Topological nodal line semimetals with chiral symmetry

Topological semimetals in three dimensions display band-touchings at points (Weyl or Dirac semimetals) or nodal lines in the Brillouin zone. Weyl semimetals can occur with internal symmetries only (time-reversal $\mathcal{T}$, charge conjugation $\mathcal{C}$, and a product of the two, called chiral/sublattice symmetry $\mathcal{S}=\mathcal{T}\mathcal{C}$). Nodal line semimetals possessing solely internal symmetries have only been classified abstractly, while those with SU(2) spin rotation or crystalline symmetries are known more explicitly. We show that chiral symmetry classes that are topologically nontrivial in three dimensions (namely class AIII, CII, CI, and DIII) always have a stable gapless phase, which is a topological nodal line semimetal. Our classification differs from previous approaches and has direct implications for gapless surface states.