To what extent can phase transitions be made rigorous?

Question

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?

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):

• Friedli and Velenik, Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction
• Georgii, Gibbs Measures and Phase Transitions
• Ruelle, Statistical Mechanics: Rigorous Results
• Simon, The Statistical Mechanics of Lattice Gases
• Prum, Stochastic Processes on a Lattice and Gibbs Measures
• Presutti, Scaling Limits in Statistical Mechanics and Microstructures in Continuum Mechanics
• Fernández, Fröhlich and Sokal, Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory
• Israel, Convexity in the Theory of Lattice Gases
• Ellis, Entropy, Large Deviations, and Statistical Mechanics
• etc.

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.