  Superfluid phase transitions are discussed from a geometrical perspective as
envisaged by Onsager. The approach focuses on vortex loops which close to the
critical temperature form a fluctuating vortex tangle. As the transition is
approached, vortex lines proliferate and thereby disorder the superfluid state,
so that the system reverts to the normal state. It is shown in detail that loop
proliferation can be described in exactly the same way as cluster percolation.
Picturing vortex loops as worldlines of bosons, with one of the spatial
coordinates interpreted as the time coordinate, a quantitative description of
vortex loops can be given. Applying a rotation (to superfluids) or a magnetic
field (to superconductors), which suppresses the formation of vortex loops and
instead can lead to open vortex lines along the field direction, is shown to be
equivalent to taking the nonrelativistic limit. The nonrelativistic theory is
the one often used to study vortex lattice melting and to describe the
resulting entangled vortex liquid.
