Magnetotransport in the presence of real and momentum space topology

This paper investigates magnetotransport in time-reversal symmetry-broken Weyl semimetals using a semiclassical Boltzmann approach, revealing that the interplay between momentum-space Berry curvature and skyrmion-induced real-space emergent magnetic fields creates distinct strong-and-weak sign-reversal regimes in magnetoconductivity and generates a unique planar Hall response that serves as a measurable signature of real-space topology.

The Gist

Imagine a bustling city where electrons are the commuters. In most materials, these commuters follow simple, predictable traffic rules. But in a special class of materials called Weyl Semimetals, the city is built on a strange, twisted geometry.

This paper explores what happens when you put these special electrons under two different kinds of "traffic pressure" at the same time: one coming from the map of the city itself, and one coming from the actual streets they are driving on.

Here is the breakdown of the research using simple analogies:

1. The Two Types of "Twists"

The researchers are studying a material where two distinct topological "twists" exist simultaneously:

• Twist A: The Map (Momentum Space Topology)
  Imagine the city's map has hidden shortcuts and dead ends that only exist in the idea of the city, not on the ground. In physics, this is called Berry Curvature. It's like the map tells the electron, "Hey, if you turn left here, you actually end up moving right." This is a fundamental property of the material's electronic structure.
• Twist B: The Streets (Real Space Topology)
  Now, imagine the actual streets of the city are winding, swirling, and twisting like a tornado. This is caused by a magnetic pattern called a Skyrmion (think of it as a tiny, swirling magnetic tornado frozen in the material). As electrons drive through these swirling streets, they feel an invisible "wind" pushing them sideways. This is called the Emergent Magnetic Field.

2. The Experiment: Mixing the Twists

The researchers asked: What happens when you drive through a city where the map is twisted AND the streets are swirling?

They used a mathematical model (like a traffic simulation) to see how the electrons move when you apply a real magnetic field (like a strong wind blowing through the city). They looked at two main things:

• Longitudinal Conductivity: How easily electricity flows with the wind.
• Planar Hall Conductivity: How much electricity gets pushed sideways by the wind.

3. The Surprising Results

The study found that the two "twists" don't just add up; they play a complex game of tug-of-war that creates three distinct behaviors:

A. The "Strong Reversal" (The Map Takes Over)

If the "swirling streets" (Skyrmions) are calm, but the "traffic jams" between different lanes (Intervalley Scattering) are heavy, the electricity flow flips direction.

• Analogy: Imagine a river flowing downstream. If you throw enough rocks (scattering) into it, the water suddenly starts flowing upstream against the current. This is a known phenomenon in these materials, but the researchers confirmed it happens here too.

B. The "Weak Reversal" (The Streets Take Over)

When the "swirling streets" (Emergent Field) are active, they don't flip the flow entirely. Instead, they shift the whole pattern.

• Analogy: Imagine a parabolic slide (a U-shape). Usually, the bottom of the slide is at the center (zero magnetic field). The swirling streets push the whole slide to the left or right. Now, the bottom of the slide isn't at zero anymore. The electricity flow changes sign, but the shape of the curve stays the same. It's like the "zero point" of the system has moved.

C. The "Strong-and-Weak" Combo (The Best of Both Worlds)

This is the most exciting discovery. When both effects are present, you get a slide that is both flipped upside down (Strong) and shifted to the side (Weak).

• The Takeaway: The researchers realized that the "Map" (momentum space) controls the shape of the curve, while the "Streets" (real space) control the position of the curve. They are independent knobs you can turn to tune the material's behavior.

4. The "Asymmetry" Surprise

Usually, if you blow wind from the North, the traffic behaves the same as if you blow from the South (just mirrored). But because the "swirling streets" (Skyrmions) have a specific direction, they break this symmetry.

• Analogy: Imagine a windmill. If the wind blows from the North, it spins one way. If it blows from the South, it spins the other way. But if the windmill blades are bent (the Skyrmion texture), blowing from the North might make it spin fast, while blowing from the South makes it spin slowly. The response isn't perfectly symmetrical anymore. This asymmetry is a "fingerprint" that proves the real-space topology is there.

5. Why Does This Matter?

The paper concludes that the "Emergent Magnetic Field" (from the Skyrmions) is not just a background noise; it is a tuning knob.

• Before: Scientists thought you could only tune these materials by changing the external magnetic field or the material's strain.
• Now: We know that by changing the magnetic texture (the Skyrmions), you can independently shift the electrical response without changing the material's fundamental map.

In simple terms: The researchers found a new way to control electricity in exotic materials. They discovered that the "swirls" in the magnetic texture act like a steering wheel that shifts the entire electrical response, while the "map" acts like the engine that determines how fast it goes. This opens the door to designing new electronic devices where we can manipulate current flow using magnetic textures, potentially leading to faster, more efficient, and smarter electronics.

Technical Summary

1. Problem Statement

The paper investigates the magnetotransport properties of Weyl Semimetals (WSMs) where two distinct forms of topology coexist:

1. Momentum-space topology: Arising from Weyl nodes (monopoles of Berry curvature, $\Omega_k^\chi$) which drive phenomena like the chiral anomaly.
2. Real-space topology: Arising from non-trivial magnetic textures, specifically skyrmions, which generate an emergent magnetic field ($B_{emer}$) via the real-space Berry phase.

While previous studies have focused on either momentum-space Berry curvature or strain-induced axial fields, this work addresses the mixed regime where an external magnetic field ($B$) and a skyrmion-induced emergent field ($B_{emer}$) act simultaneously. The central question is how the interplay between these two fields, mediated by intervalley scattering, modifies longitudinal magnetoconductivity (LMC) and planar Hall conductivity (PHC).

2. Methodology

The authors employ a semiclassical Boltzmann transport formalism incorporating Berry-curvature corrections.

• Model: A minimal low-energy Hamiltonian for a time-reversal symmetry-broken, untilted WSM with a pair of Weyl nodes (chirality $\chi = \pm 1$).
• Emergent Field: The skyrmion texture is modeled as a uniform emergent magnetic field $B_{emer} = \frac{\hbar}{2e} \mathbf{n} \cdot (\partial_x \mathbf{n} \times \partial_y \mathbf{n})$, which acts on charge carriers analogously to an external field but originates from real-space spin textures.
• Equations of Motion: The dynamics are governed by modified equations of motion where the total magnetic field is $B_{tot} = B + B_{emer}$. Crucially, the phase-space factor is modified to $D^\chi = [1 + \frac{e}{\hbar}(B + B_{emer}) \cdot \Omega_k^\chi]^{-1}$, introducing a cross-term $B_{emer} \cdot \Omega_k^\chi$.
• Scattering: The collision integral includes both intra-node and inter-node (intervalley) scattering. The relative strength of intervalley scattering is controlled by a parameter $\alpha = U^{\chi\chi'}/U^{\chi\chi}$.
• Calculation: The authors solve the linearized Boltzmann equation to determine the non-equilibrium distribution function and subsequently calculate the current density $J$. They analyze the longitudinal conductivity ($\sigma_{zz}$) and planar Hall conductivity ($\sigma_{xz}$) as functions of magnetic field magnitude, direction, and scattering strength.

3. Key Contributions

• Identification of Mixed Topological Regimes: The paper establishes that the coexistence of real-space (skyrmion) and momentum-space (Weyl) topology creates a unique transport regime distinct from systems with only one type of topology.
• Decoupling of Geometric Features: The authors demonstrate that intervalley scattering and the emergent field control distinct geometric features of the magnetoconductivity curve:
  • Intervalley scattering controls the curvature (sign of the quadratic term).
  • The emergent field controls the shift of the parabola's vertex.
• Discovery of a New Transport Signature: They show that a finite Planar Hall Conductivity (PHC) can arise solely from the emergent field $B_{emer}$ when its direction is varied, even in the absence of an external magnetic field. This serves as a direct transport signature of real-space topology.
• Angular Asymmetry: The study reveals that $B_{emer}$ breaks the effective inversion symmetry of the angular magnetotransport response, leading to pronounced asymmetry in the angular dependence of both LMC and PHC.

4. Key Results

A. Longitudinal Magnetoconductivity (LMC)

The LMC is analyzed using a quadratic fitting form $\sigma(B) = \sigma^{(0)} + (B-B_0)^2 \sigma^{(2)}$. Three distinct regimes are identified:

1. Strong Sign Reversal: Occurs when intervalley scattering is strong ($\alpha > \alpha_c$). The curvature $\sigma^{(2)}$ becomes negative, inverting the parabola. This is driven by the competition between chiral anomaly pumping and scattering relaxation.
2. Weak Sign Reversal: Occurs when $B_{emer}$ is finite. The vertex of the parabola shifts ($B_0 \neq 0$), but the curvature remains positive. This is caused by the linear coupling of $B_{emer}$ to the Berry curvature.
3. Strong-and-Weak Sign Reversal: A unique regime where both effects occur simultaneously. The parabola is inverted (strong) and shifted (weak). This arises from the cooperative interplay of strong intervalley scattering and the skyrmion-induced field.

• Angular Dependence: The presence of $B_{emer}$ introduces asymmetry in the angular dependence of LMC around $\gamma = \pi$ (where $\gamma$ is the angle of the magnetic field). This is because $B_{emer}$ has the same sign at both Weyl nodes, whereas the intrinsic Berry curvature changes sign with chirality, breaking the effective inversion symmetry.

B. Planar Hall Conductivity (PHC)

• Standard Behavior: Without $B_{emer}$, PHC is governed by momentum-space topology and follows a $\sin(2\gamma)$ dependence. Increasing intervalley scattering suppresses the magnitude but does not cause sign reversal.
• Emergent Field Effect: When $B_{emer}$ is present, the PHC parabola shifts, leading to a weak sign-reversal regime.
• Pure Real-Space Signature: Crucially, the authors find that if the external field $B=0$ but the direction of $B_{emer}$ is varied, a finite PHC is generated. This confirms that real-space topology alone can induce a measurable transverse response, scaling as $\sin(2\gamma_{emer})$.

5. Significance and Implications

• Unified Topological Description: The work provides a unified framework describing how the product of two topological charges (Weyl monopole charge and skyrmion winding number) drives transport phenomena. It bridges the gap between momentum-space Berry curvature and real-space spin textures.
• Experimental Predictions: The paper proposes clear experimental signatures for magnetic Weyl semimetals (e.g., kagome materials like $Co_3Sn_2S_2$) hosting skyrmions:
  1. A shifted parabolic LMC curve that does not peak at $B=0$.
  2. Angular asymmetry in magnetotransport measurements.
  3. A finite Planar Hall effect driven purely by the rotation of magnetic textures (skyrmion density).
• Tunability: The results suggest that the emergent field acts as an independent topological tuning parameter. By manipulating skyrmion density (via temperature or magnetic history), one can tune the transport response without changing the external magnetic field.

In conclusion, this study demonstrates that the interplay between real-space and momentum-space topology in Weyl semimetals leads to rich, measurable transport phenomena that cannot be explained by considering either topology in isolation. The "strong-and-weak" sign reversal regime and the skyrmion-driven Planar Hall effect offer new avenues for detecting and characterizing mixed topological states in quantum materials.