From the point of view of non linear dynamics where self similarity plays an important role if the attractor is a fractal I would say that the difference is one between continuous and discrete transformations.

A self similar transformation like the one producing the Cantor set or the Sierpinski triangle proceeds by discrete stages. The fractal which is the limit when the number of stages N tends to infinity shows self similarity (e.g is identical to itself) only for a discrete number of stages.

For instance when zooming on the Sierpinski triangle, one may not zoom anywhere and by any zooming factor. One has to zoom only with a factor 1/3 and center the zoom on the symmetry axis of the triangle. So basically the number of self similar objects is an integer and has for characteristic the self similarity dimension which is a number D such as N=L^D where N is the number of copies produced by changing the size by L.

As for scale invariance which is not so largely used, it is a statement that f(µx) = µ^D.f(x) with some constant D. The property is continuous and true for every x. Fractal attractors are generally not exactly scale invariant - they have often 2 or several different scalings.

Hence from this point of view the self similarity and scale invariance may only be identical in a discrete number of points for simple fractals which have a unique scaling factor. I am aware that this does not adress spin lattices but it answers the question in the frame of the chaos theory.

This post imported from StackExchange Physics at 2014-03-09 15:49 (UCT), posted by SE-user Stan Won