It seems a lot of physical intuition in statistical mechanics, for example phase transitions, critical temperature, scaling hypothesis, renormalization group methods etc. should have a purely mathematical formulation; my question is: to what extent can this be done? Can we prove statements from a standard textbook (say Statistical mechanics by Huang) in a mathematically rigorous way?

A more specific example: it is well know that the 2D ising model with no external magnetic field has a 2nd order phase transition, can this be proven rigorously?

  • $\begingroup$ Are you looking for exactly solvable systems with phase transitions? The free bose gas is solvable in terms of polylogarithms and has a phase transition. $\endgroup$
    – AfterShave
    May 27, 2022 at 6:21

1 Answer 1


Yes, a lot can be proven rigorously, at least for lattice systems.

There are entire books on this topic. Here are a few (the first one can be legally downloaded for free):

Concerning the specific list of topics you mention, let me mention that rigorous results about the critical behavior remain scarce. Exceptions are:

  • in some planar models (in particular, the Ising model), a lot of progress has been made in the last 15 years, since the introduction of the Schramm-Löwner evolution made it possible to prove conformal invariance; (one example)
  • in some cases, it is possible to implement rigorous renormalization group methods to obtain information about the critical behavior of nonintegrable perturbations of integrable models; (one example)
  • it is also possible to implement rigorously the renormalization group method in sufficiently high dimensions; (one example)
  • in sufficiently high dimensions (at the very least, above the upper critical dimension), an alternative way of obtaining detailed information about the critical behavior of a variety of models (including Ising) is the lace expansion. (one example)

As to your question about the order of the phase transition in the Ising model, I don't think any of the books I mention above proves the fact that the latter is of second order. In dimension $2$, this follows, for instance, from Onsager's exact expression for the free energy. In higher dimensions, continuity of the magnetization at $\beta_{\rm c}$ is proved in this paper.


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