## Monday, September 10, 2018

### What can students learn from an Ising model simulation?

Computer simulations can provide significant insight into different physical phenomena. Two decades ago the best one could do in a class or seminar was show screen shots of simulations and try and explain what was going on. Now one can show a simulation live and even vary parameters in real time to provide insight. I have done this quite a bit with Solid State Simulations.

One simulation I like but have never used effectively is that of the Ising model.
See for example, Daniel Schroeder's simulation or James Sethna or Matt Bierbaum.
What does it help me understand?
The main ideas are the concept of symmetry breaking, the correlation length, and the divergence of the correlation length at the critical point.

1. Watching the different configurations changing with time illustrates the notion of an ensemble.
2. At high temperatures one sees the paramagnetic phase where the spins are independent of each other and so there are no domains.
3. As the temperature approaches the critical temperature (T=2.27J) from above the correlation length increases and large fluctuating domains form.
4. Below the critical temperature large domains form and fluctuate less and less as the temperature lowers.
5. The ferromagnetic ground state (blue or yellow, up or down spin) in zero external field depends on the history. This illustrates symmetry breaking

Any other things?

1. I'm not sure it's the appropriate place to introduce these topics but: Scale invariance and RG flow. See https://www.youtube.com/watch?v=MxRddFrEnPc

2. My own version, which I use in teaching fourth-year undergraduate and first-year graduate classes:
https://www.nottingham.ac.uk/~ppzscp/MC/

I try to emphasise that the magnetisation is discontinuous across the first-order transition line (going from positive to negative h for T<Tc). You can "prove" this in the low-T limit, but I think it's still quite surprising.

3. The Ising model can also be useful to nucleation processes (there is a nice JCP from Binder: https://aip.scitation.org/doi/pdf/10.1063/1.4959235).
You could also consider variants of the Ising model where specific spins are fixed either up or down to create walls that can help students to understand heterogeneous nucleation.

4. On the theory part: I really enjoyed all the possible ways of solving the problem, from Onsager's to Feynman's method. The dependence on dimensions. All the approximate solutions. The link with QFT and RG, although I think it's not a complete analogy since it's a first order transition. The correspondence with the bosonic string...

Regarding the simulation, I liked things which are more related to computer science, like noticing it's a Markov chain.

5. Btw, the history is cool: Ising solved the 1D case in his Phd, left academia, physicists ignored the 2D model but engineers used it, the Peirls got an approximate solutions and then the rest of physicists had their attention on it...