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

This paper theoretically computes the direction-dependent magnetoelectric conductivity of gapped nodal-line semimetals in various planar-Hall configurations, demonstrating how the interplay between Berry curvature and orbital magnetic moment reveals unique topological signatures in the resulting response.

The Gist

The Big Picture: A New Kind of Traffic Jam

Imagine a city where the roads (which represent the energy paths electrons take) aren't just straight lines or simple curves. In this specific city, the roads form a giant, floating hula hoop in the sky. This is what physicists call a Nodal-Line Semimetal.

Usually, electrons flow smoothly. But in this material, the "hula hoop" creates a special traffic pattern. The authors of this paper wanted to know: What happens to the electric traffic if we push it with an electric field (a battery) and pull it with a magnetic field (a magnet), all while the hula hoop is tilted at different angles?

The Cast of Characters

1. The Electrons: Think of them as tiny cars driving on the hula hoop.
2. The Gap (The "Mass"): In a perfect hula hoop, the road is continuous. But in this study, the authors added a tiny "speed bump" or a small gap in the ring. This is the "mass gap." It's like putting a small fence around the hula hoop so the cars can't fall through the center, but they can still drive around the ring.
3. The Berry Curvature (The "Magnetic Whirlwind"): This is a weird, invisible wind that exists only in the space where the cars drive (momentum space). It doesn't blow on the cars directly; it makes the cars feel like they are turning, even if the road is straight. It's like driving on a road that feels like it's curving because of a strong crosswind.
4. The Orbital Magnetic Moment (The "Spinning Top"): Imagine each electron-car isn't just a point; it's a spinning top. When you put a magnet near a spinning top, it wobbles. This "wobble" changes how the car drives. The paper argues that ignoring this wobble is a huge mistake.

The Experiment: Three Different Angles

The researchers set up three different "traffic scenarios" (Set-ups I, II, and III) by changing the angle of the magnetic field relative to the hula hoop.

• Set-up I: The magnetic field is in the same flat plane as the hula hoop.
• Set-up II: The magnetic field is tilted up, pointing partly out of the plane of the hoop.
• Set-up III: The electric field is pushed from the top, while the magnetic field is tilted.

The Surprising Discoveries

1. The "Ghost" Currents (Planar Hall Effect)

Usually, if you push cars forward (Electric Field) and pull them sideways with a magnet, they drift sideways. This is the Hall Effect.
But in this "Planar Hall" setup, the magnetic field is in the same plane as the electric field. You'd think nothing special would happen.
The Result: The cars do drift sideways!
Why? Because of the "Whirlwind" (Berry Curvature) and the "Spinning Top" (Orbital Magnetic Moment). The paper shows that the spinning top effect is just as important as the whirlwind. If you ignore the spinning top, your map of where the cars go is completely wrong.

2. The "Tilt" Matters

The direction of the magnetic field changes the traffic flow completely.

• If the magnetic field is parallel to the hula hoop, you get a strong sideways current.
• If the magnetic field is perpendicular to the hoop, the sideways current vanishes.
  Analogy: Imagine trying to spin a hula hoop. If you push it from the side, it spins. If you push it from the top, it just wobbles but doesn't spin the same way. The material "remembers" the shape of the hula hoop and reacts differently depending on how you poke it.

3. The "Lorentz Force" is More Than Just a Push

In standard physics, a magnet pushes a moving charge sideways (the Lorentz force). The authors found that in these strange materials, this force does something extra. It creates a recursive chain reaction (like a snowball rolling down a hill getting bigger).
The Metaphor: It's not just a single push; it's a push that triggers a series of smaller pushes, creating new currents in directions you wouldn't expect. The paper calculates these "hidden" currents, which are just as strong as the main ones.

Why Does This Matter?

1. The "Tiny Gap" is the Key:
If the hula hoop had no gap (no mass), the "Whirlwind" and "Spinning Top" effects would cancel out or behave differently. The fact that there is a tiny gap changes the rules of the road. This helps scientists identify which materials are actually these "hula hoop" materials.

2. Don't Ignore the Spin:
For a long time, physicists calculated these effects by only looking at the "Whirlwind" (Berry Curvature). This paper proves that if you ignore the "Spinning Top" (Orbital Magnetic Moment), your calculations are wrong. You need both to get the right answer.

3. Finding New Materials:
The authors provide a "cheat sheet" (mathematical formulas) for experimentalists. If you build a device with these materials and measure the electricity, you can look at the results. If the numbers match their "Planar Hall" predictions, you know you've found a gapped nodal-line semimetal.

The Takeaway

This paper is like a detailed traffic report for a city built on a giant floating hula hoop. It tells us that:

• The shape of the road (the nodal line) creates invisible winds.
• The cars (electrons) spin like tops, which changes how they drive.
• The direction you push the cars matters immensely.
• To understand the traffic, you must account for both the invisible wind and the spinning tops.

By understanding these rules, scientists can design better, faster, and more efficient electronic devices in the future, using these exotic materials as the foundation.

Technical Summary

1. Problem Statement

The paper addresses the need to understand the magnetoelectric transport properties of gapped nodal-line semimetals (NLSMs) under the combined action of static electric ($\mathbf{E}$) and magnetic ($\mathbf{B}$) fields. While the topological properties of gapless NLSMs (such as drumhead surface states) are well-studied, the linear response of gapped NLSMs (where a small mass gap $\Delta$ is introduced, often via spin-orbit coupling) in planar-Hall configurations remains less explored.

Specifically, the authors aim to:

• Compute the magnetoelectric conductivity tensor for ideal NLSMs with a finite but tiny mass gap.
• Investigate how the relative orientation between the nodal-ring plane and the plane spanned by $\mathbf{E}$ and $\mathbf{B}$ affects transport (anisotropy).
• Determine the specific roles of two topological vector fields: the Berry Curvature (BC) and the Orbital Magnetic Moment (OMM), and whether the OMM can be neglected in these calculations.
• Analyze contributions from the Lorentz-force operator ($\check{L}$), which generates currents beyond the standard Hall effect, often ignored in previous linear response studies of semimetals.

2. Methodology

The authors employ a semiclassical Boltzmann transport formalism within the relaxation-time approximation ($\tau$).

• Model Hamiltonian: They utilize a minimal two-band model for an NLSM with a circular nodal ring in the $k_x k_y$-plane. The Hamiltonian is $H_0(\mathbf{k}) = \mathbf{d}(\mathbf{k}) \cdot \boldsymbol{\sigma}$, where the gap $\Delta$ opens a small energy gap at the nodal line.
• Coordinate Transformation: To handle the toroidal Fermi surface resulting from the gapped nodal ring, they transform momentum space into toroidal coordinates $(\kappa, \phi, \Phi)$. This allows for efficient integration near the Fermi surface.
• Topological Quantities: They explicitly calculate the Berry Curvature ($\Omega_s$) and Orbital Magnetic Moment ($m_s$) for the conduction band. A key finding in the derivation is the relation $m_s = e \epsilon_s \Omega_s$ for two-band models, though they treat them as distinct contributions to the response.
• Linear Response Expansion:
  • The conductivity is expanded up to order $O(|\mathbf{B}|^2)$ (quadratic in magnetic field) to satisfy Onsager-Casimir reciprocity relations for the in-plane components.
  • The distribution function is expanded in a Taylor series with respect to the Zeeman-like energy correction induced by the OMM ($\varepsilon^{(m)} = -\mathbf{B} \cdot \mathbf{m}$).
  • The conductivity tensor is decomposed into:
    1. Drude term: Field-independent.
    2. Berry Curvature (BC) term: Arises from the phase-space modification ($D_s$).
    3. Orbital Magnetic Moment (OMM) term: Arises from the modification of the band velocity and energy dispersion.
    4. Lorentz-force ($\check{L}$) term: Derived from the recursive operation of the Lorentz force operator on the distribution function, generating both in-plane and out-of-plane currents.
• Three Distinct Set-ups: The authors analyze three specific configurations of $\mathbf{E}$ and $\mathbf{B}$ relative to the nodal ring (lying in the $k_x k_y$-plane):
  • Set-up I: $\mathbf{E} \parallel \hat{x}$, $\mathbf{B}$ in the $xy$-plane (nodal plane).
  • Set-up II: $\mathbf{E} \parallel \hat{x}$, $\mathbf{B}$ in the $xz$-plane.
  • Set-up III: $\mathbf{E} \parallel \hat{z}$, $\mathbf{B}$ in the $xz$-plane.

3. Key Contributions

1. Inclusion of Orbital Magnetic Moment (OMM): The paper demonstrates that the OMM contributes terms of comparable magnitude to the Berry Curvature terms. Neglecting the OMM leads to incorrect signs and magnitudes in the total conductivity, particularly for the longitudinal components.
2. Lorentz-Force Operator ($\check{L}$) Analysis: Unlike many previous studies that only consider the $\check{L}$-induced conventional Hall effect (out-of-plane), this work explicitly calculates the in-plane longitudinal and transverse components generated by the $\check{L}$ operator. These terms are shown to be significant and comparable to the standard magnetoelectric terms.
3. Directional Anisotropy: The study provides a comprehensive map of how the conductivity tensor changes based on the orientation of the $\mathbf{E}$-$\mathbf{B}$ plane relative to the nodal ring. It highlights that components perpendicular to the nodal ring's plane (e.g., involving $B_z$) vanish or behave differently due to the specific vortex structure of the BC and OMM.
4. Analytical Expressions: The authors derive explicit, closed-form analytical expressions for all non-zero components of the conductivity tensor for the three set-ups, valid in the low-temperature ($T \to 0$) and weak-field limits.

4. Key Results

• Sign Reversal and Cancellation: In the longitudinal conductivity ($\sigma_{xx}$ or $\sigma_{zz}$), the BC-only contribution and the OMM contribution often have opposite signs. In the regime where the Fermi surface is toroidal ($\Delta, \mu \ll k_0$), the OMM contribution dominates, leading to a sign reversal of the total response compared to calculations that ignore the OMM.
• Vanishing Transverse Components:
  • In Set-up II and Set-up III, where the magnetic field has a component perpendicular to the nodal ring ($B_z$), the in-plane transverse conductivity vanishes. This is because the BC and OMM have zero components along the $z$-axis (they form vortices in the $xy$-plane).
  • In Set-up I (where $\mathbf{B}$ is in the nodal plane), a non-zero in-plane transverse conductivity (Planar Hall Effect) exists, proportional to $B_x B_y$.
• Anomalous Hall Effect (AHE): The out-of-plane transverse conductivity (AHE) is non-zero only when the OMM is included. The BC alone yields a vanishing AHE in this specific gapped NLSM model; the OMM correction is essential to generate the observed AHE.
• Lorentz-Force Contributions: The $\check{L}$-induced terms generate significant in-plane currents. For Set-up I, the $\check{L}$-induced longitudinal and transverse components almost coincide with the OMM-dominated terms, reinforcing the total signal.
• Parameter Dependence: The conductivity scales with material parameters such as the gap $\Delta$, chemical potential $\mu$, and nodal ring radius $k_0$. The results show a strong dependence on the ratio of velocities ($v_z/v_0$).

5. Significance

• Experimental Guidance: The derived analytical expressions provide specific signatures (e.g., specific dependencies on $B_x, B_y, B_z$ and sign changes) that experimentalists can use to identify gapped NLSMs in materials like CuTeO$_3$, Ca$_3$P$_2$, or SrAs$_3$.
• Theoretical Necessity of OMM: The work establishes that for gapped nodal-line systems, the Orbital Magnetic Moment is not a minor correction but a fundamental component of the transport response. Ignoring it leads to qualitative errors (wrong sign) in predictions.
• Beyond Conventional Hall: By quantifying the in-plane contributions of the Lorentz-force operator, the paper corrects a common oversight in the literature, showing that "Hall-like" physics in these topological materials is more complex and multidimensional than previously thought.
• Topological Distinction: The results distinguish gapped NLSMs from Weyl semimetals. While Weyl nodes have intrinsic BC/OMM even when gapless, NLSMs require a finite gap to exhibit these specific transport signatures, and their anisotropy is strictly tied to the planar nature of the nodal line.

In conclusion, this paper provides a rigorous theoretical framework for understanding the linear magnetoelectric response of gapped nodal-line semimetals, emphasizing the critical interplay between Berry curvature, orbital magnetic moments, and the Lorentz force in determining anisotropic transport properties.