Why do vortices in a superconductor (i.e. magnetic flux-tubes) form triangular lattices? In one of the articles I found, I read that a square lattice would cause repulsion but a triangular lattice will cause attraction. Can anyone explain it?
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The vortex lines in a superconductor (or a neutral superfluid) form a regular crystal. Just like in a solid the precise arrangement depends on the interaction between the vortices, and is the one that minimizes the energy (or free energy) of the system.
The profile of a single vortex line and the interaction between them can be obtained from the Landau-Ginzburg theory. For two parallel vortices with the same circulation the interaction is $$ E_{int} = \frac{\Phi_0}{2\pi\lambda^2}K_0(r/\lambda) $$ where $K_0$ is a modified Bessel function, $r$ is the separation, $\lambda$ is the penetration length, and $\Phi_0$ is the flux quantum. The is a repulsive force, which decays exponentially with distance.
It is now a straightforward exercise, first undertaken by Abrikosov, to compare the energy of a square lattice and a triangular lattice. The result (as described in the paper) is that the triangular arrangement is favored.
Thomas
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