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Closed-form evaluation of the in-plane clean conductance integral

Derive a closed-form analytical expression for the in-plane (parallel) zero-energy dimensionless conductance g_parallel and conductivity σ_parallel of the clean nodal line semimetal described by Hamiltonian (1) with b oriented along the x-direction, by analytically evaluating the momentum-space integral given in Eq. (Conductance_Parallel_Transport).

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Background

The authors obtain an exact analytical expression for out-of-plane transport in the clean system but, for transport in the plane of the nodal line (b → b e_x), they present the conductance as a two-dimensional momentum integral (Eq. (Conductance_Parallel_Transport)).

They report that they were unable to evaluate this integral analytically and therefore computed it numerically, revealing oscillatory behavior in σ_parallel with system length L and a large-L asymptote differing from the Kubo result.

References

Since we cannot find an analytical solution of this integral, we integrate~eq:Conductance_Parallel_Transport numerically and show the resulting conductivity $\sigma_\parallel = g_\parallel L/W2$ in Fig.~\ref{fig:Numerical_Integral_PlaneNodalLine}.

— Conductivity scaling and absence of localization in disordered nodal line semimetals (2401.07906 - Paiva et al., 15 Jan 2024) in Section 'Model and transport in clean systems', after Eq. (Conductance_Parallel_Transport)