Topological media are systems whose properties are protected by topology and thus are robust to deformations of the system. In topological insulators and superconductors the bulk-surface and bulk-vortex correspondence gives rise to the gapless Weyl, Dirac or Majorana fermions on the surface of the system and inside vortex cores. Here we show that in gapless topological media, the bulk-surface and bulk-vortex correspondence is more effective: it produces topologically protected gapless fermions without dispersion – the flat band. Fermion zero modes forming the flat band are localized on the surface of topological media with protected nodal lines[1, 2] and in the vortex core in systems with topologically protected Fermi points (Weyl points) [3]. Flat band has an extremely singular density of states, and we show that this property may give rise in particular to surface superconductivity which could exist even at room temperature.

Excitations in vortex cores in superconductors and other Fermi superfluids are single-particle excitations with a peculiar energy spectrum. These excitations are responsible for many important thermodynamic properties such as specific heat, London penetration length, etc. They also determine the dynamic characteristics of superconductors and superfluids through their interaction with vortices. Flux flow resistance, the Hall effect in type II superconductors and the mutual friction in superfluids are the most important phenomena which strongly depend on vortex core excitations. These phenomena determine the electromagnetic responses of type II superconductors and the hydrodynamic behaviour of superfluids and are of great significance for practical applications of superconducting devices and for understanding the most fundamental properties of correlated electrons and other Fermi particles. In this review we consider the dynamic properties of superconductors and superfluids and outline the basic ideas and results on the vortex dynamics in clean superfluid Fermi systems. The forces acting on moving vortices are discussed including the problem of the transverse force which was a matter of confusion for quite some time. We formulate the equations of the vortex dynamics, which include all the forces and the inertial term associated with excitations bound to the moving vortex.

Лев Петрович Горьков (К шестидесятилетию со дня рождения), Абрикосов А.А., Андреев А.Ф., Боровик-Романов А.С., Бычков Ю.А., Воловик Г.Е., Гинзбург В.Л., Грибов В.Н., Дзялошинский И.Е., Ивлев Б.И., Иорданский С.В., Копнин Н.Б., Минеев В.П., Питаевский Л.П., Покровский В.Л., Халатников И.М., Щеголев И.Ф., Элиашберг Г.М.

Time-dependent equations for a d-wave superconductor are derived for temperatures close to Tcunder the gapless condition τΔ(T) « 1. They differ from the usual time-dependent Ginzburg–Landau (TDGL) theory by an additional term which describes a diffusive relaxation of nonequilibrium excitations. These equations are applied to the problem of flux flow. The longitudinal conductivity is found to differ considerably from the Bardeen and Stephen model; the implication of this result for resistive measurements of the upper critical field is discussed. The Hall conductivity, however, coincides with the usual TDGL expression.

In a recent Letter Simon and Lee suggested a scaling law for thermodynamic and kinetic properties of superconductors with lines of gap nodes. However their crossover parameter between the bulk dominated regime and the vortex dominated regime is different from that found in our paper (N.B. Kopnin and G.E. Volovik, JETP Lett., {\bf 64}, 690 (1996); see also cond-mat/9702093). We discuss the origin of the disagreement.

An analysis is made of experimental data from measurements of the specific heat, the resistivity, the critical magnetic fields, the paramagnetic susceptibility above Tc, the Hall effect, and of other properties of the two new types of superconducting materials: lanthanum cuprates and the 1-2-3 compounds. The results of this analysis are discussed from the point of view of applying the Fermi-liquid picture to the description of the normal and superconducting properties of these materials, the role of fluctuations near the critical temperature, and the dimensionality of the superconductivity in them. Estimates of the width of the conduction band and of some other microscopic parameters indicate that there is a rather wide (about 0.7 eV) delocalized band. The fluctuations region near Tc is narrow, but nonetheless wider than that in ordinary superconductors. The superconductivity near Tc is three-dimensional, although sufficiently far from Tc the layered nature of the structure may be important.

The dynamics of superfluid helium-3 in flow channels with transverse sizes smaller than the mean free path of quasiparticles with respect to collisions with each other is considered, taking into account the diffusive reflection of quasiparticles from the walls. For quasiclassical Green functions the boundary conditions obtained by Ovchinnikov for the similar problem in superconductors have been used. Equations are derived defining the behavior of the difference between chemical potentials of normal and superfluid components of helium-3. These equations describe a phenomenon similar to the branch imbalance (or charge imbalance) in superconductors, and determine the relaxation depth of the pressure gradient in superfluid helium-3. The time-dependent Ginzburg-Landau equations are also obtained for the order parameter in the case when the transverse size of the channel is close to the critical value when the superfluid transition temperature goes to zero. The approach makes it possible to study theoretically effects related to the overcritical flows of superfluid helium-3 through narrow channels under pressure.