Generalised rigidity is a key concept

What are some of the most important concepts in condensed matter physics?
In a recent comment on this blog Gautam Menon suggested that one of them is that of generalised rigidity, i.e. the elasticity of order parameters associated with broken symmetry phases.

 A while ago I wrote a post trying to introduce Phil Anderson's discussion of the concept.
Thinking about this made me appreciate just how important and useful the concept is.

Basically, generalised rigidity quantifies how the free energy of a system varies when introduces spatial variations in the order parameter. These variations can result from boundary conditions, fluctuations, or topological defects.

Depending on the type of broken symmetry there are just a few parameters, maybe only one, involved in defining the rigidity. One is looking at "linear" response and so symmetry determines how many different terms one can write down that are second order in a gradient operator.

A concrete example is the Frank free energy density associated with non-chiral nematic crystals.

Here n is a unit vector (the order parameter) and there are just three parameters and K1, K2, and K3. The three terms represent pure splay, bend, and twist, respectively. Spatial uniformities are at the heart of liquid crystal displays.

Historical aside. It is impressive that Frank wrote this down in 1958, without any reference to Landau.

For s-wave superconductivity and superfluids such as 4He, there is just one parameter, known as the superfluid stiffness or superfluid density. This is the coefficient of the gradient term in the Ginzburg-Landau theory and determines the superconducting "coherence length".

The generalised rigidity is also important because it is central to the renormalisation group theory of critical phenomena, which start with Ginzburg-Landau-Wilson functionals (effective actions). For most cases, one discovers that higher order gradient terms are "irrelevant'' to long wavelength properties and the rigidities are renormalised by fluctuations.

The spin stiffness is the only parameter (energy scale) that appears in a non-linear sigma model treatment of ferromagnetism and antiferromagnetism. The model is sufficient to describe all of the long-wavelength and low-energy properties.

The XY model also involves just one parameter, the rigidity.  From this model one gets the Kosterlitz-Thouless transition.

Note that for both the non-linear sigma model and XY model in one and two dimensions there is no long-range order (symmetry breaking) [Mermin-Wagner theorem] yet the rigidity still has meaning.

The rigidity also determines the emergent length scales in these systems, including the size of topological defects such as vortices, skyrmions, disinclinations,....

A nice detailed discussion of much of the above is in Chaikin and Lubensky, particularly chapter 6.