Intraband circular photogalvanic effect in Weyl semimetals

This paper demonstrates that existing semiclassical theories, including Berry curvature dipole, side jumps, and skew scattering, fail to quantitatively match full quantum-mechanical results for the intraband circular photogalvanic effect in Weyl semimetals, indicating the necessity of incorporating additional microscopic mechanisms.

The Gist

The Big Picture: A Mismatch in the Rules of the Road

Imagine you are trying to predict how a crowd of people will move through a city when a specific type of music starts playing. You have two ways to do this:

1. The "Traffic Cop" Method (Semiclassical Approach): You treat the people like individual cars. You look at the road, the traffic lights, and how cars bump into each other to predict the flow.
2. The "Choreographer" Method (Quantum-Mechanical Approach): You treat the movement as a complex dance where every step is a probability wave. You calculate the exact interaction of every dancer with the music and the other dancers.

In most cities (standard materials), both methods give you the exact same prediction for how the crowd moves. However, in this paper, the authors looked at a very special, exotic city called a Weyl Semimetal.

They found that when they tried to predict a specific type of movement called the Circular Photogalvanic Effect (CPGE)—which is essentially a direct electric current generated when you shine a spinning (circularly polarized) light on the material—the two methods gave completely different answers.

The Exotic City: Weyl Semimetals

To understand why this is weird, you need to know what a Weyl Semimetal is.

• The Terrain: Imagine a landscape where the ground (energy levels) touches the sky at specific points, leaving no gap. These are called "Weyl nodes."
• The Residents: The particles living here are "Weyl fermions." They are like high-speed, ghostly runners who carry a special "spin" or twist.
• The Effect: When you shine a spinning flashlight (circularly polarized light) on them, these runners start moving in a specific direction, creating an electric current. This is the CPGE.

The Two Methods of Prediction

The authors tried to calculate exactly how strong this current would be using two different rulebooks.

1. The Traffic Cop Method (Semiclassical)

This method uses "rules of the road" that include some special quantum tricks. The authors looked at three specific tricks that usually explain how particles move in these materials:

• The Berry Curvature Dipole: Imagine the road has invisible magnetic hills that push the runners sideways.
• Side-Jumps: Imagine that every time a runner bumps into a rock (a defect), they don't just bounce; they take a tiny, involuntary step to the side.
• Skew Scattering: Imagine that when runners hit a rock, they are more likely to bounce left than right.

When the authors added up the effects of these three tricks, they calculated a specific strength for the current. They found a value they called $\gamma = -1$.

2. The Choreographer Method (Quantum-Mechanical)

This method looks at the raw physics of light hitting the particles. It considers the light as a photon being absorbed, which kicks the runner from one spot to another, often involving a "virtual" detour through a different energy level.

When the authors did the full, complex math for this method, they found something shocking: The current should be zero.

• They found two parts to the calculation that were equal in size but opposite in direction (like two people pushing a car with equal force from opposite sides).
• One part pushed the current to $+4/3$.
• The other part pushed it to $-4/3$.
• They canceled each other out perfectly, leaving $\gamma = 0$.

The Great Discrepancy

Here is the problem:

• Traffic Cop says: "The current is strong (value of -1)."
• Choreographer says: "There is no current at all (value of 0)."

In normal materials, these two methods always agree. In this special Weyl Semimetal, they disagree completely.

The authors tested this disagreement under many different conditions:

• What if the "rocks" (disorder) in the material are very small?
• What if the rocks are spread out over a large area?
• What if the scattering is uneven?

They found that no matter how they changed the conditions, the two methods never agreed. The "Traffic Cop" method always predicted a current, while the "Choreographer" method predicted a different current (which changed slightly with conditions but never matched the Traffic Cop).

The Conclusion: Missing the Puzzle Piece

The authors conclude that the "Traffic Cop" method (the semiclassical approach) is missing a piece of the puzzle.

They know that the "Side-Jumps," "Berry Curvature," and "Skew Scattering" are real physical effects. However, in this specific gapless material, these known effects are not enough to explain the full picture.

The Takeaway:
There is a hidden, microscopic mechanism that the current "Traffic Cop" rules don't know about yet. To get the right answer for how Weyl Semimetals react to light, we need to discover and add this missing rule to our physics toolbox. Until then, our two best ways of calculating this effect will continue to give us different, conflicting results.

Technical Summary

Problem Statement
The paper addresses a theoretical discrepancy regarding the Circular Photogalvanic Effect (CPGE) in Weyl semimetals under intraband absorption conditions ($\hbar/\tau \ll \hbar\omega \ll \varepsilon_F$). While two established theoretical frameworks—the semiclassical kinetic equation (supplemented by quantum corrections) and the full quantum-mechanical approach—are known to yield identical results for CPGE in gapped semiconductors, their application to gapless Weyl semimetals has produced conflicting predictions. Specifically, the semiclassical approach, which incorporates the Berry curvature dipole (BCD), side-jumps, and skew scattering, predicts a finite CPGE current. In contrast, the quantum-mechanical approach, treating light absorption as a photon-assisted scattering process involving virtual interband transitions, suggests the current should vanish. The authors aim to resolve this inconsistency by performing a rigorous quantitative comparison of both methods across different disorder potentials.

Methodology
The authors analyze a single Weyl node with topological charge $C = \pm 1$, described by the Hamiltonian $H = C\hbar v_0 \sigma \cdot k$. They consider the interaction with circularly polarized light and short-range scattering potentials.

1. Semiclassical Approach: The authors solve the Boltzmann kinetic equation for the distribution function of Weyl fermions. They incorporate three known semiclassical corrections:

   • Berry Curvature Dipole (BCD): Anomalous velocity linear in the electric field.
   • Side-jumps: Real-space shifts of electron wavepackets during scattering, which modify the collision integral.
   • Skew Scattering: Calculated to be zero due to the high rotational symmetry of the Weyl model.
     The total current is derived by summing contributions from the BCD and side-jump mechanisms.

2. Quantum-Mechanical Approach: The authors calculate the CPGE current using Fermi's golden rule for indirect intraband optical transitions ($k \to k'$) assisted by momentum scattering. The transition matrix element $M_{k'k}$ is decomposed into:

   • A term involving intermediate states in the conduction band ($M_{k'k}$).
   • A term involving virtual intermediate states in the valence band ($\delta M_{k'k}$).
     The current is computed by summing over all transitions, accounting for the interference between these matrix elements.

3. Disorder Potential Generalization: The analysis is extended from isotropic short-range scattering to anisotropic, angle-dependent scattering characterized by a Gaussian correlator with a spatial range parameter $d$. This allows the authors to test the robustness of the discrepancy across different disorder regimes.

Key Contributions and Results

• Semiclassical Calculation: The authors derive the CPGE current constant $\gamma$ for the semiclassical approach. The Berry curvature dipole contributes $\gamma_{BCD} = -2/3$, and the side-jump mechanism contributes $\gamma_{sj} = -1/3$. The total semiclassical result is $\gamma_{SC} = -1$. This result holds for isotropic scattering.
• Quantum-Mechanical Calculation: The quantum-mechanical calculation reveals two competing terms. The term proportional to $|\delta M_{k'k}|^2$ (virtual valence band processes) yields $\gamma^{(1)}_{QM} = 4/3$. However, the interference term $2\text{Re}(M^*_{k'k}\delta M_{k'k})$ yields $\gamma^{(2)}_{QM} = -4/3$. These two terms exactly cancel each other, resulting in a total quantum-mechanical current of $\gamma_{QM} = 0$ for isotropic scattering.
• Disorder Dependence: When generalized to anisotropic scattering (parameterized by $\alpha = 2(k_F d)^2$), the discrepancy persists.
  • The quantum-mechanical constant $\gamma_{QM}$ varies from $0$ (at $\alpha=0$) to $-4/3$ (at $\alpha \to \infty$), but remains non-zero for any finite $\alpha$.
  • The semiclassical constant $\gamma_{SC}$ exhibits a non-monotonic dependence on $\alpha$, ranging from $-1$ to values with a minimum absolute value near $\alpha \approx 1.5$.
  • Crucially, the two approaches yield different values for $\gamma$ at all spatial ranges of the disorder potential.
• Exclusion of Other Mechanisms: The authors investigate the "interband-coherence" effect (a linear-in-$E$ correction to the scattering matrix element overlap). They demonstrate that for Weyl semimetals, this correction vanishes ($\delta_E |\langle u_{k'}|u_k \rangle|^2 = 0$), ruling it out as the missing contribution.

Significance and Claims
The paper claims that the existing quasiclassical and full quantum-mechanical approaches are fundamentally inconsistent in their description of the intraband CPGE in Weyl semimetals. Unlike in wide-band gap gyrotropic materials where these methods are equivalent, the gapless nature of Weyl semimetals leads to a persistent disagreement regardless of the disorder potential range.

The authors conclude that the standard semiclassical mechanisms (BCD, side-jumps, skew scattering) are insufficient to describe the CPGE in this system. They assert that the implementation of an additional, currently unidentified microscopic mechanism into the quasiclassical description is required to reconcile the semiclassical results with the quantum-mechanical findings. The paper does not propose a specific new mechanism but highlights the necessity of finding one to resolve the theoretical contradiction.