Interaction and disorder effects on Cooper instability in two-dimensional fractional Dirac semimetals

Using renormalization group analysis, this study reveals that in two-dimensional fractional Dirac semimetals, the emergence of Cooper instability depends on a critical interaction threshold influenced by the fractional exponent and transfer momentum, while disorder types can either promote or suppress superconductivity by respectively lowering or raising this threshold and altering the available phase space.

The Gist

Imagine a bustling city made of tiny, invisible particles called electrons. In most materials, these electrons move like cars on a highway, bumping into each other and slowing down. But in a special class of materials called Fractional Dirac Semimetals, the electrons behave like ghosts or magical spirits. They don't just move; they dance to a weird, "fractional" rhythm that doesn't follow the usual rules of physics.

The big question scientists asked in this paper is: Can these ghostly electrons ever decide to hold hands and form a superconductor?

Superconductivity is like a magical state where electricity flows with zero resistance. Usually, this happens when electrons pair up (called Cooper pairs) and move in perfect unison. In normal metals, even a tiny bit of attraction is enough to make them pair up. But in these "ghost cities," the rules are different.

Here is the story of what the researchers found, broken down into simple concepts:

1. The "Magic Threshold" (The Clean City)

Imagine the city is perfectly clean, with no potholes or obstacles.

• The Problem: In this fractional city, the electrons are so independent that they don't want to pair up easily. It's like trying to get two strangers to dance at a party when they are both very shy.
• The Discovery: The researchers found that you can't just whisper "let's dance" (a tiny bit of attraction). You have to shout it! There is a critical threshold. The attraction between electrons must be strong enough to overcome a "friction" caused by their weird fractional movement.
• The Map: They drew a map of the city based on the direction and speed of the electrons.
  • Zone I (The "No Dance" Zone): In some directions, no matter how much you shout, the electrons will never pair up. It's physically impossible there.
  • Zone II (The "Dance Floor"): In other directions, if the attraction is strong enough, they will pair up and become superconductors.
• The Twist: The shape of these zones depends on a number called $\alpha$ (alpha). Think of $\alpha$ as the "weirdness factor" of the city.
  • If the city is very weird (high $\alpha$), the "Dance Floor" (Zone II) gets bigger, making superconductivity easier to achieve.
  • If the city is only slightly weird, the "No Dance" zone dominates.

2. The "Potholes" (Disorder)

Now, imagine the city isn't perfect. It has potholes, construction zones, and random obstacles. In physics, we call this disorder. The researchers asked: Does a messy city help or hurt the dance?

They found that not all potholes are the same. They are like different types of obstacles:

• The "Good" Potholes (Types $\Delta_1$ and $\Delta_2$):
  Imagine these are like a DJ playing a beat that makes people want to dance. These specific types of disorder actually help the electrons pair up. They lower the "shout volume" needed to get the dance started. They expand the "Dance Floor" (Zone II), making superconductivity more likely.

  • Analogy: It's like a chaotic party where the chaos actually breaks the ice and gets people talking.

• The "Bad" Potholes (Types $\Delta_0$ and $\Delta_3$):
  Imagine these are like a security guard shouting "STOP!" or a heavy fog. These types of disorder hinder the pairing. They make the electrons even more stubborn, raising the "shout volume" needed to get them to pair up. They shrink the "Dance Floor."

  • Analogy: It's like a bouncer who refuses to let anyone into the VIP dance area.

3. The "Team Battle" (Mixed Disorder)

What happens if the city has both the "Good" potholes and the "Bad" potholes at the same time? It becomes a tug-of-war.

• The Result: The "Bad" potholes (the security guards) are generally stronger than the "Good" ones (the DJs). Even if you have a few helpful obstacles, if you have a mix of all types of disorder, the "Bad" ones usually win. They suppress the superconductivity, shrinking the dance floor back down.
• The Exception: If you have only the "Good" potholes, you can get superconductivity. But as soon as you add the "Bad" ones, they tend to take over and ruin the party.

The Big Picture

This paper is like a guidebook for building a superconductor out of these exotic, fractional materials.

• Key Lesson 1: You need a strong "push" (interaction) to get these electrons to pair up.
• Key Lesson 2: The "weirdness" of the material (the fractional exponent) matters a lot. The weirder it is, the easier it is to get superconductivity.
• Key Lesson 3: Not all dirt is bad. Some specific types of "messiness" (disorder) can actually help the electrons pair up, but other types will kill the effect.
• Key Lesson 4: If you have a mix of helpful and harmful messiness, the harmful stuff usually wins.

Why does this matter?
Understanding this helps scientists design better materials for future technologies. If we want to build super-fast, energy-efficient computers or power grids, we need to know exactly how to tune these "fractional" materials and control their "messiness" to get the electrons to dance in perfect harmony.

Technical Summary

1. Problem Statement

The paper addresses the fate of superconductivity in two-dimensional fractional Dirac semimetals (2D FDSMs). Unlike conventional Dirac or Weyl semimetals where energy dispersion is linear ($E \propto k$), FDSMs exhibit an anomalous fractional dispersion relation $E \propto k^\alpha$ (where $0 < \alpha < 1$).

Key open questions investigated include:

• Can Cooper instability (the precursor to superconductivity) survive in 2D FDSMs given their unique density of states and fractional statistics?
• What is the critical interaction strength required to trigger superconductivity in the presence of the fractional dispersion barrier?
• How do various types of disorder (random chemical potential, gauge potential, and mass disorders) compete with or enhance Cooper pairing in these systems?

2. Methodology

The authors employ a Wilsonian Renormalization Group (RG) approach to construct an unbiased effective low-energy theory.

• Effective Hamiltonian: The study starts with a non-interacting Hamiltonian for 2D FDSMs featuring fractional dispersion $E \propto |k|^\alpha$.
• Interactions:
  • Cooper Pairing: An attractive fermion-fermion interaction is projected onto the Cooper channel (singlet pairing).
  • Disorder: Four types of disorder scatterings are introduced using the replica technique:
    • $\Delta_0$: Random chemical potential.
    • $\Delta_1, \Delta_2$: Random gauge potentials (two components).
    • $\Delta_3$: Random mass disorder.
• RG Analysis: The authors perform a one-loop RG analysis. They integrate out fast fermionic modes within a momentum shell ($b\Lambda < |k| < \Lambda$) and rescale the remaining slow modes.
• Flow Equations: This procedure yields a set of coupled differential flow equations for the interaction strength ($\lambda$), disorder strengths ($\Delta_i$), and fermion velocity ($v_\alpha$).
• Scenarios: The analysis considers two momentum transfer scenarios:
  • Scenario-A: Finite transfer momentum $Q \neq 0$ (ZS channel).
  • Scenario-B: Finite transfer momentum $Q' \neq 0$ (ZS' channel).
  • Due to symmetry, the results for Scenario-B are analogous to Scenario-A, so the focus is primarily on Scenario-A.

3. Key Contributions and Results

A. Clean Limit (No Disorder)

In the absence of disorder, the behavior of Cooper instability is governed by the fractional exponent $\alpha$ and the transfer momentum parameters ($Q, \phi$).

• Critical Threshold: Unlike conventional metals where arbitrarily weak attraction triggers superconductivity, FDSMs require the initial pairing strength $|\lambda_0|$ to exceed a finite critical threshold $|\lambda_c|$.
• Phase Space Division: The $(Q, \phi)$ parameter space splits into two distinct regions:
  • Zone-I: Where $|\lambda_c|$ diverges. Cooper instability is strictly suppressed regardless of interaction strength.
  • Zone-II: Where $|\lambda_c|$ is finite. Cooper instability is allowed if $|\lambda_0| > |\lambda_c|$.
• Role of $\alpha$:
  • There exist two critical values, $\alpha_{c1} \approx 10^{-3}$ and $\alpha_{c2} \approx 0.61$.
  • For $\alpha \in (\alpha_{c1}, \alpha_{c2})$, Zone-I and Zone-II coexist.
  • For $\alpha < \alpha_{c1}$ or $\alpha > \alpha_{c2}$, Zone-II dominates the entire parameter space.
  • Higher $\alpha$ values generally reduce $|\lambda_c|$, enhancing the tendency toward BCS instability.
• Role of Velocity ($v_\alpha$): Lower fermion velocities reduce the critical threshold, favoring instability.

B. Effects of Single Disorder Types

Disorders are categorized based on their impact on the critical threshold $|\lambda_c|$ and the size of the superconducting phase space (Zone-II).

• Suppressive Disorders ($\Delta_0, \Delta_3$):
  • Random chemical potential ($\Delta_0$) and random mass ($\Delta_3$) increase the critical threshold $|\lambda_c|$.
  • They shrink Zone-II and expand Zone-I, effectively suppressing superconductivity.
• Promotive Disorders ($\Delta_1, \Delta_2$):
  • Random gauge potentials ($\Delta_1, \Delta_2$) decrease the critical threshold $|\lambda_c|$.
  • They expand Zone-II. Notably, $\Delta_1$ is unique in its ability to induce Cooper instability in regions (Zone-I) that were previously forbidden in the clean limit.

C. Effects of Multiple/Competing Disorders

When multiple disorder types coexist, they compete, leading to complex phase diagrams:

• Promotive + Suppressive: If a promotive disorder ($\Delta_1$ or $\Delta_2$) is present with a suppressive one ($\Delta_0$ or $\Delta_3$), the suppressive influence generally dominates. The presence of $\Delta_0$ or $\Delta_3$ prevents the transition from Zone-I to Zone-II, even if $\Delta_1$ is present.
• All Disorders Present: When all four types ($\Delta_0, \Delta_1, \Delta_2, \Delta_3$) coexist, the combined suppressive effect of $\Delta_0$ and $\Delta_3$ outweighs the promotive effects of $\Delta_1$ and $\Delta_2$. The system behaves as if the suppressive disorders are dominant, shrinking the superconducting phase space.
• Synergy of Promoters: $\Delta_1$ and $\Delta_2$ do not compete; they cooperate to lower $|\lambda_c|$ further when present together without suppressive disorders.

4. Significance

• Theoretical Framework: The paper provides a comprehensive RG framework for understanding interacting fractional Dirac materials, bridging the gap between standard Dirac physics and fractional quantum states.
• Control Parameters: It identifies the fractional exponent $\alpha$ and momentum transfer direction as tunable parameters for controlling the superconducting phase transition.
• Disorder Engineering: The study reveals that not all disorder is detrimental. Specific types of disorder ($\Delta_1, \Delta_2$) can actually enhance superconductivity in FDSMs, a counter-intuitive result that contrasts with the general "disorder destroys order" paradigm.
• Material Guidance: The findings offer critical insights for experimentalists searching for superconductivity in emerging 2D fractional Dirac semimetals, suggesting that material purity (clean limit) is not the only path to superconductivity; specific disorder engineering could be a viable strategy.

In conclusion, the paper demonstrates that Cooper instability in 2D FDSMs is a delicate balance between fractional dispersion, interaction strength, and the specific nature of disorder scatterings, with the suppressive disorders generally prevailing in complex, multi-disorder environments.