Simple model Hamiltonians can describe complexity

An important idea in condensed matter physics, both soft and hard, is that the rich phenomena seen in materials that are chemically and/or structurally complex can often be described by relatively simple model Hamiltonians that involve only a few parameters. This is particularly true when the model and system have competing interactions. This often leads to two inter-related phenomena, that I have previously described for strongly interacting quantum many-body systems.

• the emergence of new energy scales that are much smaller than the energy parameters in the Hamiltonian
• subtle competition between different ground states that involve distinct types of ordering (or not)

These phenomena also occur in classical systems. A nice example is described in this 1993 paper.

hcp Ising model in the cluster-variation approximation 
R. McCormack, M. Asta, D. de Fontaine, G. Garbulsky, and G. Ceder

The authors studied the Ising model on the hexagonal close-packed (hcp) lattice in a magnetic field. The authors are all from materials science departments and are motivated by the fact that the problem of binary alloys AxB1_x can be mapped onto an Ising model.
Rich phase diagrams result by varying the relative concentration of the atoms A and B (e.g. gold and silver), or equivalently the difference in the chemical potential between A and B, or the relative size of the interatomic interactions, or the temperature. The phase diagram can contain many competing phases with well-defined stoichiometry: A, B, AB, A2B, A3B, A2B3, A3B5, ...
Furthermore, even for a single stoichiometry, there can be multiple possible distinct orderings (and crystal structures).

The hcp lattice can be viewed as layers of two-dimensional hexagonal lattices where each layer is displaced relative to others. A unit cell is shown below on the left, where V1, V2, and V3, denote nearest-neighbour (nn), next-nearest neighbour (nnn), and nnnn interactions.
For the case of perfect packing of hard spheres V1=V2.
Note, that even when only nn interactions are present, and they are antiferromagnetic, that the system is frustrated, and for a single layer the Ising model does not order at finite temperature and has a massively degenerate ground state (i.e. non-zero entropy).
The figure on the right shows a way to represent this unit cell and the interactions in terms of two hexagonal lattices superimposed on top of each other.

 The authors show that there is a plethora (menagerie) of possible ground states and stoichiometric orderings.

We predict 32 physically realizable ground states with stoichiornetries A, AB, A2B, A3B, A, B, and A4B3. Of these structures, six are stabilized by NN pairs and eight by NNN pairs; the remaining 18 structures require multiplet interactions for their stability. 

This is a nice example of how a simple model can describe complex and rich behaviour. It is also a nice example of emergence in that many of the details don't matter such as the identity of the atoms or the form of the interaction between them.