Direction-dependent linear response for gapped nodal-line semimetals in planar-Hall configurations

TL;DR

This paper addresses direction-dependent linear transport in gapped nodal-line semimetals by deriving the magnetoelectric conductivity in planar-Hall setups using a minimal two-band model linearized near the nodal ring. The authors incorporate Berry curvature and orbital magnetic moment as topological sources and include Lorentz-force-operator contributions within a semiclassical Boltzmann framework in the weak-field limit, obtaining explicit expressions for Drude, BC/OMM, and LF parts across three distinct E–B geometries. They find that BC and OMM contributions are comparable and often compete with opposite signs, with the OMM frequently dominating in the regime $\mu,\Delta\ll k_0$, and show that topology-induced components vanish in the ungapped case while exhibiting strong anisotropy tied to the nodal-ring orientation. The results provide concrete, experimentally testable signatures of NLSMs and highlight the essential role of including OMM and LF effects alongside BC in accurately predicting magnetoelectric responses.

Abstract

We compute the magnetoelectric conductivity for ideal nodal-line semimetals (NLSMs), with a finite but tiny mass-gap, in distinct planar-Hall set-ups. Each differing configuration results from the relative orientation of the nodal-ring's plane with respect to the plane spanned by the electric ($\mathbf E $) and magnetic ($\mathbf B$) fields. The net conductivity tensor has components comprising the Drude, anomalous-Hall, in-plane (with $\mathbf E$ and $\mathbf B$) longitudinal and transverse, and Lorentz-force-operator-induced parts. Our results feature the signatures of the inherent topology of a gapped NLSM, revealed through nonzero values of the Berry curvature and the orbital magnetic moment. In particular, we show that both of these vector fields, arising in the momentum space, give rise to terms of comparable magnitudes in the resulting response. Our explicit theoretical expressions will help identify unique signatures of NLSMs in contemporary experiments.

Figures (4)

• Figure 1: Gapped nodal-line semimetal with isotropy along the $k_x k_y$-plane: (a) Dispersion against the $k_x k_y$-plane. (b) Schematics of the Fermi surfaces representing the scenarios for without and with the OMM-correction, respectively. Here, we have taken the applied magnetic field ($\mathbf B$) to be directed purely along the $y$-axis. A toroid-shaped Fermi surface deforms into a ring cyclide when a nonzero $\mathbf B$ is applied. We have assumed that $|\mathbf B|$ is low-enough so as not to cause a Lifshitz transition of the Fermi surface to a horn cyclide.
• Figure 2: (a) Vector-plot of ${\mathbf \Omega}_{s= 2} ( \mathbf k)$ in the Brillouin zone. (b) Schematics of the three set-ups that we use to investigate the planar-Hall effect in NLSMs, showing the relative alignments of the external uniform electric (red arrow) and magnetic (blue arrow) fields. We label the three scenarios as set-up I, set-up II, and set-up III, respectively. The plane containing the $\mathbf E$ and $\mathbf B$ vectors (making an angle $\theta$ with each other) in each set-up has been highlighted by a background colour-shading. The coordinates have been chosen such that the NLSM in question has its nodal line lying along the $k_x k_y$-plane (cf. Fig. \ref{['figfs']}).
• Figure 3: Set-up I: Behaviour of the three components of the conductivity tensor (in units of eV), taking into account all the contributions. We have used the parameter values from Table \ref{['tab-params']}. The values of $B \equiv |\mathbf B|$ are in units of eV$^2$.
• Figure 5: Set-up III: Behaviour of the three components of the conductivity tensor (in units of eV), taking into account all the contributions. We have used the parameter values from Table \ref{['tab-params']}. The values of $B \equiv |\mathbf B|$ are in units of eV$^2$.