Anisotropic marginal Fermi liquid for Coulomb interacting generalized Weyl fermions

Using a large-$N$ renormalization group approach, this paper demonstrates that three-dimensional generalized Weyl semimetals with monopole charge $n \ge 2$ exhibit an anisotropic marginal non-Fermi liquid phase driven by amplified Coulomb interactions, characterized by intrinsically anisotropic screening and power-law quasiparticle suppression, in contrast to the isotropic behavior found in systems with $n=1$.

The Gist

Imagine a city built on a very strange landscape. In most cities (standard metals), the roads are flat and uniform, and traffic flows smoothly. But in this specific city, called a Generalized Weyl Semimetal, the terrain is lopsided.

Here is the story of what happens when you add "traffic jams" (electrical repulsion) to this strange city, explained simply.

1. The Strange City (The Material)

Think of the electrons in this material as cars. In a normal city, if you drive north, south, east, or west, the road looks the same. But in this "Generalized Weyl" city, the roads are different depending on which way you face:

• One direction: The road is a straight, smooth highway (linear).
• The other directions: The road is a bumpy, winding hill that gets steeper the further you go (non-linear).

The paper focuses on cities where this "bumpiness" is extra strong (mathematically, where the "monopole charge" $n$ is greater than 1). Because of this weird shape, there are more "parking spots" (states) available for cars at low speeds compared to a normal city.

2. The Traffic Jam (The Coulomb Interaction)

Electrons don't like being close to each other; they repel, like magnets with the same pole. This is the Coulomb interaction.

• In a normal city, if you have a traffic jam, the police (screening) quickly clear it up, and traffic flows normally.
• In this strange city, because there are so many "parking spots" at low speeds, the traffic jam gets amplified. The repulsion between cars becomes a huge deal.

3. The Detective Work (The Study)

The authors are like detectives trying to figure out how this traffic jam changes the behavior of the cars. They used a special mathematical tool called a Renormalization Group (RG) approach.

• The Problem: Usually, when you do this math, you have to make a guess about how to cut off the infinite details of the universe. If you guess wrong, you break the "rules of the road" (gauge symmetry), and your results are fake.
• The Solution: The authors invented a very strict, "gauge-consistent" rulebook. They checked their math against a known, simple case (like a 2D version of the city) to make sure they weren't breaking any laws. This is like a carpenter using a level to make sure their wall is perfectly straight before building the rest of the house.

4. The Big Discovery: The "Anisotropic Marginal Fermi Liquid"

When they applied their strict rules to the bumpy cities ($n > 1$), they found something surprising that doesn't happen in the flat cities ($n = 1$):

The "Cylindrical" Effect:
The traffic jam doesn't clear up the same way in all directions.

• Side-to-Side: The repulsion gets "dressed up" and changes significantly.
• Up-and-Down: The repulsion stays mostly the same.
  This creates an anisotropic (direction-dependent) environment. The electrons start behaving like a "Marginal Fermi Liquid."

What does "Marginal Fermi Liquid" mean?
Think of a "Fermi Liquid" as a group of dancers moving in perfect, synchronized steps. A "Marginal Fermi Liquid" is a group of dancers who are almost in sync, but they are slightly stumbling and losing their rhythm.

• The Stumble: The electrons lose their "coherence" (their ability to act like distinct, long-lived particles).
• The Result: The "quasiparticle residue" (the strength of the electron's identity) gets suppressed. It's as if the dancers are wearing foggy masks; you can see them, but they aren't sharp.

5. The Slow Fade (The Long-Term Outcome)

Here is the twist: The authors found that this chaotic, stumbling behavior doesn't last forever.

• Eventually, the "traffic police" (screening) do win, and the repulsion fades away. The electrons return to being normal, synchronized dancers.
• However, this fade-out happens extremely slowly (logarithmically). It's like a slow-motion sunset.
• Because it takes so long to fade, there is a huge, wide window of time (intermediate energies) where the electrons are stuck in this "stumbling" state. For all practical purposes in an experiment, they act like this strange, anisotropic liquid for a very long time.

6. How to See It (Experimental Proof)

The paper suggests how scientists can spot this in the real world:

• Heat and Squeeze: If you measure how much heat the material holds or how easy it is to squeeze (compressibility), you won't see a simple curve. You will see a curve with a "fuzzy" logarithmic correction, like a smooth line with a slight, consistent wobble.
• Light: If you shine light on it, the way it conducts electricity will depend on the direction you look. It will conduct differently horizontally than vertically.
• The Microscope (ARPES): If you use a powerful camera (Angle-Resolved Photoemission Spectroscopy) to take a picture of the electrons, the "blur" on the image will be different depending on the angle. The electrons will look "fuzzier" in one direction than the other, proving they are losing their coherence.

Summary

In short, the paper says: If you take a material with a specific, lopsided shape ($n > 1$) and let the electrons repel each other, the electrons will get stuck in a weird, direction-dependent "stumbling" state for a very long time. They aren't quite normal particles, but they aren't completely broken either. They are a Marginal Fermi Liquid, and this state is so long-lasting that it dominates the material's behavior before it finally settles down.

Technical Summary

Technical Summary: Anisotropic Marginal Fermi Liquid for Coulomb Interacting Generalized Weyl Fermions

Problem Statement
The paper investigates the fate of quasiparticles in generalized Weyl semimetals (WSMs) characterized by Weyl nodes with an integer monopole charge $|n| > 1$, in the presence of instantaneous long-range Coulomb interactions. While simple WSMs ($n=1$) exhibit a "marginal Weyl-liquid" phase where Coulomb interactions are marginally irrelevant and produce only logarithmic corrections, the behavior of systems with higher monopole charges remains an open question. The intrinsic anisotropy of the dispersion relation in generalized WSMs—linear along the $k_z$ axis but nonlinear of order $n$ in the transverse $(k_x, k_y)$ plane—leads to an enhanced low-energy density of states (DOS), $\rho(E) \sim |E|^{2/n}$. This enhancement suggests that Coulomb-induced screening and self-energy renormalization may be significantly amplified, potentially leading to a distinct non-Fermi liquid phase. A key theoretical challenge is addressing this problem while simultaneously accounting for anisotropic scaling and maintaining gauge invariance, specifically the Ward-Takahashi identity, which is often violated by naive regularization schemes.

Methodology
The authors employ a Wilsonian renormalization group (RG) approach controlled by a large-$N$ expansion, where $N$ represents the number of Weyl fermion flavors. The interaction is treated within the random phase approximation (RPA) at leading order, while fermion self-energies and anomalous dimensions arise at $O(1/N)$.

A critical methodological innovation is the formulation of a gauge-consistent ultraviolet (UV) regularization. Because the Coulomb field is the temporal component of a $U(1)$ photon gauge field, the fermion self-energy and the fermion-boson vertex must satisfy the Ward-Takahashi identity. The authors benchmark their regularization against the instantaneous analog of the Schwinger model in a reduced $(1+1)$-dimensional sector (the $k_0, k_z$ plane), where gauge invariance uniquely fixes the photon mass. This benchmark dictates a specific ordering of loop integrations:

1. Integrate over frequency ($k_0$).
2. Integrate over the longitudinal momentum ($k_z$).
3. Impose the UV cutoff only on the transverse momentum ($k_\perp$) in a thin shell.

This "cylindrical prescription" eliminates spurious power-law divergences that would violate $U(1)$ gauge symmetry, ensuring the Ward-Takahashi identity is preserved throughout the RG flow.

Key Results
The study reveals a qualitative distinction between simple ($n=1$) and generalized ($n \ge 2$) Weyl semimetals:

• Anisotropic Screening: For $n \ge 2$, the interplay between dispersion anisotropy and long-range interactions renders Coulomb screening intrinsically anisotropic. The one-loop polarization generates logarithmic dressing in the transverse sector ($\delta_{12} \neq 0$) but yields no logarithmic dressing in the longitudinal sector ($\delta_3 = 0$). Consequently, the effective Coulomb interaction flows toward an emergent "cylindrical" form at long wavelengths, with a static kernel $D^{-1}_E(k) \simeq a_\perp(E) k_\perp^2 + a_z(E) |k_z|^{2/n}$.
• Anisotropic Marginal Fermi Liquid (MFL): In the intermediate energy regime, the system enters an interaction-dominated scaling regime. Fermions acquire a finite anomalous dimension ($\gamma_0 > 0$), leading to a power-law suppression of the quasiparticle residue ($Z_\psi$). This defines an anisotropic marginal Fermi liquid.
• Marginal Irrelevance and Crossover: Despite the strong interaction effects, the effective fine structure constant $\alpha_N$ remains marginally irrelevant and flows to zero as the energy scale decreases ($\ell \to \infty$). However, this flow is only logarithmically slow. As a result, the MFL phenomenology does not represent a stable strong-coupling fixed point but rather a broad crossover controlled by a slowly running coupling.
• Scaling of Observables: The slow RG flow implies that physical observables exhibit multiplicative logarithmic corrections that can mimic effective power laws over a parametrically broad intermediate-energy window.
  • Thermodynamics: Specific heat and compressibility acquire logarithmic corrections: $C(T) \sim T^{1+2/n} / [\ln(\Lambda/T)]^{p_{th}}$ and $\kappa(T) \sim T^{2/n} / [\ln(\Lambda/T)]^{p_{th}}$.
  • Transport: Optical conductivity and optical shear viscosity show direction-dependent scaling with logarithmic corrections (e.g., $\sigma_{xx}(\omega) \sim \omega / [\ln(\Lambda/\omega)]^{p_x}$).
  • Spectral Function: Angle-resolved photoemission spectroscopy (ARPES) should detect a suppressed quasiparticle residue $Z_\psi(\omega) \sim [\ln(\Lambda/\omega)]^{-\eta_\psi}$ and anisotropic linewidth broadening.

Significance and Claims
The paper claims to uncover a new state of matter, the anisotropic marginal Fermi liquid, which emerges specifically in generalized Weyl semimetals with monopole charge $n \ge 2$ due to the amplification of Coulomb interactions by the enhanced DOS and intrinsic dispersion anisotropy.

The authors emphasize that their conclusions are qualitatively distinct from previous works (Refs. [49–51]) that utilized RPA-dressed propagators but ignored frequency dependence or dynamic screening, thereby failing to capture the anomalous dimensions. Furthermore, the study highlights the necessity of a gauge-consistent regularization scheme to correctly capture the interplay between anisotropic scaling and gauge constraints, a feature previously overlooked in similar contexts.

The significance of the work lies in providing a theoretically controlled prediction for how long-range Coulomb interactions reshape screening and quasiparticle coherence in topological semimetals with higher-order nodes. The authors assert that these predictions are testable via:

1. Scaling in thermodynamic quantities (specific heat and compressibility).
2. Direction-dependent optical conductivity.
3. Anisotropic broadening of the single-particle spectral function in ARPES.

The paper concludes that while the system ultimately flows to a weak-coupling fixed point, the logarithmically slow nature of this flow ensures that the anisotropic MFL phenomenology is observable over a wide intermediate-energy window in realistic materials.