arXiv2015
Topological media are gapped or gapless fermionic systems, whose properties are protected by topology, and thus are robust to deformations of parameters of the system and generic. We discuss the class of gapless topological media, which contains the quantum vacuum of Standard Model in its symmetric phase, and condensed matter systems with zeroes in the energy spectrum, which form Fermi surfaces, Weyl and Dirac points, Dirac lines, Khodel-Shaginyan flat bands, etc. Some zeroes are topologically protected, being characterized by topological invariants, expressed in terms of Green's function. For stability of the others the ${\bf p}$-space topology must be accompanied by symmetry. Vacua with Weyl points serve as a source of effective relativistic quantum fields emerging at low energy: chiral fermions, effective gauge fields and tetrad gravity emerge together in the vicinity of a Weyl point. The accompanying effects, such as chiral anomaly, electroweak baryo-production and chiral vortical effect, are expressed via the symmetry protected ${\bf p}$-space invariants. The gapless topological media exhibit the bulk-surface and bulk-vortex correspondence: which in particular may lead to the flat band on the surface of the system or in the core of topological defects. The materials with flat band in bulk, on the surface or within the dislocations have singular density of states, which crucially influences the critical temperature of the superconducting transition in such media. While in all the known superconductors the transition temperature is exponentially suppressed as a function of the pairing interaction, in the flat band the transition temperature is proportional to the pairing interaction, and can be essentially higher. The ${\bf p}$-space topology may give us the general recipe for search or artificial fabrication of the room-temperature superconductors.