The spin-1 antiferromagnetic Heisenberg chain provides a nice example of emergence in a quantum many-body system. Specifically, there are three distinct phenomena that emerge that were difficult to anticipate: the energy gap conjectured by Haldane, topological order, and the edge excitations with spin-1/2.
An interesting question is whether anyone could have ever predicted these from just knowing the atomic and crystal structure of a specific material. I suspect Laughlin and Pines would say no.
To understand the emergent properties one needs to derive effective Hamiltonians at several different length and energy scales. I have tried to capture this in the diagram below. In the vertical direction, the length scales get longer and the energy scales get smaller.
It is interesting that one can get the Haldane gap from the non-linear sigma model. However, it coarse grains too much and won't give the topological order or the edge excitations.
It seems to me that the profundity of the emergence that occurs at the different strata (length scales) is different. At the lower levels, the emergence is perhaps more "straightforward" and less surprising or less singular (in the sense of Berry).
Aside. I spend too much time making this figure in PowerPoint. Any suggestions on a quick and easy way to make such figures?
Any comments on the diagram would be appreciated.