🔬 mesoscale physics

Electron-phonon interactions and instabilities in Weyl semimetals under magnetic fields and torsional strain

This paper investigates how the combination of external magnetic fields and torsional strain induces asymmetric pseudo-magnetic fields in type-I Weyl semimetals, utilizing renormalization group analysis to explore the resulting evolution of coupling parameters and the emergence of lattice instabilities driven by interactions between phonons and chiral Landau levels.

Original authors: Fabian Jofre Parra, Daniel A. Bonilla, Enrique Muñoz

Published 2026-02-03

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CC BY 4.0

Original authors: Fabian Jofre Parra, Daniel A. Bonilla, Enrique Muñoz

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a crystal made of a special material called a Weyl semimetal. Inside this crystal, electrons don't behave like normal particles; they act more like massless, high-speed ghosts that can only move in specific directions. These electrons gather at specific "meeting points" in the material called Weyl nodes. Think of these nodes as two distinct dance floors where the electrons spin and move.

In this paper, the researchers are asking: What happens if we twist this crystal and also put it in a strong magnetic field?

Here is the story of their discovery, broken down into simple concepts:

1. The Setup: Twisting and Magnetizing

The researchers imagined taking a rod of this special material and doing two things to it:

Twisting it: Like wringing out a wet towel, they applied a "torsional strain" (a twist). In the world of these electrons, twisting the crystal creates a "fake" magnetic field. It's not a real magnet, but the electrons feel it exactly as if one were there.
Adding a real magnet: They also applied a real, external magnetic field.

The Magic Trick: Because of the way the material is built, the "fake" magnetic field created by the twist points in opposite directions for the two different dance floors (nodes). When you add the real magnetic field to this mix, the two dance floors end up feeling different total magnetic forces. One floor gets a stronger push, and the other gets a weaker one. This breaks the perfect symmetry between the two floors.

2. The Dance Floor Effect: Landau Levels

When you put electrons in a strong magnetic field, their movement changes drastically. Instead of moving freely in 3D space, they get trapped in tight, circular orbits, like cars stuck in a roundabout. In physics, these are called Landau Levels.

The researchers focused on the "very strong field" scenario. In this case, the electrons are so tightly trapped that they are forced to live almost entirely on the lowest possible roundabout (the Lowest Landau Level). This effectively squeezes the electrons from moving in 3D space down to moving in just 1D (like beads on a string).

3. The Instability: The Peierls Instability

When electrons are forced into this tight, 1D line, they become unstable. It's like a crowd of people trying to walk in a single file line; eventually, they start to bunch up or form patterns to make it easier to move.

In this material, the electrons want to pair up with vibrations in the crystal lattice (called phonons). When they do this, they can cause the whole crystal structure to distort slightly, forming a new, ordered pattern. This is called a Peierls instability (or a Charge Density Wave). It's a phase transition where the material changes its state to become more stable.

4. The Battle of Channels

The researchers used a complex mathematical tool (Renormalization Group theory) to track how the strength of these interactions changes as they zoomed in on the physics. They found two main "channels" or ways the electrons try to pair up:

The Peierls Channel: Electrons pairing up to create the crystal distortion (the instability we want to find).
The Cooper Channel: Electrons pairing up in a different way (similar to how superconductors work).

These two channels are like two teams fighting for control. Usually, the Cooper channel tries to stop the Peierls instability from happening.

5. The Main Discovery: Controlling the Instability

The paper's big finding is about how the twist (strain) and the magnetic field work together to tip the scales in this battle.

Symmetry Breaking: Because the twist makes the two dance floors feel different magnetic forces, it breaks the symmetry between them.
The Result: This asymmetry changes the rules of the game.
- If the two dance floors were perfectly identical (mirror symmetry), the researchers found that the critical temperature (the point where the instability happens) increases when you add the twist and magnetic field.
- If the dance floors were already different (no mirror symmetry), the critical temperature decreases.

In simple terms: By twisting the material and applying a magnetic field, you can act like a "volume knob" for this instability. You can tune the material to make it more likely (or less likely) to undergo this structural change, depending on how the material is built.

Summary

The paper doesn't claim to build a new device or cure a disease. Instead, it provides a theoretical map showing that mechanical strain (twisting) combined with magnetic fields can be used to engineer the electronic properties of Weyl semimetals. It proves that by controlling the "twist," scientists can manipulate the delicate balance between different types of electron interactions, potentially allowing them to trigger or suppress specific quantum phases in these materials.

Technical Summary: Electron-Phonon Interactions and Instabilities in Weyl Semimetals under Magnetic Fields and Torsional Strain

Problem Statement
This work investigates the interplay between external magnetic fields and torsional mechanical strain in Type-I Weyl semimetals (WSMs). While the effects of intense magnetic fields on electronic properties (such as the formation of Landau levels and dimensionality reduction) and the effects of strain (modeled as pseudo-gauge fields) are individually understood, their combined influence on electron-phonon interactions and the emergence of many-body instabilities remains less explored. Specifically, the authors address how the superposition of a physical magnetic field and a strain-induced pseudo-magnetic field breaks the symmetry between Weyl nodes of opposite chirality. This asymmetry leads to distinct effective magnetic fields at each node, potentially altering the renormalization group (RG) flow of coupling constants and the conditions for Peierls instabilities (leading to Charge Density Wave, or CDW, phases).

Methodology
The study employs a field-theoretical approach based on the Kadanoff-Wilson renormalization group (RG) scheme within the continuum approximation.

Model Construction: The authors define a minimal low-energy continuum Hamiltonian for a Type-I WSM with two nodes of opposite chirality ( $\xi = \pm$ ). The system is subjected to an external magnetic field $\mathbf{B}_0$ and torsional strain, the latter modeled as a strain-induced gauge field $\mathbf{A}^S_\xi$ . The total effective field is $\mathbf{B}_\xi = \mathbf{B}_0 + \xi \mathbf{B}_S$ , where $\mathbf{B}_S$ is the pseudo-magnetic field.
Landau Level Reduction: In the regime of very strong pseudo-magnetic fields (where the strain-induced field dominates, $\epsilon = |B_0/B_S| \ll 1$ ), the electronic spectrum is dominated by the Lowest Landau Level (LLL). The authors restrict their analysis to the chiral LLL subspace ( $n=0, \lambda=-1$ ).
Interaction Terms: The many-body Hamiltonian includes electron-electron (Coulomb) and electron-phonon (deformation potential) interactions. These are expressed in the magnetic Landau eigenbasis, distinguishing between forward scattering (intra-nodal) and backward scattering (inter-nodal) processes.
RG Analysis: Using a one-loop perturbative expansion, the authors derive flow equations (beta functions) for the coupling constants: the backward scattering coupling ( $g_1$ ), the forward scattering coupling ( $g_2$ ), and the electron-phonon vertex ( $z_{\xi\bar{\xi}}$ ). The analysis considers both the Peierls channel (electron-hole pairing) and the Cooper channel (electron-electron pairing).
Symmetry Cases: The RG flows are analyzed under two conditions: (a) Mirror symmetry between nodes ( $\Delta = 0, \delta v = 0$ ) and (b) Broken mirror symmetry ( $\Delta > 0, \delta v > 0$ ), where $\Delta$ represents the momentum separation and $\delta v$ the velocity asymmetry.

Key Contributions and Results

Asymmetric Effective Field: The combination of external magnetic fields and torsional strain generates an effective pseudo-magnetic field that differs in magnitude for the two Weyl nodes ( $B_+ \neq B_-$ ). This breaks the nodal symmetry, a feature incorporated into the RG flow equations via the parameter $\epsilon = |B_0/B_S|$ .
Renormalization Flow Equations: The authors derive a closed system of differential equations describing the flow of $g_1$ , $g_2$ , and the electron-phonon vertex. They identify that the system is decoupled from the electron-phonon sector in the fermion-only sector but coupled when vertex corrections are included.
Fixed Points:
- In the fermion sector, the analysis rules out the existence of a non-trivial fixed point (a straight line of fixed points) under the assumed strain and magnetic field regimes, leaving only the trivial Gaussian fixed point.
- In the electron-phonon sector, a fixed point exists along the line $g_1^* = (\ell_G/\ell_A)^2 g_2^*$ , where $\ell_G$ and $\ell_A$ are magnetic length scales dependent on the field asymmetry.
Peierls Instability and Critical Temperature:
- The emergence of a Peierls instability (CDW phase) is signaled by the vanishing of the renormalized phonon frequency.
- Mirror Symmetric Case ( $\Delta=0$ ): The critical temperature $T_c$ is determined by a competition between Peierls and Cooper channels. The presence of strain ( $\epsilon > 0$ ) modifies the critical exponent $\gamma_\epsilon$ , leading to an increase in the critical temperature compared to the pure magnetic field case ( $\epsilon=0$ ).
- Broken Symmetry Case ( $\Delta>0$ ): When mirror symmetry is broken, the Cooper channel is suppressed. The authors find that a Peierls instability can arise even in the limit of vanishing electron-phonon coupling, driven purely by electronic interactions. However, a finite electron-phonon coupling enhances this instability. In this regime, the critical temperature decreases as $\epsilon$ increases.
Role of Channels: The study confirms that the interplay between the Peierls channel (driving CDW) and the Cooper channel (driving superconductivity or suppressing CDW) is fundamental. The strain-induced asymmetry effectively tunes the balance between these channels.

Significance and Claims
The paper claims to provide a natural extension of previous RG analyses (specifically Ref. [33]) by incorporating the effects of mechanical strain and the resulting nodal asymmetry. The primary significance lies in demonstrating that the onset of Peierls instabilities in Type-I Weyl semimetals can be controlled via "magneto-strain engineering." By tuning the ratio of the external magnetic field to the strain-induced pseudo-magnetic field ( $\epsilon$ ), one can modulate the critical temperature for CDW formation.

The authors conclude that their theoretical results highlight the essential role of electron-phonon interactions in the emergence of interacting phases in topological materials under intense fields. They explicitly state that their work opens possibilities for engineering electronic properties in these materials through the combination of strain and magnetic fields. The study remains modest regarding future implications, noting that the current analysis is restricted to the adiabatic limit and the LLL subspace, with plans to extend the work to non-adiabatic effects and higher Landau levels in future publications.