• Vector fields L and A, |L|=1;
• A is chaotic;
• L interacts with A: E = Uin * (L*A)^2;
• L has gradient energy: E = Ugr * ((dL/dx)^2 + (dL/dy)^2).

One can start from some initial distribution of L and look for the equilibrium distribution of L. Ugr is a control parameter.

There are a few non-trivial effects which can be seen here:

1. Larkin-Imry-Ma state: if gradient energy is strong then L is uniform, if gradient energy is weak, L is chaotic (following A field). Between these limits there is an intermediate state where L form some structures of size R which can be estimated in the following way:

By rotating a uniform area of size R (with N \propto R^2 impurities) you can decrease interaction energy by \propto Uin/sqrt(N) \propto Uin/R and increase gradient energy by \propto Ugr/R^2. There is an optimal R where these two energies are equal: R \propto Ugr/Uin.

2. You can choose one of two different interactions: E = Uin * (L*A)^2 or E = Uin * (L*A). If you decrease gradient energy and then increase it again, in the first case you will return to the original distribution of L (uniform or with vortices), in the second case you will have lots of random vortices. The reason is that in the first case interaction with A will never rotate L more then 90 degrees and information about original orientation will survive.

picture1: start from uniform L, gradient energy Ugr goes up-down-up: https://slazav.xyz/tmp/lim1u.gif

picture2: start from L with a vortex, Ugr goes up-down-up. The vortex is stable: https://slazav.xyz/tmp/lim1v.gif

picture3: start from L with a gradient of phase: https://slazav.xyz/tmp/lim2s.gif

picture4: start from L with a few vortices: https://slazav.xyz/tmp/lim3s.gif