Parity Anomaly in the non-linear response of Nodal-Line Semimetals

Nodal-line semimetals are topological semimetals characterized by one-dimensional band-touching loops protected by the combined symmetry of inversion $mathcal{P}$ and time-reversal $mathcal{T}$ in absence of spin-orbit coupling. These nodal loops can be understood as a one-parameter family of Dirac points exhibiting the parity anomaly associated to $mathcal{P}*mathcal{T}$ symmetry. We find that the parity anomaly also appears in the non-linear optical response of these systems in an analogous way to the linear response transport. We analyze the presence of a tilting term in the Hamiltonian as an element that does not spoil $mathcal{P}*mathcal{T}$ symmetry: while it is $mathcal{P}*mathcal{T}$-symmetric, it breaks separately both $mathcal{P}$ and $mathcal{T}$ symmetries, allowing for the potential experimental observability of the linear and non-linear Hall conductivities in appropriate nodal-line semimetals.