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We identify a topological Z index for three dimensional chiral insulatorswith P*T symmetry where two Hamiltonian terms define a nodal loop. Such systemsmay belong in the AIII or DIII symmetry class. The Z invariant is a windingnumber assigned to the nodal loop and has a correspondence to the geometricrelation between the nodal loop and the zeroes of the gap terms. Dirac coneedge states under open boundary conditions are in correspondence with thewinding numbers assigned to the nodal loops. We verify our method with thelow-energy effective Hamiltonian of a three-dimensional material of topologicalinsulators in the Bi$_2$Te$_3$ family.

Chiral topological insulating phases from three-dimensional nodal loop semimetals