One example of particular current interest is the type II superconductor in a magnetic field. As we discussed earlier, flux penetrates these materials in the form of flexible lines. At very low fields, it's natural to attempt an independent-vortex description as we did earlier. At larger fields, however, flux lines certainly interact, and prefer to arrange themselves into a periodic lattice, as first realized by Abrikosov. In a film one has a two-dimensional lattice of points, while in a bulk crystal one has instead a three-dimensional lattice of lines.
A great advantage of this system is that the density and interactions
between vortices can be to some extent tuned by varying the magnetic
field. Up to small deviations depending upon the geometry, the vortex
density is simply proportional to the applied flux. The stiffness of
the resulting lattice is strongly dependent on this density, because
of the presence of a large length scale. This length is 
,
the magnetic penetration depth. For 
, currents and
fields in the superconductor are screen, leading to exponential ( 
) inter-vortex interactions. For 
,
however, currents flow as in an uncharged (and hence unscreened)
superfluid, and the inter-vortex interaction energies become
logarithmic. Since 
in most
superconductors of interest, for fields B > 1 T many vortices are
contained in an area of order 
, and thus can interact
logarithmically.