Poisson Brackets Scheme for Vortex Dynamics in Superfluids and Superconductors and Effect of Band Structure of Crystal

Poisson brackets for the Hamiltonian dynamics of vortices are discussed for 3 regimes, in which the dissipation can be neglected and the vortex dynamics is reversible: (i) The superclean regime when the spectral flow is suppressed. (ii) The regime when the fermions are pinned by crystal lattice. This includes also the regime of the extreme spectral flow of fermions in the vortex core: these fermions are effectively pinned by the normal component. (iii) The case when the vortices are strongly pinned by the normal component. All these limits are described by the single parameter $C_0$, which physical meaning is discussed for superconductors containing several bands of electrons and holes. The effect of the Fermi-surface topology on the vortex dynamics is also discussed.