Besides topological band insulators, which have a full bulk gap, there are also gapless phases of matter that belong to the broad class of topological materials, such as topological semimetals and nodal superconductors. We systematically study these gapless topological phases described by the Bloch and Bogoliubov-de Gennes Hamiltonians. We discuss a generalized bulk-boundary correspondence, which relates the topological properties in the bulk of gapless topological phases and the protected zero-energy states at the boundary. We study examples of gapless topological phases, focusing in particular on nodal superconductors, such as nodal noncentrosymmetric superconductors (NCSs). We compute the surface density of states of nodal NCSs and interpret experimental measurements of surface states. In addition, we investigate Majorana vortex-bound states in both nodal and fully gapped NCSs using numerical and analytical methods. We show that different topological properties of the bulk Bogoliubov-quasiparticle wave functions reflect themselves in different types of zero-energy vortex-bound states. In particular, in the case of NCSs with tetragonal point-group symmetry, we find that the stability of these Majorana zero modes is guaranteed by a combination of reflection, time-reversal, and particle-hole symmetries. Finally, by using K-theory arguments and a dimensional reduction procedure from higher-dimensional topological insulators and superconductors, we derive a classification of topologically stable Fermi surfaces in semimetals and nodal lines in superconductors.
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