1. Spin Glass Theory

1.1 Replica calculations in the Sherrington-Kirkpatrick (SK) model (feat. replica symmetry breaking)

1.2 Replica Method for the Machine Learning Theorist

A very visual guide to the replica method for people interested in solving problems in ML/optimization etc. Co-wrote with Blake Bordelon, Abdul Canatar, Boaz Barak, and Cengiz Pehlevan.

General Statistical Physics

Where does the Gibbs distribution (Boltzmann distribution) come from? A derivation from first principles (kind of).

Comments and requests (via email) are needed!

I wrote these notes with the following principles:

  1. Respect the equal sign! No term is dropped and no constant is removed without explanation. If I take an approximation, I always state so.
  2. Respect notations! I try to always explicitly define new variables, subscripts and so on.
  3. Avoid physics jargons as much as possible. If I take the “beta to infinity limit”, I won’t say I’m doing the “zero temperature limit”.
  4. Avoid explaining physics with physics.

I’ always looking for comments on these notes. If you have anything to share, please email me at hshan at g dot harvard dot edu.

One more comment:

My impression is that in these calculations, while the math is usually elementary (calculus and linear algebra mainly), the amount and complexity of it can be very frustrating. If you are feeling this way, remember that C.N. Yang also felt the same way:

I was thus led to a long calculation, the longest in my career. Full of local, tactical tricks, the calculation proceeded by twists and turns. There were many obstructions. But always, after a few days, a new trick was somehow found that pointed to a new path. The trouble was that I soon felt I was in a maze and was not sure whether in fact, after so many turns, I was anywhere nearer the goal than when I began. This kind of strategic overview was very depressing, and several times I almost gave up. But each time something drew me back, usually a new tactical trick that brightened the scene, even though only locally. Finally, after about six months of work off and on, all the pieces suddenly fitted together, producing miraculous cancellations, and I was staring at the amazingly simple final result … Since there were some limiting procedures in my calculation that were not rigorous, I did not feel quite secure …

Selected Papers, 1945-1980, with Commentary by C.N.Yang, 1982.

Here, Yang was describing his calculations of the spontaneous magnetization in the 2D Ising model. Last updated Apr 20 2020