Parity anomaly in the nonlinear response of nodal-line semimetals

Abstract

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 nonlinear 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 nonlinear Hall conductivities in appropriate nodal-line semimetals. We also propose effective actions for both the linear and nonlinear electromagnetic responses of tilted nodal-line semimetals. We find that the linear Hall-like response of tilted nodal-line semimetals is an axion response due to the parity anomaly, extending the class of systems that display such electromagnetic response.