In our discussion of one-dimensional physics, more than once we have encountered situations when some relevant interaction scales to strong coupling. In such a situation the original description becomes inapplicable at low energies and must be replaced by some other description. In all previous cases I have restricted the discussion by some qualitative analysis, promising to provide more details later. Now the time is ripe to fulfil the promise.

In fact, almost all our understanding of strong coupling physics comes from exact solutions of just a few models. The most ubiquitous among them is the sine-Gordon model, which we have already encountered in this book many times. Anyone who thoroughly understands this model and possibly a few others (such as the O(3) nonlinear sigma model and the off-critical Ising model) may consider themself an expert in the area of strongly correlated systems.

The sine-Gordon model belongs to a category of integrable field theories. This means that it has an infinite number of constants of motion. On the one hand this fact makes it possible to solve this model exactly and describe its thermodynamics and correlation functions; on the other hand it makes the results less general than one might wish. First of all, not all interesting models are integrable, and one may wonder how and in what respect their behaviour may differ from the behaviour of integrable models. Integrable models possess some exceptional physical properties (such as ballistic transport) which are destroyed when the integrability is violated.

There are several important review articles which one can read to familiarize oneself with the subject.