Chiral topological insulating phases from three-dimensional nodal loop semimetals

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We identify a topological $\mathbb{Z}$ index for three-dimensional chiral insulators with $P*T$ symmetry where two Hamiltonian terms define a nodal loop. Such systems may belong in the AIII or DIII symmetry class. The $\mathbb{Z}$ invariant is a winding number assigned to the nodal loop and has a correspondence to the geometric relation between the nodal loop and the zeros of the gap terms. Dirac cone edge states under open boundary conditions are in correspondence with the winding numbers assigned to the nodal loops. We verify our method with the low-energy effective Hamiltonian of a three-dimensional material of topological insulators in the ${\mathrm{Bi}}_{2}{\mathrm{Te}}_{3}$ family.