Spin and orbital mixing of edge states in a quantum Hall system proximitized by a superconductor

This paper numerically investigates chiral Andreev edge states in a quantum Hall system proximitized by a superconductor, demonstrating how Andreev reflection induces edge mode mixing, how Zeeman splitting preserves spin orthogonality, and how Rashba spin-orbit coupling combined with magnetic fields drives complex spin mixing and robust transmission degeneracies.

The Gist

The Big Picture: A Dance on the Edge

Imagine a crowded dance floor (the material) where people (electrons) are forced to move in a specific direction because of a giant magnet (the magnetic field). They can't move backward or sideways; they can only march along the very edge of the room. This is the Quantum Hall effect.

Now, imagine one side of this dance floor is lined with a special "mirror" that doesn't just reflect people—it swaps them. If a dancer steps toward the mirror, they bounce back as their opposite (a "hole" instead of an electron). This mirror is a superconductor.

When these two things meet, something magical happens. The dancers don't just bounce back and forth; they get stuck in a loop, constantly turning into their opposite and back again. The scientists call these loops Chiral Andreev Edge States (CAES). Think of them as a special kind of "ghost train" running along the edge of the room, made of half-electron and half-hole.

The paper investigates what happens when you add two new ingredients to this dance: Spin (which direction the dancer is facing) and Orbit (how they spin while moving).

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1. The Simple Dance (No Spin Mixing)

First, the scientists looked at a simple scenario where all dancers are identical twins (spin-degenerate).

• The Result: When there are only two lanes of traffic (low filling factor), the dancers interfere with each other like waves in a pond. Sometimes they cancel out, sometimes they boost each other. This creates a predictable pattern of "wiggles" in how easily electricity flows.
• The Twist: When they added more lanes of traffic (higher filling factors), the dance got messy. In a normal room, if you start in Lane 1, you must finish in Lane 1. But here, because of the special mirror (superconductor), a dancer starting in Lane 1 could end up in Lane 2. The mirror mixes the lanes together. This is a rule-breaking behavior that doesn't happen in normal materials.

2. The "Spin" Split (The Zeeman Effect)

Next, they introduced a strong magnetic force that makes the dancers care about which way they are facing (Spin).

• The Result: The dancers split into two distinct groups: "Left-Facing" and "Right-Facing."
• The Analogy: Imagine the dance floor is now divided by a wall. The Left-Facing dancers can only dance with other Left-Facing dancers, and the Right-Facing ones stay in their own group. They never mix.
• The Consequence: Because the groups stay separate, the complex lane-mixing from the previous step disappears. The dance becomes simple again. If the magnetic field gets too strong, one group disappears entirely, and the special "ghost train" stops running.

3. The Spin-Orbit Twist (Rashba Coupling)

Finally, they added a new rule: Spin-Orbit Coupling.

• The Analogy: Imagine that the dancers' direction of facing is now tied to how fast they are running. If they speed up, they are forced to turn their heads. This creates a "wobble" in their spin.
• The Result: This wobble breaks the wall between the Left-Facing and Right-Facing groups. Even though the magnetic field tries to keep them separate, the "wobble" forces them to mix.
• The Surprise: When they combined this wobble with a magnetic field pointing sideways (in-plane), the dance floor became chaotic again. All four lanes of traffic mixed together. The simple patterns of the past were replaced by complex, new oscillations. The "ghost train" became a tangled web of all possible paths.

4. The Hidden Symmetry (The Magic of Numbers)

The most fascinating discovery was a hidden mathematical rule governing the dance.

• The Observation: No matter how chaotic the mixing got, the probability of a dancer taking a specific path was always exactly the same as the probability of them taking a "mirror image" path.
• The Analogy: Imagine you have a deck of cards. If you shuffle them randomly, you might expect any card to end up anywhere. But in this system, if the Ace of Spades ends up in the top spot, the King of Hearts must end up in the bottom spot with the exact same likelihood.
• Why? This isn't a coincidence. It's a fundamental law of physics (called Unitarity and Particle-Hole Symmetry) that acts like a rigid rulebook. Even when the dancers are spinning, wobbling, and mixing lanes, the universe forces the math to balance out perfectly.

Summary

The paper tells the story of how electrons behave at the edge of a superconductor.

1. Without spin: The lanes mix up.
2. With spin: The lanes separate and stay pure.
3. With spin-orbit coupling: The lanes mix up again, but in a more complex way.
4. The Golden Rule: No matter how complex the dance gets, the probabilities of where the electrons end up always follow a strict, symmetrical pattern dictated by the laws of quantum mechanics.

The authors did not claim this leads to immediate medical cures or new computers; they simply mapped out these rules to understand the fundamental physics of these "ghost trains" of electricity.

Technical Summary

Technical Summary: Spin and Orbital Mixing of Edge States in a Quantum Hall System Proximitized by a Superconductor

Problem Statement
The integration of semiconductor-superconductor hybrid nanostructures within the quantum Hall (QH) regime offers a platform for realizing novel topological phases, including chiral Majorana fermions. While the proximity effect at normal-superconductor interfaces induces Chiral Andreev Edge States (CAES) through Andreev reflection, the interplay between the QH effect, induced superconductivity, and spin interactions (Zeeman splitting and Rashba spin-orbit interaction) remains complex. Previous studies have largely focused on spin-degenerate regimes or specific signatures of Majorana states. This work addresses the specific mechanisms of spin and orbital mixing in multimode, spinful systems, investigating how these interactions alter the transmission symmetries and conductance properties of CAES.

Methodology
The authors theoretically model a two-dimensional electron gas (2DEG) in a Hall-bar geometry proximitized by a superconductor. The system is described using the Bogoliubov-de Gennes (BdG) equations on a square lattice (lattice constant $a=5$ nm), incorporating:

• Hamiltonian Components: Kinetic energy, chemical potential, superconducting pairing potential ($\Delta$), Rashba spin-orbit interaction (SOI), and Zeeman interaction (with arbitrary magnetic field orientation).
• Parameters: Simulations utilize InSb material parameters ($m^*=0.014m$, $g=-50$, $\alpha=50$ meVnm) and a superconducting gap of $\Delta=2$ meV.
• Transport Calculation: The Landauer-Büttiker formalism is employed to calculate non-local conductance and transmission probabilities. The scattering matrix is computed using the Kwant package, with conductance maps generated via the Adaptive Python package. The analysis focuses on zero-temperature, zero-bias conditions.

Key Contributions and Results

1. Orbital Mixing in Spin-Degenerate Systems:

   • In the spin-degenerate regime, the study confirms that at low filling factors ($\nu=2$), conductance oscillations arise from the interference of two CAES modes, consistent with simple analytical models.
   • Crucially, at higher filling factors (e.g., $\nu=4$), the authors demonstrate that Andreev reflection induces orbital mixing between distinct quantum Hall edge modes. Unlike clean electronic systems where edge modes remain distinct, electrons injected into one edge mode can escape via a different mode. This mixing prevents the treatment of CAES as independent processes from separate edge channels, fundamentally altering the interpretation of conductance oscillations.

2. Spin Orthogonality and Zeeman Splitting:

   • When Zeeman interaction is included, the energy bands split into orthogonal spin species. The authors show that CAES form from electron-hole pairs with opposite spins (consistent with s-wave pairing).
   • This spin orthogonality prohibits mixing between opposite spin sectors. Consequently, the system behaves as two uncoupled sets of CAES, and the simple two-mode interference model (Eq. 5) remains valid for each spin sector individually.
   • At strong magnetic fields, Zeeman splitting can isolate a single spin species below the Fermi energy, suppressing Andreev reflection and CAES formation entirely, resulting in a constant conductance of $e^2/h$.

3. Complex Mixing via Spin-Orbit Interaction (SOI) and In-Plane Fields:

   • The inclusion of Rashba SOI alone causes slight spin depolarization but does not significantly mix spin sectors in the presence of a perpendicular magnetic field.
   • However, the combination of SOI with an in-plane magnetic field drives complex spin mixing. The SOI disrupts the ideal spin polarization induced by the Zeeman effect, re-enabling Andreev reflection even in regimes where it would otherwise be suppressed.
   • This leads to the hybridization of all four CAES bands (two orbital modes $\times$ two spin states), resulting in complex conductance oscillations that cannot be described by simple uncoupled models.

4. Transmission Degeneracies and Symmetries:

   • The paper reveals robust degeneracies in electron transmission probabilities in the presence of both SOI and magnetic fields. Specifically, distinct transport processes (e.g., electron mode 1 to CAES mode 2 vs. electron mode 2 to CAES mode 4) exhibit identical probabilities.
   • These degeneracies are not coincidental but emerge as direct consequences of the unitarity of the scattering matrix and particle-hole symmetry at zero bias. The authors derive these symmetries using Jacobi's complementary minor formula applied to the transmission matrix, showing that the specific constraints of the QH system (no backscattering) combined with PHS enforce these equalities.

Significance
The paper claims to provide a detailed theoretical understanding of how spin and orbital degrees of freedom interact in proximitized QH systems. It establishes that:

• Andreev reflection is a mechanism for orbital mixing in multimode systems, a phenomenon absent in purely electronic QH systems.
• Spin orthogonality in Zeeman-split systems preserves mode separation, whereas the combination of SOI and in-plane fields induces complex, all-mode mixing.
• The observed transmission degeneracies are fundamental properties arising from the symmetry constraints of the scattering matrix in magnetic fields with spin-orbit coupling.

The work does not propose new experimental setups or applications but clarifies the underlying transport symmetries and mixing mechanisms that must be accounted for when interpreting experimental data in semiconductor-superconductor QH devices.