Ising Model

Important Things to Note:

You can solve simple ising model problems using the canonical ensemble. It’s important to look at the derivation of the ising model and understand the statements about it being a “mean-field theory.” Harder ising model problems can be solved by using this thing called a “transfer matrix.” You’ll need to learn this too.

Review:

The ising model uses the canonical ensemble. So it’s not a bad idea to refresh yourself on how the canonical ensemble works. Make sure to reread the last part of the notes.

New Things to Learn:

Mean-field theory:    Read Chapter 3 in Equilibrium Statistical Physics by Bergesen (sections 3.1-3.6).  It’s a pretty mediocre book, but I haven’t really found much better for this topic.

Transfer Matrix:   Read these excellent lecture notes from Hiram College.

Most important Problems & Solutions:

Basic Ising Model Problems (Just do Canonical Ensemble):

  • Problem #2 (solution)
  • U Illinios Quals Problem (I don’t have a solution written up but it’s easy)

Transfer matrix Ising model:

  • Do previous problems using the transfer matrix and compare your answer (it should be the same).
  • Problem #2 (you can skip the stuff about density matrices for now)

Harder Ising Model Problems:

  • Problem #3 in 2015 Fall Comprehensive Exam
  • Problem 3.14 (the Potts Chain) of the “Equilibrium Statistical Physics” (I found this to be very helpful in understanding the transfer matrix) I’ll upload my scanned solutions soon.
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