The Devil is not in the details

Condensed matter physics aims to understand different and describe states of matter. Each state (phase) is associated with a particular type of order and phase diagrams encode the external parameters (temperature, pressure, chemical composition, magnetic field,...) that are necessary for each of the possible orderings to be stable.

Phase diagrams of even the simplest systems, such as binary alloys, can be quite rich, with many competing phases. Nevertheless, in many cases, simple microscopic models, with just a few degrees of freedom and a few parameters can describe these rich diagrams. Often a key is for the model to involve competing interactions, which can arise from different forces or from geometric frustration. Earlier, I wrote about how an Ising model on a hexagonal close-packed lattice could describe the plethora of distinct orderings that are observed in binary alloys. There the orderings are defined by the ordering wave vector and the composition of the unit cell. Another example occurs in spin-crossover materials and is described in recent work by my UQ colleagues.

Structure–property relationships and the mechanisms of multistep transitions in spin crossover materials and frameworks, Jace Cruddas and Ben J. Powell

A 2016 chemistry paper was First Step Towards a Devil's Staircase in Spin‐Crossover Materials

Another rich example is where there are two different spatial scales associated with interactions between the components of the system. In a lattice system, this can lead to ordering wavevectors that are incommensurate with the lattice. About forty years Per Bak wrote a nice series of papers that explored this situation.

Ising model with solitons, phasons, and "the devil's staircase", Per Bak and J. von Boehm 

This is an elegant and clear study of a simple Ising model in three dimensions with a frustrating interaction J_2 in the vertical direction. This is an example of an ANNNI model.

A mean-field theory for a state with a sinusoidally varying magnetisation (with wavevector 2 pi q) gives the phase diagram on the right above. The point P is a Lifshitz point, a tricritical point where the incommensurate (modulated) phase becomes stable. 

[Aside: in fermion models, a Lifshitz point is quite different: where the volume of the Fermi surface vanishes].

However, there is much more to the story. The authors then construct a mean-field theory where the magnetisation is commensurate, allowing for large unit cells. This leads to the phase diagram below.

The fractions p/q correspond to states with wavevector 2 pi p/q.

For example, the 1/4 state is below.

What does this have to do with a "devil's staircase"?

If for fixed J_2/J_1 the wavevector is plotted as a function of temperature it has steps of varying size and width.

A paper by Bruinsma and Bak considers an AFM Ising chain with 1/n^2 interaction, in a magnetic field, at zero temperature.

Note: In the figure below q is NOT the wavevector but rather the ratio of up to down spins. 

The magnetisation vs field curve has a fractal structure.

Bak also wrote a Physics Today article (that compares the phenomena to frequency mode locking) and a general review that includes examples of experimental realisations, ranging from magnets to atoms on surfaces.

This illustrates an important point that is often made in complexity science. Simple rules (theoretical models) can produce complex behaviour. 

This also illustrates characteristics of emergent phenomena. A wide range of physical systems can exhibit the same phenomena. Many of the details do not matter.

The devil is not in the details.