It has been realized over the past two decades that topological nontriviality can be present not only in insulators but also in gapless semimetals, the most prominent example being Weyl semimetals in three dimensions. Key to the topological classification schemes are the three “internal” symmetries, time reversal $\mathcal{T}$, charge conjugation $\mathcal{C}$, and their product, called chiral symmetry $\mathcal{S}=\mathcal{T}\mathcal{C}$. In this work, through explicit calculations, we show that robust topological semimetal phases occur in $d=3$ without invoking crystalline symmetries other than translations. These semimetals appear as not only intermediate gapless phases between the topological and the trivial gapped phases in symmetry classes that have nontrivial topology, but also in symmetry classes that have only trivial gapped phases in $d=3$. We show that all classes that lack chiral symmetry always have topologically stable and robust phases with Weyl nodes, classified by their Chern number on an enclosing surface. Furthermore, all classes that possess chiral symmetry always have topologically protected and robust nodal-line semimetal phases, classified by the winding number of a linking loop. Just as point nodes imply Fermi arcs on the surface Brillouin zone (BZ), a nonzero winding number on a nodal loop implies robust gapless drumhead states on the surface BZ.

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• Nodal-line semimetals
• Semimetals
• Topological materials
• Free-electron model
• Topology