6 - Generalized elasticity

Summary

In the preceding several chapters, we have seen that the order established below a phase transition breaks the symmetry of the disordered phase. In many cases, the broken symmetry is continuous. For example, the vector order parameter m of the ferromagnetic phase breaks the continuous rotational symmetry of the paramagnetic phase, the tensor order parameter Q_ij of the nematic phase breaks the rotational symmetry of the isotropic fluid phase, and the set of complex order parameters ρG of the solid phase breaks the translational symmetry of the isotropic liquid. In these cases, there are an infinite number of equivalent ordered phases that can be transformed one into the other by changing a continuous variable θ. If rotational symmetry is broken, θ specifies the angle (or angles) giving the direction of the order parameter; if translational symmetry is broken, θ specifies the origin of a coordinate system. Uniform changes in θ do not change the free energy. Spatially non-uniform changes in θ, however, do. In the absence of evidence to the contrary, one expects the free energy density f to have an analytic expansion in gradients of θ. Thus we expect a term in f that is proportional to (∇θ)² for θ varying slowly in space. We refer to this as the elastic free energy, f_el, since it produces a restoring force against distortion, and we will refer to θ as an elastic or hydrodynamic variable.