Spatial TexturesEarlier, we generalized the Landau Free energy to account for spatial variations in the texture of the order parameter, by allowing to vary as a function of space. However, this generalization turned out rather fruitless, because it turned out that any spatial variation would increase the free energy, meaning that the best field configuration was the one that didn't change in space! So we were just as good assuming that was uniform. Well, when do spatial variations in actually matter? On this page, we'll give a quick sketch of situations where we'll actually need to take into account the gradient term in the free energy:
Surprisingly, in both of these situations, the characteristic lengthscale for variations in the order parameter is the same correlation length . In these simple field models, everything changes in space over same lengthscale – the size of local fluctuation patches, the ‘‘skin depth’’ from the boundary, you name it. Well, perhaps it's not too surprising that all spatiallyvaryingphenomena happen on the same lengthscale, because there's only one lengthscale that enters into the problem. We'll derive it via dimensional analysis, and then if time permits, we'll take a more formal approach by looking at how the field responds to a deltafunction impulse. Finally, we'll take a look at spatial variations of order parameters describing a continuously broken symmetry, such as a spin that can point in any direction on a circle. If we impose contradictory boundary conditions on such fields, we find that the equilibrium configuration is not a domain wall solution (as it was for discrete symmetries), but rather, a continuous twisting sort of solution which is energetically cheaper. The existence of such Goldstone modes highlight a key difference between phases that involve broken discrete and continuous symmetries. Let's begin. Situating OurselvesLeave a Comment Below!Comment Box is loading comments...
