: W 3-4pm, PSB 425A