Physics 563: Phase Transitions and the Renormalization Group

Spring 2013

Instructor: Nigel Goldenfeld

Time: 1.00-2.50pm, Mondays, Wednesdays and Fridays
Place: Loomis Laboratory of Physics Room 136



Please register for the email list for the class, so that last minute announcements, class cancellations etc. can be sent to you.  This applies even if you are not taking the class for credit.   Thanks.

The gradebook for the class is now on the web.

Term essay archives

Term essays for Spring 2013

Term essays for Spring 2012

Term essays for Spring 2010

Term essays for Spring 2008

Term essays for Fall 2005.

Term essays for Spring 2000.  These cover topics ranging from the behaviour of biological membranes, the formation of black holes, and physics of traffic jams.

Term essays from 1998

Now available ...

Course Handout

Homework 1 and photographs of the first atomic explosion

Homework 2

Homework 3

Homework 4

Homework 5

Homework 6

Homework 7


Software and Experimental Data


For HW 4:

Here is a set of real experimental data on the penetration depth of the high temperature superconductor YBCO, which were published in S. Kamal, D. Bonn, Nigel Goldenfeld, P. Hirschfeld, Ruixing Liang and W. N. Hardy. Penetration depth measurements of 3D XY critical behaviour in YBa2Cu3O6.95 crystals. Phys. Rev. Lett. 73, 1845-1848 (1994). This paper is frequently cited, being the first really strong evidence for XY scaling in a three dimensional superconductor.

Incidentally, this is one of my favourite articles for many reasons. I discovered the critical behaviour in the data late one night, but the data had been taken to test a different prediction Peter Hirschfeld and I had made about d-wave superconductivity. At that time, I was trying to see if YBCO was a d-wave superconductor, and our work had arrived at a sharp test of the d-wave scenario, based on the behaviour of the low temperature penetration depth with impurities present. This turned out to be crucial for the identification of the pairing state of YBCO, because earlier we had discovered that the best evidence for s-wave pairing, a fit to BCS theory for the penetration depth, was misanalyzed, and the correct analysis showed strong evidence for a power law at low temperatures, whose most likely interpretation was d-wave pairing with impurity scattering.

The data set was so complete that it became apparent that there was a deviation from mean field theory at high temperature, near the superconducting critical temperature. The measurements were redone in that region and we were able to report excellent agreement with several predictions of theory for the critical phenomena, including the Harris criterion (I am not sure that I have ever seen any other experimental test of this). Thus, our studies of the pairing state also led unexpectedly to a sharp observation of critical phenomena.

These measurements were later extended to ultra-pure YBCO samples, and cover what is to my knowledge the greatest range of scaling near a critical point in a solid material.

One of the morals of this story relates to what we discussed in class: try to find data where there is no background substraction required.

I also like the paper because I got to write most of it, and the structuring and exposition are all carefully done. I like to recommend this paper to students as a good model.

Here is a set of real experimental data for the friction factor experienced by a turbulent fluid as it flows through a pipe. The friction factor is basically the drag force exerted on the wall by the fluid as it flows past the wall of the pipe, but normalized by the mean kinetic energy density of the fluid. The data were reported in a famous publication by Nikuradse in 1933, and describe how the friction factor depends on the speed of the fluid (represented by the Reynolds number) and by the relative roughness of the wall compared to the pipe diameter. The format for the data file is described in the README file. After you have done the homework assignment, you can read about some of the Illinois group's work on this turbulence problem.


There are several computer simulations of the Ising model and other simple statistical mechanics model available on the WWW. With these you can visualise equilibrium behaviour of correlations, dynamic critical behaviour and even the kinetics of the approach to equilibrium. These include the following links:

Miscellaneous Information

An important area of statistical mechanics is using computers to evaluate thermal averages.  Although we do not have time in this course to discuss this, let me point out some useful resources:


Exact solutions of the two dimensional Ising model

Here is a recent article that reviews briefly analytic solutions of the two-dimensional Ising model, and in particular the mapping to a system of free fermions which is what enables the analytic solutions.  The article has references in it to much earlier work, especially the seminal review article of Schultz, Mattis and Lieb, Rev. Mod. Phys. 36, 856 (1964).  Some historical context and a brief discussion of the way in which the Onsager solution has influenced subsequent developments is given in this article, written on the 50th anniversary of Onsager's solution.

Term Paper Instructions

Instead of a final exam, this course will have a term paper assignment. The subject matter can be any topic in physics which is related to Phase Transitions in some sense.  Since many interesting phenomena exhibit or are a manifestation of Phase Transitions, you have unusual latitude in your choice of topic.  It need not be restricted to condensed matter but can cover the many recent and exciting developments in other areas of science, including, but not limited, to: high energy physics, cosmology, even biology.   I hope many of you will chose topics in these non-condensed matter areas.

I have no objection to people choosing to write about their own research area if it falls within the scope of the course. Some of you may, however, feel the need to broaden your outlook by literature searching a topic that is new to you.

You may not reuse an essay from another course to satisfy the term paper requirement.

You should be careful to note the requirements of originality: to forestall any questions, please see this note about plagiarism.

I can provide you with a guide to the literature for many topics if you come and ask me. However, I really don't want to do this. Part of this assignment is to give you an opportunity to develop the skills in doing a literature survey and digging up information from the library.  Other important components of this assignment are that you demonstrate good taste, curiosity and ambition in your choice of subject, and that you are capable of distilling the most important and essential details from very technical papers.

Some hints: look at the back of the current Reviews of Modern Physics where you will find a listing of topics that have been reviewed in the last ten years or so. These are always a good starting point. Similarly for journals such as Advances in Physics and Reports of Progress in Physics. Use internet keyword searches on archives and search engines such as Google.

Don't be restricted by the topics that we addressed in class. We didn't have time to cover the huge variety of phase transition phenomena or techniques that arise in the real world.

Detailed Directions

The purpose of your essay is to explain why the problem is interesting, what has been done, and what are the conclusions. Don't go into unnecessary technical details. The amount that you personally chose to work through the technical details is up to you; my goal is that you understand the broad issues. Hopefully you will find your topic sufficiently interesting that you will wish to delve deeper (and perhaps even think for yourself about the subject).  You should imagine that you are writing the paper for a reader who is like you were before you started thinking about your topic.  Every essay must include some sort of discussion of experiment or observations: these can either be the focus of the essay, or at least must be mentioned specifically with regard to how they demonstrate, provide counter-examples to, or otherwise inform theory.  Essays which are purely theoretical may receive a relatively lower grade than others.  More detailed suggestions about format are given below.

The final exam/paper is due by 4:30 PM, Thursday, May 9 2013. No excuses for lateness will be accepted unless there are extenuating circumstances or previous arrangements have been made, in accord with University regulations. It should not be less than 8 pages long, single spaced 12 point font, including figures.  It should be no more than 12 pages long, single spaced 12 point font, including figures. I will not read more than 12 pages of an essay.  You have been warned!

Your essay must be written in an electronic format, including the figures, if any.  Ultimately, all essays will be posted on the WWW. The format for submission will ONLY be an Acrobat PDF file. Each essay should include a cover page which will consist of the following (a) Title and author's name. (b) abstract. The essays should also have a decent set of references, at least 6, which should include particularly good review articles.

Submission instructions:

(A) Convert your essay to a PDF file.

(B) Name your essay according to the following scheme

<Your last name>.pdf

(Example:    goldenfeld.pdf)

(C) Email to Nigel Goldenfeld <>

Make the subject heading of your email

563 final:   <Your last name>.pdf

and have in the body of the email your name, the title and a brief  abstract of your essay.





Subject: 563 final:   smith.pdf


Author: Fred Smith

Title: Critical dynamics of the superconducting transition


This essay describes the observations, computer simulations, and analytic theory of critical fluctuation contributions to the electrical and thermal conductivity near the superconducting transition of the high temperature superconductors YBCO and BSCCO.



Your paper should have approximately the following structure, but feel free to modify it to fit your chosen topic.  Here are some suggestions for the sorts of questions your paper should address to make it most useful to the reader.  As you will see, the purpose is not to focus too much on technical details.

Introduction and Background:

What hypotheses are being tested in this paper?

What information induced the authors to perform the experiments/theory?

What new methods or insights brought to bear on the problem?

Why did you chose to write about this topic? 

Why is this interesting or important?


What are the critical methods of the paper?

What enabling technologies are used?

What are the weaknesses of the methods used?

Are there other or better approaches that could be used?

Results and Discussion

What are the primary conclusions of the paper?

Did the authors prove their hypotheses?

What novel information or directions come from this work?

What control experiments were performed? (If appropriate)

What assumptions still remain in the work?

How could these assumptions be tested?

What other explanations for the observations are still possible?

What would you do next to advance this field?

Nigel Goldenfeld

Telephone: (217)-333-8027

Office: 3-113 ESB
Nigel Goldenfeld's Home Page

Updated January 2013