  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 and in the vortex core in systems
with topologically protected Fermi points (Weyl points). 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.
