London theory across superconducting phase transitions: application to   UPt$_3$
  For multi-component superconductors, it is known that the presence of symmetry breaking fields can lead to multiple superconducting phase transitions. This phenomenon is best illustrated in UPt$_3$. Motivated by recent small angle neutron scattering experiments on the vortex state of UPt$_3$, the London theory in the vicinity of such phase transitions is determined. It is found that the form of this London theory is in general quite different than that for conventional superconductors. This is due to the existence of a diverging correlation length associated with these phase transitions. One striking consequence is that non-trivial vortex lattices exist arbitrarily close to H$_{c1}$. It is also found that the penetration depth develops a novel temperature dependence and anisotropy. Results of this theory for UPt$_3$ are derived. Possible applications to CeIn$_3$, U$_{1-x}$Th$_x$Be$_{13}$, electron doped cuprate superconductors, Sr$_2$RuO$_4$, and MgCNi$_3$ are also discussed. 
D.F. Agterberg and Matthew J.W. Dodgson