In addition to the acoustic (phonon) excitations described above, specific excitations called ‘quantum vortices’ may exist in superfluid Bose systems.

Attempts to incorporate the vortex excitations have already been made in the original version of Landau's theory (Landau, 1941). The existence of a periodical lattice of quantum vortices in a superconductor located in a magnetic field was conjectured by Abrikosov (1952, 1957). The idea of quantum vortices occurring in rotating helium below the λ-point was introduced independently by Onsager (1949) and Feynman (1955).

In this section we present the description of quantum vortices in the functional integral formalism. We will also discuss the role of quantum vortices in the phase transition for the superfluid to the normal state.

Our starting point is the functional integration scheme over fast and slow variables. We will apply this formalism mainly to two-dimensional Bose systems and will try to take into account vortex-like configurations of the Bose field. We thus come to the conclusion that the system of vortices is equivalent to the system of charged particles interacting via a new field which play the role of electromagnetic field. At small temperatures there are only pairs with opposite signs, bounded by the long-range logarithm potential. It turns out that the phase transition in the two-dimensional Bose system is nothing but the dissociation of bounded pairs. A similar approach to the three-dimensional systems leads to the conclusion that phase transition is accompanied here by creation of long vortex filaments.