How relevant are solvable models such as 2D Ising to real systems?

[petergreat]

It surprises me how much mathematics is needed to solve the innocent-looking 2D Ising model, and the Heisenberg model is quite an exercise even in 1D. I begin to doubt whether solving these models is really worth the effort, given that these systems don't look very realistic, and computer simulations give you answers much more quickly. Some calculations are useful due to universality, but others are not. So what's your opinion regarding whether solvable models of statistical mechanics are important to understanding real-world physics?

 

[kanato]

I'd say these are highly relevant.

Computer simulations generally introduce approximations of some sort, and it can be difficult to accurately assess the effects of these approximations. Monte Carlo simulations can give more-or-less exact results within statistical error, but may behave badly under certain conditions. For instance, most thermodynamic Monte Carlo simulations have difficulties at low temperatures, requiring very long computation times. Hirsch-Fye QMC behaves very badly when the interaction parameter is large. There is also the infamous sign problem for many types of QMC calculations. On the other hand, when models are exactly solved, you gain knowledge of things like the fact that the 2D isotropic Heisenberg model has no phase transition at finite temperature. It eliminates any debate about the accuracy of certain approximation methods, etc. and whether results obtained are accurate or even qualitatively correct. Also it prevents people from spending time on trying to just get a more accurate answer (xxx is only known to 3 decimal places, let's throw a big supercomputer at it and see if we can get up to 5 decimal places!). Also, an answer which is exactly known can be used to benchmark a particular approximation technique, which is especially useful if that technique can be applied to both a model for which the exact solution is known and another model for which there is no known exact solution.

Many materials display qualities that can be reasonably modeled by 2D or 1D models. The cuprates for instance have planes of CuO layers that are separated vertically by other atoms, and one can rather accurately take a 2D model of the CuO planes for examining certain types of properties. There are several materials that have chains of atoms magnetically coupled, and so are effectively modeled by a 1D model.

Just yesterday I was looking at some calculations I have done for an unusual system, the model for which under certain conditions can be reduced to look like a simple Heisenberg model. I found it very useful to be able to look up exact results for such models to compare to without having to consider any ambiguities that may arise to the the calculation methods other people used.

As is always the case with research, one cannot accurately claim a priori that a particular research endeavor will be useless. But I think these sorts of things have proven their worth. Also, keep in mind, the tremendous amount of effort that goes into an exact solution is only done once. After that, for everyone else in the future the effort is just a literature search.