Pathways from a chiral superconductor to a composite Fermi liquid

Motivated by recent experiments on rhombohedral graphene, this paper demonstrates that the evolution between a chiral superconductor and a composite Fermi liquid generally proceeds through an intermediate stable Landau Fermi liquid phase for weak attractive interactions, rather than via a direct continuous transition.

The Gist

Imagine you are a traffic controller for a massive city of electrons. In most metals, these electrons flow like cars on a highway, following predictable rules. This is what physicists call a Landau Fermi Liquid (LFL). It's the "normal" state of matter.

But in some exotic materials, like the special graphene layers mentioned in this paper, things get weird. The electrons stop acting like individual cars and start acting like a synchronized dance troupe. This is the Composite Fermi Liquid (CFL). Here, the electrons are so strongly connected that they form new, hybrid particles that move in a complex, topological way, often creating "fractional" effects (like having a charge that is a fraction of an electron).

The paper by Zhang, Shackleton, and Senthil asks a fascinating question: What happens when you try to turn this "weird dance" (CFL) into a "super-highway" (a Superconductor)?

The Setup: Two Worlds Colliding

Recent experiments have found that in certain graphene setups:

1. Without a "Moiré Pattern" (no honeycomb grid): The material becomes a Chiral Superconductor. This is a state where electricity flows with zero resistance, but it breaks the symmetry of time (it prefers to flow in one direction, like a one-way street).
2. With a "Moiré Pattern" (aligned with a substrate): The superconductivity disappears, and instead, you get Fractional Quantum Anomalous Hall (FQAH) states. This is the "weird dance" (CFL) territory.

The scientists wanted to know: If you slowly turn a knob to change the material from the "weird dance" (CFL) to the "super-highway" (Superconductor), what happens in between? Do they switch instantly? Or is there a stopover?

The Big Discovery: The "Stable Stopover"

Intuitively, you might think: "If I have a metal that wants to be a superconductor, and I add a little bit of attraction between the electrons, it should immediately become a superconductor."

The paper says: No, not so fast.

The authors used a mathematical tool called the Renormalization Group (think of it as a high-powered microscope that zooms in and out to see how rules change at different energy levels) to study this transition. They found something surprising:

The "Weird Dance" (CFL) is actually a bodyguard for the Superconductor.

In the CFL state, the electrons are coupled to invisible, fluctuating "gauge fields" (imagine them as a chaotic, swirling fog). This fog is so effective at disrupting the electrons that it suppresses their desire to pair up and become a superconductor. Even if you add a weak attractive force (the "glue" needed for superconductivity), the fog keeps the electrons apart.

The Two Pathways

Because of this "fog," the transition from the CFL to the Superconductor isn't a direct jump. It depends on how strong the "glue" (attractive interaction) is:

Pathway 1: Weak Glue (The Detour)

If the attraction between electrons is weak, the system cannot jump directly from the CFL to the Superconductor.

• The Journey: The system must first pass through an intermediate Landau Fermi Liquid (LFL) phase.
• The Analogy: Imagine trying to cross a river. The CFL is a deep, turbulent current that keeps you from swimming to the other side (Superconductor). The "glue" isn't strong enough to fight the current. So, you have to swim to a calm, shallow island in the middle (the LFL). Once you are on this island, the current is gone, and then you can finally swim to the Superconductor side.
• The Surprise: This intermediate "island" (LFL) is surprisingly stable. Even though it's a metal that should want to be a superconductor, the memory of the nearby "turbulent current" (the CFL) keeps it from superconducting until the attraction gets stronger.

Pathway 2: Strong Glue (The Magic Bridge)

If the attraction between electrons is very strong, the system takes a different route.

• The Journey: Instead of stopping at the calm island, the strong glue forces the "weird dance" (CFL) to change its steps entirely. It transforms into a Moore-Read State.
• The Analogy: This is like the dancers suddenly realizing they can hold hands in a very specific, complex knot. This knot is a Non-Abelian Topological State. It's a special kind of quantum state that is incredibly robust and has "exotic" properties (useful for quantum computing).
• The Result: From this special knot (Moore-Read), the system can then transition directly into the Chiral Superconductor.

Why Does This Matter?

This paper solves a puzzle about how nature transitions between two very different quantum states.

1. It explains the "Missing Link": It predicts that if you look closely at these graphene materials, you shouldn't see a direct jump from the fractional state to the superconductor. You should see a "normal" metal phase in between (for weak interactions) or a very exotic topological state (for strong interactions).
2. It highlights the power of "Fog": It shows that the strange, non-Fermi liquid behavior (the CFL) isn't just a weird phase; it actively protects the system from becoming a superconductor until the forces are strong enough to break through.
3. It guides future experiments: Scientists can now look for this "intermediate island" (the stable LFL) or the "magic knot" (Moore-Read state) as they tune their materials, knowing exactly what to expect.

In a Nutshell

Think of the electrons as a crowd of people.

• CFL: A chaotic, swirling mosh pit where everyone is holding hands in a complex pattern.
• Superconductor: A perfectly synchronized marching band.
• The Paper's Lesson: You can't just tell the mosh pit to instantly become a marching band. If the music (attraction) is too soft, the crowd will first calm down into a regular, orderly line (LFL) before they can start marching. If the music is loud and intense, the mosh pit will twist into a complex, magical formation (Moore-Read) before finally marching. The "mosh pit" is surprisingly good at resisting the urge to march until the conditions are just right.

Technical Summary

1. Problem Statement and Motivation

Recent experiments on $n$-layer rhombohedral graphene ($n=4,5,6$) have revealed two distinct quantum phases depending on the presence of a moiré potential:

• Moiré-less limit: A chiral superconductor (time-reversal symmetry broken) emerges from a normal metallic state.
• Moiré-aligned limit (with hBN): The superconductivity is suppressed, replaced by Fractional Quantum Anomalous Hall (FQAH) states.

The central theoretical problem addressed is the nature of the phase transition between these two regimes. Specifically, the authors investigate how a system evolves from a Composite Fermi Liquid (CFL) (the parent state of FQAH states) to a Landau Fermi Liquid (LFL) (the parent state of the chiral superconductor) in the presence of attractive electron-electron interactions.

The naive expectation is that weak attractive interactions would render the LFL unstable to superconductivity, potentially allowing a direct continuous transition from the CFL to the chiral superconductor. The paper challenges this intuition by analyzing the stability of the CFL-LFL critical point against pairing.

2. Methodology

The authors employ a combination of parton decomposition, effective field theory, and Renormalization Group (RG) analysis.

• Parton Decomposition: The physical electron $c$ is decomposed into a boson $\Phi$ and a fermion $f$ ($c = \Phi f$), coupled to an emergent $U(1)$ gauge field $a_\mu$.
  • The LFL corresponds to the condensation of $\Phi$ (superfluid), which Higgses the gauge field, leaving $f$ as a standard Fermi liquid.
  • The CFL corresponds to $\Phi$ being in a $\nu=1/2$ Laughlin state, where $f$ forms a Fermi surface coupled to a gapless gauge field.
• Critical Field Theory: The transition between CFL and LFL is described by the condensation of $\Phi$. Near the critical point, the theory is mapped to a Fermi surface of fermions coupled to gapless gauge fluctuations (Landau-damped). The authors utilize the framework developed in prior works (e.g., Metlitski et al.) involving a $1/N$ expansion and an $\epsilon$-expansion to study non-Fermi liquid criticality.
• RG Analysis of Pairing: An attractive interaction (BCS channel) is introduced. The authors derive the RG flow equations for the attractive coupling $V$ and the gauge coupling $\alpha$ to determine the stability of the phases against superconductivity. They analyze the flow in three regimes:
  1. Deep in the CFL phase.
  2. Deep in the LFL phase.
  3. At the CFL-LFL critical point.

3. Key Contributions and Results

A. Stability of the Critical Point and CFL against Pairing

The most significant finding is that pairing is suppressed at the CFL-LFL critical point and within the CFL phase for weak attractive interactions.

• Mechanism: The gauge fluctuations at the critical point (and in the CFL) generate a repulsive feedback loop that counteracts the attractive BCS interaction.
• RG Flow: The RG flow equations show an attractor line at $V = \sqrt{\tilde{\alpha}}$ (where $\tilde{\alpha}$ is the dimensionless gauge coupling). Unless the bare attractive interaction $V$ is stronger than a critical threshold ($V < -\sqrt{\tilde{\alpha}}$), the flow drives the system to a stable fixed point with no pairing instability.
• Implication: The non-Fermi liquid state at the critical point is stable against weak superconductivity.

B. Emergence of an Intermediate LFL Phase

Because the critical point and the CFL are stable against weak pairing, but the deep LFL is unstable (standard BCS), the evolution from CFL to Chiral Superconductor cannot be direct for weak interactions.

• Result: As the system is tuned from the CFL to the LFL (e.g., by increasing the moiré potential strength), it must pass through an intermediate stable Landau Fermi Liquid phase where superconductivity is suppressed due to proximity to the quantum critical point (QCP).
• Phase Diagram: The paper constructs a phase diagram where the path from CFL to Superconductor (SC) goes through an intermediate LFL region for weak interactions.

C. Strong Interaction Pathway: The Moore-Read State

For stronger attractive interactions (where $V < -\sqrt{\tilde{\alpha}}$), the stability is broken:

• CFL Pairing: The CFL undergoes a pairing transition of composite fermions, leading to a non-Abelian fractional quantum Hall state, specifically the Moore-Read (Pfaffian) state.
• Direct Transition: This Moore-Read state can undergo a direct continuous transition to the chiral superconductor (via the condensation of the bosonic parton $\Phi$).
• Alternative Pathway: For strong interactions, the evolution bypasses the stable LFL and proceeds: CFL $\to$ Moore-Read State $\to$ Chiral Superconductor.

D. Universal Scaling

The authors characterize the universal critical properties:

• Gap Scaling: Near the critical separatrix, the pairing gap exhibits an unusual scaling form: $\Delta \sim \Lambda_\omega \delta V^{-1/(16 \log \delta V)}$, distinct from standard BCS exponential scaling.
• LFL Stability Region: They derive the precise boundary in parameter space (interaction strength vs. distance from criticality) where the LFL remains stable against superconductivity.

4. Significance and Impact

• Resolving Experimental Discrepancies: The paper provides a theoretical framework explaining why chiral superconductivity and FQAH states appear in similar parameter ranges in rhombohedral graphene but are separated by a metallic phase in the absence of strong pairing. It predicts that a "metallic" region stable against superconductivity should exist between the FQAH and SC phases.
• Non-Fermi Liquid Physics: It demonstrates that non-Fermi liquid critical points (specifically CFL-LFL transitions) can act as a "shield" against superconductivity, a counter-intuitive result given that critical fluctuations often enhance pairing in other contexts (like nematic QCPs).
• Topological Order: The identification of the Moore-Read state as a potential intermediate phase connects the physics of moiré materials to non-Abelian anyons and topological quantum computing.
• Experimental Guidance: The work suggests that tuning parameters (like displacement field or pressure) to move between these phases will reveal a stable metallic window before superconductivity sets in, offering a clear experimental signature to test the theory.

In summary, the paper establishes that the transition between chiral superconductivity and fractional quantum Hall states is not generically direct. Instead, it is mediated by either a stable Fermi liquid (for weak interactions) or a non-Abelian topological state (for strong interactions), fundamentally altering our understanding of the interplay between topology and strong correlations in moiré materials.