Non-Fermi liquid and Weyl superconductivity from the weakly interacting 3D electron gas at high magnetic fields

This paper revisits interacting 3D electron gases in strong magnetic fields to demonstrate that generalized interactions and symmetry breaking can stabilize nematic charge density waves and catalyze a novel layered Weyl superconducting state, thereby expanding the understanding of non-Fermi liquid stability and field-resistant superconductivity in high-field regimes.

The Gist

Imagine a crowded dance floor where everyone is trying to move to the music. In most materials, electrons (the dancers) zip around freely, bumping into each other, but generally behaving like a predictable, fluid crowd. This is called a "Fermi liquid."

But what happens if you put this dance floor in a super-strong magnetic field?

The magnetic field acts like a strict bouncer who forces everyone to dance in specific, rigid lanes. They can still move forward and backward along the lane, but they can't move side-to-side. They are trapped in "lanes" (called Landau levels). In this scenario, the usual rules of the dance floor break down.

This paper explores what happens when these trapped electrons start interacting with each other in this high-magnetic-field environment. The authors, Nandagopal Manoj, Valerio Peri, and Jason Alicea, act like detectives trying to figure out the new rules of the dance.

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

1. The Great Dance-Off: Waves vs. Pairs

When electrons interact, they usually have two main ways to organize themselves:

• The Wave (CDW): Imagine the dancers suddenly deciding to form a static, wavy pattern, like a frozen ripple in the crowd. They stack themselves into layers.
• The Pair (Superconductivity): Imagine the dancers deciding to hold hands and spin together in pairs, moving as a single unit without friction. This is superconductivity.

The Old Theory: Previous research suggested that if the electrons repel each other (push away), they form the Wave. But if they attract each other (pull together), they should form Pairs (Superconductivity).

The Surprise: The authors found that even when electrons attract each other, they often don't form superconducting pairs. Instead, they get stuck in a weird, chaotic state called a Non-Fermi Liquid (NFL). It's like a dance floor where the music is so confusing that no one can form a pattern or a pair; everyone just jitters in a unique, unpredictable way.

2. The "Tilted" Layers (Nematic CDW)

The authors discovered that if you tweak the rules slightly (by changing how the electrons interact over distance), the "Wave" state changes.

• The Analogy: Imagine a stack of pancakes (the electron layers). In the old theory, the pancakes were perfectly flat and stacked straight up.
• The New Finding: In this new "Nematic" state, the pancakes spontaneously tilt. They lean over at an angle.
• Why it matters: This tilt creates a strange electrical effect. If you try to push electricity through the stack, it doesn't flow straight up; it flows sideways in a way that defies normal expectations. It's like a staircase that forces you to walk diagonally instead of straight up.

3. The "Dipole" Rule (Why Superconductivity Fails)

Why don't the electrons just pair up and become superconductors when they attract each other?

• The Analogy: Imagine a rule in the dance hall that says, "You can only move if you carry a specific type of balloon with you." In this physics world, that "balloon" is called a dipole moment.
• In a perfect, empty magnetic field, the electrons are forced to conserve these "balloons." This rule is so strict that it prevents them from forming the superconducting pairs they want to make. The "balloon rule" locks them in that chaotic Non-Fermi Liquid state.
• The authors proved that as long as this "balloon rule" exists, the electrons stay in this weird, non-superconducting state, no matter how much they try to attract each other.

4. Breaking the Rules to Create Superconductivity

So, how do we get superconductivity back? We have to break the "balloon rule."

• The Trick: The authors introduced a periodic potential (like a wavy floor or a series of hills and valleys) across the dance floor.
• The Result: This wavy floor breaks the strict "balloon rule" in one direction. Suddenly, the electrons are free to pair up!
• The New Superconductor: But it's not a normal superconductor.
  • The "Island" Effect: The electrons form superconducting "islands" along the magnetic lanes. Inside each island, they flow perfectly (superconductivity).
  • The "Insulator" Effect: Between the islands, they are stuck. They cannot jump from one island to the next because the "balloon rule" still holds in that specific direction.
  • The Result: The material acts like a superconductor in one direction but an insulator (a wall) in the other. It's a Layered Superconductor.

5. The "Weyl" Magic

The most exciting part of this new superconductor is what happens to the particles inside it.

• The Analogy: Usually, in a superconductor, all the energy gaps are closed, and particles move smoothly. But here, the authors found "holes" in the energy map.
• Weyl Nodes: These are like portals or wormholes in the energy landscape. They are points where the energy is zero, and particles can zip through them without resistance.
• Why it's cool: These "portals" (called Weyl nodes) are topologically protected, meaning they are very stable and hard to destroy. They suggest that the surface of this material might have special "highways" for electricity that don't exist anywhere else.

Summary: What does this mean for the real world?

This paper is a theoretical blueprint. It tells us:

1. High magnetic fields create weird states: Electrons don't just behave normally; they can get stuck in chaotic "Non-Fermi Liquid" states.
2. Symmetry is key: The reason they don't superconduct is due to hidden conservation laws (the "balloon rule").
3. We can engineer new materials: By adding a patterned potential (like a wavy floor), we can break those rules and create a new type of superconductor.
4. The Future: This could help scientists design materials that can conduct electricity perfectly even in strong magnetic fields (which usually kill superconductivity). This is crucial for things like powerful MRI machines, fusion energy reactors, and future quantum computers.

In short, the authors found a way to trick electrons out of a chaotic jam and into a new, exotic form of superconductivity that flows like water in one direction but acts like a wall in another, all while hosting magical "portals" for energy.

Technical Summary

1. Problem Statement

The paper investigates the fate of a three-dimensional (3D) electron gas subjected to a strong magnetic field (the "ultra-quantum" limit) when weak interactions are introduced.

• Physical Context: In this regime, the cyclotron energy exceeds the Fermi energy. The kinetic energy is quenched in the plane perpendicular to the magnetic field ($xy$-plane), leading to the formation of Landau levels (LLs). The system effectively reduces to a set of 1D dispersive bands (along the field direction $z$) with a completely flat Fermi surface in the transverse momentum space.
• The Conflict: A flat Fermi surface supports a continuous family of nesting vectors, leading to a competition between:
  1. Charge Density Wave (CDW) instabilities: Driven by repulsive interactions, leading to "self-layering" into 2D integer quantum Hall (IQH) states.
  2. Superconducting (BCS) instabilities: Driven by attractive interactions.
• The Puzzle: Previous work (Yakovenko, 1993) established that for contact interactions in the Lowest Landau Level (LLL), repulsion leads to a CDW, but attraction leads to a Non-Fermi Liquid (NFL) rather than a superconductor. The mechanism for the stability of this NFL phase and the conditions required to recover superconductivity in 3D were not fully understood, particularly regarding the role of symmetries and interaction range.

2. Methodology

The authors employ a combination of analytical and numerical techniques to generalize Yakovenko's functional renormalization group (fRG) approach:

• Continuum Model: They define a Hamiltonian for spinless (and later spinful) fermions in a magnetic field, projecting the interaction onto the LLL.
• Functional Renormalization Group (fRG):
  • They analyze the flow of the coupling function $h(k_{y1}, k_{y2})$ at the one-loop level.
  • The flow equation involves two competing diagrams: the CDW channel (particle-hole) and the BCS channel (particle-particle).
  • Unlike 1D Luttinger liquids where these diagrams cancel exactly, in this 3D problem, they do not cancel, requiring numerical integration to determine the flow.
• Generalized Interactions: The authors move beyond simple contact interactions to include:
  • Higher-order derivative terms (generalized local interactions).
  • Projections into higher Landau levels.
  • Inclusion of spin degrees of freedom.
  • Explicit breaking of translation symmetry via a periodic potential.
• Symmetry Analysis: A crucial aspect of the methodology is the identification of effective dipole conservation symmetries inherent to the LLL. In the LLL, magnetic translation symmetry implies the conservation of dipole moments ($D_x \sim P_y$, $D_y \sim P_x$).

3. Key Contributions and Results

A. Stability of the Non-Fermi Liquid (NFL)

The authors confirm and extend the finding that attractive interactions in the LLL do not spontaneously break $U(1)$ charge conservation to form a superconductor. Instead, the system flows to a gapless NFL phase.

• Robustness: The NFL phase is found to be stable even when:
  • Interactions include higher-derivative terms (generalized local interactions).
  • The system is projected into the second Landau level.
  • Continuous rotation symmetry is explicitly broken.
• Connection to 1D: In the spinful case, the phase diagram of the 3D LLL system quantitatively matches that of a 1D spinful Luttinger liquid. This suggests the NFL is a 3D analog of a Luttinger liquid, potentially exhibiting spin-charge separation.
• Symmetry Origin: The stability of the NFL is attributed to the effective dipole conservation symmetries. These symmetries forbid the spontaneous breaking of $U(1)$ charge conservation (superconductivity) in the absence of explicit symmetry breaking, analogous to modified Hohenberg-Mermin-Wagner theorems.

B. Nematic Topological Charge Density Wave (CDW)

For repulsive interactions, the authors identify a new phase distinct from the standard "normal" topological CDW (stack of IQH layers).

• Mechanism: By tuning the interaction range (specifically the ratio of contact to derivative coupling strengths), the system stabilizes a nematic CDW.
• Phenomenology: In this phase, the integer quantum Hall layers spontaneously "tilt" relative to the magnetic field. This breaks rotational symmetry ($U(1)$ in the $xy$ plane) and results in an unconventional Hall response where the Hall conductivity depends on the tilt angle.

C. Weyl Superconductor via Explicit Symmetry Breaking

The most significant result regarding superconductivity is achieved by explicitly breaking the translation symmetry in the transverse plane using a periodic potential $V(x) = -V_0 \cos(2\pi x/\lambda)$.

• Mechanism: The periodic potential destroys the continuous CDW nesting condition (by curving the Fermi surface) while preserving the dipole conservation symmetry along the $x$-direction. This removes the competition from the CDW channel, allowing weak attractive interactions to drive superconductivity.
• Layered Structure: The system forms a layered superconductor consisting of "islands" of superconductivity localized at the potential minima/maxima.
  • Intra-layer: Superconductivity exists within each island with phase coherence along $y$ and $z$.
  • Inter-layer: Due to the remaining dipole conservation symmetry along $x$, conventional Josephson coupling is forbidden. Instead, a higher-order "coordinated" hopping process occurs, leading to zero phase stiffness in the transverse ($x$) direction.
  • Transport: The bulk behaves as a Bose-Einstein Insulator (BEI) in the $x$-direction (insulating) while superconducting in the $y$-$z$ plane.
• Weyl Nodes: The quasiparticle spectrum of this bulk superconductor hosts gapless Weyl nodes.
  • The pairing function $\Delta(k_y)$ decays exponentially with momentum, leading to nodes in the Bogoliubov spectrum.
  • These nodes are topologically protected, implying the existence of Bogoliubov-Fermi arcs on the surface.
• Vortex Lattice: The order parameter forms a vortex lattice (rectangular or triangular depending on phase choices) with a unit cell area fixed by the magnetic flux, containing two vortices per flux quantum.

4. Significance and Implications

• Resolution of the NFL Puzzle: The paper provides strong evidence that the NFL phase in high-field 3D electron gases is a robust, stable phase protected by dipole conservation symmetries, rather than a fine-tuned artifact.
• New State of Matter: It predicts a novel Weyl superconductor that is anisotropic: superconducting in-plane but insulating out-of-plane (BEI behavior), with topologically protected Weyl quasiparticles.
• Experimental Relevance:
  • The results are relevant for low-carrier-density materials (e.g., Weyl semimetals) that can reach the ultra-quantum limit at accessible magnetic fields.
  • The "nematic CDW" offers a potential explanation for unconventional Hall responses in high-field experiments.
  • The findings may inform the design of field-resistant superconductivity in materials where carrier density is low, suggesting that breaking translational symmetry (e.g., via moiré potentials or strain) could be a route to stabilizing superconductivity against competing density waves.
• Theoretical Framework: The work bridges the gap between 1D Luttinger liquid physics and 3D bulk phenomena, demonstrating how higher-moment conservation laws (dipole symmetry) fundamentally alter the phase diagram of interacting fermions.

In summary, the paper demonstrates that while dipole conservation stabilizes a Non-Fermi Liquid against superconductivity in translation-invariant systems, explicitly breaking translation symmetry can catalyze a unique, anisotropic Weyl superconducting state with exotic topological properties.