What is a simple example illustrating the concept of "commensurate" and "incommensurate" order in condensed matter physics? In a wide range of contexts in condensed matter physics, e.g this paper, the concepts of commensurate and incommensurate orders are invoked to describe particular ordered phases. I think I have some intuition for what this means - that somehow lattice translation symmetry is inconsistent with the structure of the ordered phase - but I would like to refine this notion. A clear formal definition of these terms would also be useful.
 A: Commensurate means that the ration of two parameters is an integer or, more generally, a ratio of integers.
For example, if we consider a one-dimensional lattice with period $a$, and we impose on top of it a perturbation potential $$V(x)=V_0\cos(kx)$$ with spatial period $$\lambda=\frac{2\pi}{k}.$$
The sutuations where $\frac{\lambda}{a}=n$ or $\frac{\lambda}{a}=\frac{1}{n}$ or $\frac{\lambda}{a}=\frac{n}{m}$, where $n,m$ are integers, can be all referred to as commensurate perturbation, although the results would be somewhat different. Such a perturbation preserves some of the translational symmetries (which could be treated as another definition of the term commensurate).  On the other hand, a general periodic perturbation would not fall in any of these classes and is usually more difficult to treat.
A: I am very new at this and have seen these two terms come up in various topics within CM physics so the answer depends on what you are talking about. I'll provide an answer in different words than what was stated in the previous answer.
Commensurate means integer multiple of the crystal lattice constant.
Incommensurate means not an integer multiple of the crystal lattice constant.
Near-commensurate is, of course, somewhere in between.
Example:
A commensurate charge density wave has a wavelength that is exactly an integer multiple of the lattice constant.
See the second paragraph in Charge Density Wave Wikipedia
