Localization and universality of three-dimensional pseudospin-$s$ fermions

This paper establishes a unified theory showing that while the Drude conductivity of three-dimensional disordered fermions depends on pseudospin, their leading quantum interference correction exhibits a striking universality where the sign of localization or antilocalization is determined solely by the parity of the pseudospin, though interband scattering can suppress antilocalization in specific cases like $s=3/2$.

The Gist

Imagine a crowded dance floor where thousands of dancers (electrons) are trying to move from one side of the room to the other. In a perfect room, they glide smoothly. But in a real room, there are obstacles—chairs, pillars, and random people standing in the way (impurities). This is what happens in a disordered metal.

This paper is about how these dancers behave when they have a very specific, complex internal "twist" or "spin" (called pseudospin), and how that twist changes the way they get stuck or move freely.

Here is the breakdown of their discovery, using simple analogies:

1. The Dancers with Different "Spins"

In the old days, physicists thought of electrons as simple dancers with just two possible spins (like spinning clockwise or counter-clockwise). This is like a pseudospin-1/2 dancer.

However, in new, exotic materials (like CoSi or RhSi), electrons act like they have more complex internal structures. They can be pseudospin-1, pseudospin-3/2, or even higher.

• The Analogy: Imagine a simple dancer (spin-1/2) who can only do a simple spin. Now imagine a breakdancer (spin-3/2) who can do complex, multi-layered spins. The more complex the spin, the more "internal space" the dancer has to move around in.

2. The Two Ways to Get Stuck: WL vs. WAL

When these dancers hit obstacles, they scatter. Sometimes, they take a path, hit a wall, and bounce back. Quantum mechanics says that if a dancer takes a path and then takes the exact reverse path, the two journeys can interfere with each other.

• Weak Localization (WL): Imagine two dancers taking the same path in opposite directions. If they meet up and their moves add together (constructive interference), they get stuck in a loop. They are more likely to return to where they started. This makes the material a worse conductor (more resistance).

  • Who does this? Dancers with integer spins (1, 2, 3...). They are "clumsy" and tend to get stuck in loops.

• Weak Antilocalization (WAL): Now, imagine the dancers have a special "twist" in their internal structure. When they take the reverse path, their internal twist causes them to cancel each other out (destructive interference). They don't get stuck; they actually avoid returning to the start. This makes the material a better conductor.

  • Who does this? Dancers with half-integer spins (1/2, 3/2, 5/2...). They are "agile" and avoid the loops.

3. The Big Surprise: The "Universal" Rule

The authors asked: "Does the complexity of the dancer (the size of the spin) change how much they get stuck or avoid getting stuck?"

The Answer:

• The Size of the Effect is the Same: Surprisingly, whether the dancer is a simple spin-1/2 or a complex spin-100, the amount of interference (the size of the correction) is exactly the same. It's like saying a simple dancer and a breakdancer both get stuck in a loop with the exact same probability, regardless of how many spins they can do.
• The Direction Changes: The only thing that changes is the sign (positive or negative).
  • If the spin number is a whole number (Integer), they get stuck (WL).
  • If the spin number is a half number (Half-integer), they avoid getting stuck (WAL).

It's like a universal rule: "If you are an integer, you get stuck. If you are a half-integer, you don't." The complexity of the spin doesn't change the magnitude of the effect, only the direction.

4. The "Mixing" Problem (The Chaos Factor)

So far, we assumed the dancers stay in their own lane. But in real materials, obstacles can knock a dancer from one lane to another (interband scattering) or from one side of the room to the other (intervalley scattering).

The authors studied the Spin-3/2 dancer specifically. They found that when you allow these dancers to mix lanes:

• The "avoidance" behavior (WAL) gets weaker.
• If the mixing is strong enough, the dancers stop avoiding the loops and start getting stuck (switching from WAL to WL).
• The Key Insight: The stronger the "backscattering" (the chance of hitting a wall and bouncing straight back), the easier it is to switch from "avoiding loops" to "getting stuck."

5. Why Does This Matter?

This paper provides a unified rulebook for all these exotic materials.

• Before: Scientists had to study every new material (spin-1, spin-3/2, etc.) as a completely unique puzzle.
• Now: They have a single formula. If you know the spin number, you know exactly how the material will behave in a magnetic field.
  • Integer Spin? Expect a dip in conductivity (Weak Localization).
  • Half-Integer Spin? Expect a peak in conductivity (Weak Antilocalization).

Summary Analogy

Imagine a game of "Pinball."

• The Ball: The electron.
• The Flippers: The obstacles in the material.
• The Spin: The shape of the ball.

The paper says: "No matter how weird the shape of the ball is (spin-1, spin-3/2, etc.), the physics of how it bounces off the flippers follows a simple rule: Round balls (integers) tend to get stuck in corners. Spiky balls (half-integers) tend to bounce out. And the force of that bounce is the same for all of them; only the direction changes."

This helps engineers design better electronic devices by predicting exactly how these new, exotic materials will conduct electricity when they are dirty or disordered.

Technical Summary

1. Problem Statement

The paper addresses the gap in understanding quantum localization and interference effects in multifold chiral fermions. While the phenomena of Weak Localization (WL) and Weak Antilocalization (WAL) are well-established for conventional Schrödinger electrons (spin-1/2) and Dirac-Weyl fermions (pseudospin-1/2), their behavior in systems with higher pseudospin ($s > 1/2$) remains largely unexplored. These higher-order fermions, characterized by multifold band degeneracies (e.g., 3-fold, 4-fold, 6-fold) found in materials like CoSi, RhSi, and AlPt, possess an enlarged internal Hilbert space. The central questions are:

• How does the semiclassical transport (Drude conductivity) scale with arbitrary pseudospin $s$?
• Does the magnitude and sign of the leading quantum interference correction follow a universal pattern across the pseudospin hierarchy?
• How do interband and intervalley scattering channels, which become relevant for $s \geq 3/2$, affect the localization behavior?

2. Methodology

The authors develop a unified analytical framework for semiclassical transport and quantum interference in 3D disordered systems with arbitrary pseudospin $s$.

• Hamiltonian and Basis: They start with a rotationally invariant low-energy Hamiltonian $H_s(\mathbf{k}) = \hbar v \mathbf{k} \cdot \mathbf{S}$, where $\mathbf{S}$ represents the spin-$s$ generators of SU(2). The eigenstates are constructed as helicity eigenstates using Wigner rotation functions ($D^s_{ms'}$).
• Disorder Formalism: Disorder is modeled as a general short-range matrix potential $U(\mathbf{r}) = \sum u_0 \delta(\mathbf{r}-\mathbf{R}_i) M$, where $M$ is a $(2s+1) \times (2s+1)$ matrix. The disorder is expanded in terms of irreducible tensor operators, allowing for a unified treatment of scalar and matrix disorder.
• Semiclassical Transport:
  • Elastic Scattering Rates: Derived using the first Born approximation, incorporating both intraband and interband scattering.
  • Vertex Corrections: Solved via the Bethe-Salpeter equation in the particle-hole channel (ladder approximation) to satisfy the Ward identity. The vertex renormalization factor $\eta_{s'}$ is derived for arbitrary $s$ and helicity $s'$.
• Quantum Interference:
  • Cooperon Propagator: Calculated in the particle-particle channel. For scalar disorder, the authors identify that only one diffusive mode survives in the long-wavelength limit ($q \to 0$).
  • Multichannel Analysis: For $s=3/2$, the problem is extended to include interband and intervalley scattering. The Cooperon becomes a matrix object in the combined band/valley space. The authors solve the resulting coupled Bethe-Salpeter equations using an iterative ansatz to determine how channel mixing alters the interference correction.

3. Key Contributions and Results

A. Universality of Quantum Interference Magnitude

A striking result is the universality of the magnitude of the leading quantum interference correction to conductivity ($\sigma_{qi}$).

• Despite the complex internal structure and multiple bands, the magnitude of the correction for arbitrary pseudospin $s$ is identical to that of conventional diffusive metals and standard Weyl fermions.
• The formula is:
  $$\sigma_{qi} = -\frac{e^2}{h} \frac{1}{\pi^2} \left( \frac{1}{\ell} - \frac{1}{\ell_\phi} \right) e^{i \pi \alpha}$$
  where the magnitude is independent of $s$.

B. Symmetry Class and Sign Determination

While the magnitude is universal, the sign (determining WL vs. WAL) is strictly determined by the parity of $2s$:

• Half-integer pseudospin ($s = 1/2, 3/2, \dots$): The time-reversal operator satisfies $T^2 = -1$. The system belongs to the symplectic class, leading to destructive interference and Weak Antilocalization (WAL).
• Integer pseudospin ($s = 1, 2, \dots$): The time-reversal operator satisfies $T^2 = +1$. The system belongs to the orthogonal class, leading to constructive interference and Weak Localization (WL).
• This establishes a universal alternation of localization behavior as $s$ increases, independent of disorder strength or specific band details.

C. Pseudospin Dependence of Drude Conductivity

In contrast to the universal interference correction, the Drude (semiclassical) conductivity is strongly dependent on $s$.

• As $s$ increases, geometric suppression of backscattering occurs.
• The transport lifetime $\tau_{tr}$ and the vertex renormalization factor $\eta_{s'}$ increase with $s$.
• For the outermost band ($s'=s$), the conductivity scales as $\sigma \propto (2s+1)(s+1)$. This implies that higher-pseudospin materials exhibit significantly enhanced conductivity compared to standard Weyl semimetals.

D. Effect of Interband/Intervalley Scattering ($s=3/2$ Case)

The authors specifically analyzed the $s=3/2$ case (four-fold degenerate) to study the fragility of universality when multiple scattering channels exist.

• Channel Mixing: When intervalley or interband scattering is introduced, the Cooperon becomes a matrix. This mixing suppresses the WAL channel.
• Crossover: As the scattering strength between channels ($\eta_I$) increases, the system undergoes a crossover from WAL to WL.
• Critical Scattering Strength: The crossover occurs at a critical value $\eta_I^c$. The authors found an approximate inverse relationship between the critical scattering strength and the backscattering probability ($P$) of the specific channel: $\eta_I^c \propto 1/P$. Channels with high backscattering probability require weaker inter-channel mixing to drive the system into the WL regime.

4. Significance

• Unified Theory: The paper provides the first comprehensive analytical framework covering the entire pseudospin hierarchy ($s=1/2$ to arbitrary $s$), moving beyond isolated case studies.
• Experimental Predictions: It offers clear experimental signatures for multifold chiral semimetals (e.g., CoSi, RhSi). Researchers can expect:
  1. A strong variation in baseline Drude conductivity depending on the dominant Fermi surface pocket (pseudospin sector).
  2. A robust, universal magnitude for the low-field magnetoconductivity cusp, with the sign flipping between positive (WAL) and negative (WL) as the dominant carrier type changes between half-integer and integer pseudospin sectors.
• Fundamental Physics: The work highlights the interplay between internal geometry (pseudospin texture), symmetry classes (time-reversal properties), and transport universality, demonstrating that while geometric details dictate classical transport, quantum interference retains a deep universality dictated solely by symmetry.

5. Conclusion

The authors conclude that while the Drude response is highly sensitive to the pseudospin magnitude and band structure, the leading quantum interference correction exhibits a remarkable universality in magnitude, with its sign serving as a direct probe of the underlying symmetry class (orthogonal vs. symplectic). However, this universality is fragile in the presence of strong interband/intervalley scattering, which can drive a crossover from WAL to WL, a phenomenon explicitly quantified for $s=3/2$ systems.