Semiclassical theory of frequency dependent linear magneto-optical transport in Weyl semimetals

This paper develops a semiclassical Boltzmann theory to demonstrate how frequency-dependent magneto-optical transport in Weyl semimetals is governed by the complex interplay of orbital magnetic moments, cone tilt, and intervalley scattering, revealing distinct regimes where strong scattering can suppress the chiral anomaly and where tilt orientation dictates the symmetry and sign of the longitudinal magneto-optical conductivity.

The Gist

Imagine a bustling city where the roads are not flat, but shaped like perfect cones. In this city, the "cars" are electrons, and the "traffic rules" are governed by the strange laws of quantum physics. This city is a Weyl Semimetal, a special type of material where electrons behave like massless particles (Weyl fermions) moving at incredible speeds.

This paper is like a traffic report for this city, but with a few very specific twists:

1. The Weather: There is a constant, strong wind blowing (a static magnetic field).
2. The Music: The city is being shaken by a rhythmic beat, like a speaker playing a song (an electromagnetic wave or light).
3. The Roadblocks: Sometimes the cars crash into each other or hit potholes (scattering).
4. The Spin: The cars have a tiny, internal gyroscope (Orbital Magnetic Moment) that makes them wobble when the wind blows.

Here is what the researchers discovered about how traffic flows in this city under these conditions.

1. The "Chiral Anomaly": A One-Way Street That Gets Clogged

In normal metals, if you push traffic in one direction, it flows. But in this Weyl city, there's a weird rule called the Chiral Anomaly. If the wind (magnetic field) and the music (electric field) are aligned just right, the traffic suddenly speeds up in the direction of the wind. It's like a highway that magically expands when the wind blows.

However, the researchers found that this magic trick depends heavily on how fast the music is playing (the frequency).

• Slow Music (Low Frequency): If the beat is slow, the cars have plenty of time to crash into each other and switch lanes (intervalley scattering). If there are too many crashes, the "magic speed-up" gets canceled out. In fact, the traffic can actually slow down or even reverse direction! It's like a traffic jam caused by too many lane changes.
• Fast Music (High Frequency): If the beat is super fast (like a high-pitched electronic song), the cars are moving so quickly that they don't have time to crash and switch lanes before the beat changes. The "magic speed-up" stays strong and positive. The traffic jam never forms because the chaos doesn't have time to settle.

The Takeaway: By changing the speed of the music (frequency), scientists can tell if the electrons are crashing into each other or flowing freely. This is a new way to "listen" to the health of the material.

2. The "Wobble" (Orbital Magnetic Moment)

The researchers also looked at the cars' internal gyroscopes (the Orbital Magnetic Moment). When the wind blows, these gyroscopes make the cars wobble.

• This wobble adds a steady, linear push to the traffic flow, regardless of how fast the music is.
• It's like adding a gentle, constant tailwind that shifts the whole traffic pattern slightly, making the "One-Way Street" effect even more complex.

3. Tilted Roads (The Shape of the Cones)

In some versions of this city, the cone-shaped roads aren't standing straight up; they are tilted.

• Tilted Sideways: If the road leans sideways, the traffic flow becomes symmetrical. It doesn't matter if you lean left or right; the flow looks the same.
• Tilted Forward/Backward: If the road leans in the direction of the wind, things get weird. The traffic flow becomes asymmetrical.
  • The Surprise: In this specific "forward tilt" scenario, the traffic can actually flow backwards (negative conductivity) even without the "wobble" (gyroscope) effect! This is a rare phenomenon that only happens when the road is tilted just right.

4. The "Spin" of the Light

The researchers tested this with two types of music:

• Linear Polarization: Like a sound wave vibrating up and down.
• Circular Polarization: Like a sound wave spiraling like a corkscrew.

They found that for the main traffic flow (Longitudinal Conductivity), it didn't matter if the music spiraled or vibrated straight up and down. The result was the same. This is good news for experiments because it means the material behaves consistently no matter how you shine the light on it.

Why Does This Matter?

Think of this research as building a sensitive microphone for the quantum world.

• By shining light of different frequencies (from slow radio waves to fast terahertz waves) on these materials, scientists can now "hear" how the electrons are interacting.
• They can detect if the electrons are crashing (scattering), if the roads are tilted, and if the internal gyroscopes are wobbling.

In a nutshell: This paper gives us a new toolkit to understand and control the flow of electricity in these exotic materials. It shows that by simply changing the "tempo" of the light hitting the material, we can switch the material's behavior from a traffic jam to a super-highway, or even reverse the flow entirely. This could be crucial for building future ultra-fast computers and sensors that use light and magnetism to process information.

Technical Summary

1. Problem Statement

Weyl semimetals (WSMs) exhibit unique transport phenomena driven by band topology, Berry curvature, and the chiral anomaly. While linear magnetotransport (DC) in WSMs is well-studied, a comprehensive theoretical framework for frequency-dependent linear magneto-optical transport is lacking. Specifically, there is a need to understand how the interplay of the following factors influences conductivity in the MHz–THz regime:

• Chiral Anomaly: The non-conservation of chiral charge in parallel electric ($E$) and magnetic ($B$) fields.
• Orbital Magnetic Moment (OMM): The coupling of Bloch wave packet self-rotation to external fields, modifying band energy and velocity.
• Weyl Cone Tilt: The breaking of Lorentz invariance, leading to anisotropic Fermi surfaces.
• Intervalley Scattering: The relaxation mechanism between Weyl nodes of opposite chirality.
• Frequency Dependence: How the driving frequency ($\omega$) relative to the scattering rate ($1/\tau$) alters the chiral imbalance buildup.

Current theories often neglect momentum-dependent relaxation times or fail to consistently integrate OMM, tilt, and finite-frequency effects within a single semiclassical framework.

2. Methodology

The authors develop a semiclassical Boltzmann transport theory incorporating the following key elements:

• Effective Hamiltonian: They utilize a low-energy effective Hamiltonian for WSMs that includes a generic tilt term ($t_x, t_z$) and describes the quasiparticle excitations near Weyl nodes.
• Equations of Motion: The dynamics are governed by semiclassical equations of motion modified by Berry curvature ($\Omega_k$) and the orbital magnetic moment ($m_k$). The phase-space measure is corrected by the factor $D_k = (1 + eB \cdot \Omega_k / \hbar)^{-1}$.
• Scattering Matrix Formalism:
  • The collision integral includes both intranode and intervalley scattering.
  • Crucially, they employ a momentum-dependent relaxation time ($\tau_k$) derived from a scattering matrix approach, rather than a constant relaxation time approximation. This accounts for the deformation of the Fermi surface due to tilt and OMM.
  • The scattering potential distinguishes between non-magnetic and magnetic impurities.
• Frequency-Dependent Driving: The system is driven by a time-dependent electric field $E(t) = E e^{-i\omega t}$ (representing linearly or circularly polarized light). The distribution function is expanded to first order in $E$, leading to a frequency-dependent correction term $g_k(\omega)$.
• Conservation Laws: The solution enforces particle number conservation ($\sum_k g_k = 0$), which is critical for correctly determining the intervalley scattering contribution to the chiral anomaly.
• Regimes of Analysis: The study categorizes results based on the dimensionless parameter $\omega\tau$:
  • Weak AC limit: $\omega\tau \ll 1$ (low frequency).
  • Moderate AC limit: $\omega\tau \sim 1$.
  • Strong AC limit: $\omega\tau > 1$ (high frequency).

3. Key Contributions

• Unified Framework: The paper provides the first comprehensive derivation of the full frequency-dependent conductivity tensor ($\sigma_{\alpha\beta}(\omega)$) in WSMs that simultaneously accounts for Berry curvature, OMM, generic band tilt, and momentum-dependent intervalley scattering.
• OMM and Chiral Anomaly Interplay: It explicitly quantifies how the Orbital Magnetic Moment introduces linear-in-$B$ corrections to conductivity, distinct from the quadratic-in-$B$ response of the chiral anomaly.
• Frequency-Dependent Sign Reversal: The work establishes that the sign reversal of Longitudinal Magneto-Optical Conductivity (LMOC), typically associated with strong intervalley scattering in DC limits, is suppressed at high frequencies.
• Tilt Anisotropy: It reveals that the orientation of the Weyl cone tilt (parallel vs. transverse to $B$) and the relative orientation of tilts between nodes (identical vs. opposite) drastically alter the angular and magnitude profiles of the conductivity.

4. Key Results

A. Untilted Weyl Cones

• Sign Reversal of LMOC: In the weak AC limit ($\omega\tau \ll 1$) with strong intervalley scattering, the LMOC undergoes a sign reversal (becomes negative) due to the competition between the chiral anomaly current and intervalley relaxation. This is consistent with DC results.
• Suppression in Strong AC Limit: In the strong AC limit ($\omega\tau > 1$), the driving period is shorter than the intervalley scattering time. The chiral imbalance cannot relax within a single optical cycle, preventing the sign reversal. The LMOC remains strictly positive and scales as $B^2$.
• OMM Contribution: The OMM induces a linear-in-$B$ contribution to both Longitudinal (LMOC) and Transverse (TMOC) conductivities. This creates a measurable offset in the angular dependence profiles.
• Polarization Independence: The linear response (LMOC) is found to be independent of the helicity (circular polarization) of the incident light, unlike nonlinear responses.

B. Tilted Weyl Cones

• Transverse Tilt ($t_x \perp B$):
  • The LMOC exhibits a symmetric, non-monotonic dependence on tilt strength.
  • Sign reversal in the weak AC limit requires the presence of OMM.
• Parallel Tilt ($t_z \parallel B$):
  • The response is asymmetric and nearly monotonic.
  • Intrinsic Negative LMOC: For oppositely tilted cones ($t_z = -t_{-z}$), the LMOC becomes negative even without OMM in the absence of strong intervalley scattering. This is a unique feature of parallel tilt.
  • Frequency Dependence: In the strong AC limit, the LMOC becomes monotonic and insensitive to intervalley scattering strength, scaling quadratically with tilt magnitude.
• Angular Profiles: The angular dependence of LMOC follows a $\sin^2 \gamma$ profile (where $\gamma$ is the angle between $E$ and $B$), while TMOC follows $\sin 2\gamma$. The OMM shifts these profiles but preserves their functional forms.

5. Significance

• Experimental Probe: The frequency-dependent conductivity serves as a sensitive probe for chiral charge relaxation times. By tuning the frequency from MHz to THz, experimentalists can distinguish between regimes where chiral imbalance builds up versus where it is dynamically suppressed.
• Material Characterization: The distinct signatures of tilt orientation (parallel vs. transverse) and the presence of OMM provide clear experimental fingerprints to characterize the specific topological and geometric properties of Weyl materials (e.g., TaAs, NbP).
• Theoretical Validation: The results validate the necessity of including momentum-dependent relaxation times and OMM in semiclassical theories to accurately predict magneto-optical responses, correcting previous models that relied on constant relaxation times.
• Future Directions: The framework sets the stage for investigating non-linear magneto-optical effects and the role of OMM in those regimes.

In summary, this paper bridges the gap between static magnetotransport and high-frequency optical responses in topological semimetals, highlighting that the "chiral anomaly" is not a static phenomenon but a dynamic one heavily dependent on the timescale of the driving field relative to scattering processes.