Kohn-Luttinger Superconductivity of Weyl Fermi Arcs in PtBi$_2$

This paper proposes that the unconventional superconductivity observed on the surface of the Weyl semimetal PtBi$_2$ arises from a Kohn-Luttinger mechanism mediated by repulsive interactions on Weyl Fermi arcs, which robustly leads to a topological $i$-wave pairing state featuring a node at the center of each arc.

The Gist

Imagine a crystal called PtBi2 (Platinum-Bismuth) as a bustling city. Inside this city, electrons usually move in a chaotic, crowded way. But on the surface of this specific crystal, something magical happens: the electrons behave like massless, ghostly particles called Weyl fermions.

Think of these surface electrons not as a solid crowd, but as travelers moving along specific, winding highways known as Fermi arcs. These arcs are like bridges connecting two distant points in the city's map.

The Mystery: Superconducting Ghosts

Recently, scientists noticed that these surface highways are turning into superconductors. In a superconductor, electrons pair up and move without any friction or resistance, like dancers gliding across a perfectly smooth ice rink.

However, there was a puzzle. Some experiments suggested these surface electrons were pairing up in a very strange, "nodal" way (meaning the superconducting power drops to zero at specific points, like a donut with a hole in the middle). Others weren't sure if this was even happening. The big question was: What is the invisible force making these electrons pair up?

The Solution: The "Kohn-Luttinger" Dance

This paper proposes a solution using a theory called Kohn-Luttinger.

In everyday terms, imagine the electrons on the surface are a group of people who really dislike each other (they have a "repulsive" force, like magnets with the same pole facing each other). Usually, you'd think they would run away from each other.

But the Kohn-Luttinger theory suggests that because these electrons are moving in a specific, crowded environment (the Fermi arcs), their mutual dislike actually creates a complex, indirect "dance." They push against each other in a way that, surprisingly, creates a rhythm that allows them to pair up. It's like a group of people who hate being close to one another suddenly finding a way to hold hands because the room is shaped in a specific circle.

The Discovery: The "i-wave" Shape

The researchers built a mathematical model of this crystal and ran simulations to see what kind of "dance" the electrons would choose.

They found that in a large area of their model, the electrons naturally chose a specific pairing style called i-wave symmetry.

• The Analogy: Imagine the Fermi arc is a curved bridge. The "i-wave" pairing means the electrons pair up strongly at the ends of the bridge, but right in the center of the bridge, the pairing power drops to zero. It's like a bridge that is solid at the supports but has a tiny, invisible gap right in the middle.
• Why it matters: This "gap in the middle" (a node) matches exactly what some recent experiments (using a technique called ARPES) have seen on the surface of PtBi2.

The Robustness of the Finding

The team tested their theory by changing the "rules" of their model:

• Changing the crowd size (Chemical Potential): Even when they added or removed electrons, the "i-wave" dance remained the most popular choice, especially when the electrons were near the center of the highway.
• Changing the strength of the dislike (Interaction strength): Even when they made the electrons more repulsive, the i-wave state held strong.
• The "No Solution" Zone: They found that if the electrons were too far from the center of the highway, a different dance (called "nodal s-wave") took over, but the i-wave was still the dominant leader in the most relevant conditions.

The Bottom Line

This paper argues that the strange superconductivity seen on the surface of PtBi2 isn't caused by vibrations in the crystal (phonons) or some external force. Instead, it is driven purely by the repulsion between the electrons themselves on the surface highways.

The result is a highly specific, topological state called i-wave superconductivity, which features a "hole" or node right in the center of the electron paths. This provides a strong theoretical explanation for the experimental data we already have, suggesting that the surface of this crystal is a unique playground where repulsive electrons learn to dance in a very specific, exotic pattern.

Technical Summary

Problem Statement
Recent experimental observations in the noncentrosymmetric Weyl semimetal PtBi2 suggest the presence of unconventional superconductivity hosted by topological surface states, specifically Weyl Fermi arcs. These experiments indicate a surface critical temperature ($T_c \sim 5-14$ K) distinct from the non-superconducting or weakly superconducting bulk ($T_c \sim 0.5$ K). While Scanning Tunneling Microscopy/Spectroscopy (STM/STS) and Angle-Resolved Photoemission Spectroscopy (ARPES) have reported evidence for a highly nodal superconducting state with a node at the center of each Fermi arc, the nature of this state and its pairing mechanism remain enigmatic. It is unclear whether the instability is driven by phonons, electronic interactions mediated by the Fermi arcs, or a combination thereof. Furthermore, the density of states near the Fermi level in bulk Weyl semimetals is suppressed, making intrinsic bulk pairing difficult, whereas surface Fermi arcs provide a finite density of states, potentially allowing for higher-temperature surface superconductivity. The specific symmetry of the superconducting order parameter on these arcs and the role of repulsive electronic interactions in driving such a state require theoretical clarification.

Methodology
The authors employ a two-pronged theoretical approach to investigate superconducting pairing on Weyl Fermi arcs:

1. Phenomenological Patch Model: The authors construct a general phenomenological model treating the six Fermi arcs as twelve patches (two per arc, labeled by bulk Weyl point chirality). They analyze scattering processes in the center-of-mass frame for zero-momentum electron pairs. The interactions are categorized into intra-chirality ($g_\theta$) and inter-chirality ($w_\theta$) scattering, where $\theta$ denotes the scattering angle. By constructing a $12 \times 12$ pairing matrix and solving the resulting eigenvalue equation, they map the phase diagram of potential superconducting instabilities as a function of interaction parameters. This model accounts for fermionic indistinguishability and time-reversal symmetry, which constrain the allowed scattering processes and parity of the gap function.

2. Microscopic Kohn-Luttinger Calculation: To determine the effective interactions from first principles, the authors utilize a Kohn-Luttinger approach. They model the electronic structure of PtBi2 using a minimal tight-binding Hamiltonian with two spinful orbitals per site, invariant under the $C_{3v}$ point group, preserving time-reversal symmetry, and breaking inversion symmetry. Interactions are modeled via a Hubbard-Hund-Kanamori Hamiltonian with on-site repulsion $U$ and exchange coupling $J$. Within a weak-coupling scenario, they calculate the effective pairing vertex in the Cooper channel by dressing the bare repulsive interaction with particle-hole fluctuations (second-order diagrammatic contributions). The linearized gap equation is then solved numerically for the Fermi arc states to identify the leading superconducting instability as a function of interaction strength ($U, J$) and chemical potential ($\mu$).

Key Contributions and Results

• Phase Diagram of Instabilities: The phenomenological 12-patch model reveals a rich phase diagram dependent on the ratio of intra- and inter-chirality interactions. When intra-chirality interactions are similar, the system favors doubly degenerate $d$- and $g$-wave solutions (likely chiral $d+id$ and $g+ig$). However, reducing the intra-chirality interaction relative to the inter-chirality interaction suppresses these states and gives rise to nodal $i$-wave and fully gapped $s$-wave pairing instabilities. The model also predicts a region where no solution exists (no positive eigenvalue) and a narrow region of nodal $s$-wave pairing separating the $i$-wave and gapped $s$-wave phases.
• Parity Constraints: The study confirms that for noncentrosymmetric Weyl semimetals preserving time-reversal symmetry, odd-parity solutions (p-, f-, h-wave) are forbidden due to fermionic indistinguishability and anti-commutativity. The intra-band gap function must be of even parity. The authors clarify that "singlet-triplet mixing" in such systems is a kinematic consequence of spin-momentum locking during scattering, not a true mixing of parity-distinct gap functions.
• Dominance of the $i$-wave State: The microscopic Kohn-Luttinger calculation for PtBi2 demonstrates that for small values of the bare repulsive interaction ($U < 1.5$) and a chemical potential near the Weyl point ($\mu = 0$), the leading superconducting instability is an $i$-wave state. This state features a precise node at the midpoint of each Fermi arc, consistent with recent ARPES observations.
• Chemical Potential Dependence: As the chemical potential moves away from the Weyl point, the nodal $s$-wave state (with two nodes per arc) becomes increasingly prominent. A fully gapped $s$-wave state emerges at larger interaction strengths ($U$) and is robust against changes in chemical potential.
• Quantitative Agreement: The authors show that extracting phenomenological scattering amplitudes from the microscopic Kohn-Luttinger calculation quantitatively reproduces the phase diagram obtained from the patch model. This confirms that the formation of the $i$-wave state is favored by repulsive interactions between surface projections of Weyl points of opposite chirality.

Significance
The paper provides a possible mechanism for the observation of topological $i$-wave superconductivity on the surface of PtBi2, driven purely by electronically mediated repulsive interactions via the Kohn-Luttinger mechanism. The results suggest that the highly nodal state observed experimentally is a robust feature of the phase diagram for Weyl Fermi arcs, particularly when inter-chirality scattering is strongly repulsive. The findings imply that the specific symmetry of the superconducting order parameter on Weyl semimetal surfaces is highly sensitive to the chemical potential and interaction parameters, which could vary with surface termination. The study establishes a framework for understanding superconducting instabilities arising from repulsive interactions on the surfaces of Weyl semimetals beyond PtBi2, potentially impacting the understanding of transport characteristics like chiral anomaly-induced magnetoresistance in these materials. The authors note that while their minimal model captures the essential physics, a more realistic density-functional theory-based model would be necessary to fully address the interplay between surface and bulk states and the effects of specific surface terminations.