🔬 mesoscale physics

Can Majorana zero modes in quantum Hall edges survive edge reconstruction?

This paper demonstrates that edge reconstruction in a $\nu=1$ fractional quantum Hall system proximitized by superconductors and ferromagnets generates a $\nu=1/3$ side strip that doubles the topological sectors to create $Z_2 \times Z_2$ degeneracy and a $4\pi$ Josephson periodicity, with distinct signatures appearing in the fractional Josephson current when edge velocities differ.

Original authors: Kishore Iyer, Amulya Ratnakar, Sumathi Rao, Sourin Das

Published 2026-05-01

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CC BY 4.0

Original authors: Kishore Iyer, Amulya Ratnakar, Sumathi Rao, Sourin Das

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Building a Better Quantum Lego Brick

Imagine you are trying to build a super-stable computer using tiny, magical building blocks called Majorana zero modes. These blocks are special because they are "non-abelian," meaning if you swap their positions, the system remembers the swap in a way that protects the information inside. This is the holy grail for making fault-tolerant quantum computers.

Usually, scientists try to build these blocks using a specific type of quantum fluid called a Quantum Hall system (specifically at a "filling factor" of $\nu=1$ ). Think of this system as a one-lane highway for electrons.

However, in the real world, the edges of these highways aren't perfectly sharp. They are "smooth," which causes a phenomenon called edge reconstruction. In the past, scientists worried that this smoothing would ruin the delicate Majorana blocks, destroying the quantum information.

The main claim of this paper is: Even when the edge gets "smooth" and reconstructs itself, the Majorana blocks survive. In fact, the reconstruction creates a slightly more complex situation where two pairs of these blocks exist at the same spot, but they remain distinct and stable.

The Analogy: The Highway and the Detour

To understand what happens, let's use a traffic analogy.

1. The Original Highway (The $\nu=1$ Edge)
Imagine a single-lane highway where cars (electrons) drive in one direction. This is the standard, sharp edge. It's simple and predictable.

2. The Smooth Edge and the Detour (Edge Reconstruction)
Now, imagine the road becomes smooth and wide. Because of traffic rules (quantum physics laws), the cars can't just stay on the main lane. They spill over into a side strip next to the highway.

The main highway still exists.
But now, there is a side strip (a $\nu=1/3$ fractional quantum Hall state) running parallel to it.
This side strip has its own traffic rules and creates a "detour" for the electrons.

3. The Traffic Police (Superconductors and Ferromagnets)
To create the magic Majorana blocks, scientists place "traffic police" along the road. Some are Superconductors (SC) and some are Ferromagnets (FM).

The SCs force the cars to pair up (like dance partners).
The FMs force the cars to spin in a specific direction.
Where an SC meets an FM, a "gap" forms in the traffic flow. This gap is where the Majorana zero modes hide.

The Surprise: Two Sets of Blocks, Not One

When the road was a simple single lane, scientists expected to find one pair of Majorana blocks at the gap between the SC and FM.

But with the "smooth edge" and the "side strip" (the reconstruction), the situation gets interesting:

The main highway and the side strip both interact with the traffic police.
This creates two different types of electron traffic (Type A and Type B).
Consequently, at every gap between the SC and FM, you don't just get one pair of Majorana blocks. You get two separate, decoupled pairs.

The Key Finding:
You might think having two pairs means you get "Parafermions" (a more complex, higher-level version of Majoranas). However, the paper argues that the laws of physics (specifically the constraints of the main highway) force these two pairs to behave like two separate Majorana pairs rather than merging into a complex Parafermion.

Think of it like this: You have two separate keys (Majoranas) sitting on the same keyring. They don't merge into a "super-key." They stay as two distinct keys that happen to be in the same spot.

How Do We Know This? (The Josephson Current)

How can we tell if these two pairs are really there? The paper proposes looking at the Josephson current.

Imagine the Superconductors are two bridges connected by a tunnel. Electrons can tunnel through this tunnel, creating a current.

If the traffic on the main road and the side strip moves at the same speed: The current looks normal. It's hard to tell the two pairs apart; they look like a single, messy signal.
If the traffic moves at different speeds: The two pairs of Majorana blocks leave distinct "fingerprints" on the current. The current oscillates in a very specific way (a "fractional Josephson current") that reveals the presence of both pairs.

The paper shows that even though the edge is reconstructed, the current still has a 4π periodicity (a specific rhythm that repeats every 4 units of phase). This rhythm is the signature of Majorana blocks. The fact that the rhythm persists proves the blocks survived the reconstruction.

Summary of the "Story"

The Problem: Real-world quantum edges are "smooth," which causes a side strip of electrons to form (reconstruction). Scientists feared this would destroy the Majorana blocks needed for quantum computing.
The Discovery: The Majorana blocks do not die. Instead, the side strip creates a second set of blocks alongside the original ones.
The Result: You end up with two decoupled Majorana pairs at each interface. They don't merge into something else; they stay as two distinct Majoranas.
The Proof: By measuring the electrical current (Josephson current) and noticing how it changes when the electron speeds differ, we can see the unique signature of these two surviving pairs.

In short: Edge reconstruction is not a disaster for Majorana zero modes. It's more like a traffic jam that creates a second lane, but the "magic blocks" are still there, just doubled up and ready to be detected.

1. Problem Statement

The paper addresses the robustness of Majorana Zero Modes (MZMs) and Parafermion Zero Modes (PZMs) in the context of edge reconstruction in Quantum Hall (QH) systems.

Context: Engineering non-abelian anyons (like MZMs and PZMs) is a primary route toward fault-tolerant topological quantum computing. A leading proposal involves proximitizing Fractional Quantum Hall (FQH) edges with alternating superconductors (SC) and ferromagnets (FM).
The Challenge: In real experimental systems, the edge potential is rarely sharp. Smooth edge potentials in FQH systems (specifically $\nu=1$ ) lead to edge reconstruction. This phenomenon creates counter-propagating edge modes and deposits a thin $\nu=1/3$ FQH side strip along the original $\nu=1$ edge.
The Question: Does this edge reconstruction destroy the topological protection of the zero modes? Specifically, does the presence of the $\nu=1/3$ strip convert the system into a Parafermion system (which offers richer qubit operations but is harder to stabilize), or do the constraints of the bulk $\nu=1$ state force the system back to Majorana physics?

2. Methodology

The authors employ bosonization techniques and chiral Luttinger liquid theory to model the reconstructed edge.

System Setup:
- Two concentric $\nu=1$ QH systems with opposite spins ( $\uparrow$ and $\downarrow$ ).
- Edge reconstruction creates a $\nu=1/3$ side strip, resulting in two downstream charged bosonic modes with conductances $1/3 (e^2/h)$ and $2/3 (e^2/h)$ , plus an upstream neutral mode.
- The edges are proximitized by alternating SC and FM regions (Fig. 1).
Hamiltonian Formulation:
- The system is described by a bosonized Hamiltonian $H = H_0 + H_{SC} + H_{FM}$ .
- $H_0$ describes the kinetic energy of the bosonic modes ( $\phi_1, \phi_2$ ).
- $H_{SC}$ $H_{S C}$ and $H_{FM}$ $H_{F M}$ contain pairing and backscattering terms derived from the most relevant electron operators on the reconstructed edge:
  - $\psi_A \sim e^{-i(\phi_1 + \phi_2)}$ (related to the original $\nu=1$ electron).
  - $\psi_B \sim e^{-i3\phi_1}$ (related to the $\nu=1/3$ bulk).
  - $\psi_C \sim e^{-i3/2\phi_2 - i/2\phi_3}$ (involving the neutral mode, ignored in the gapped limit).
- The authors analyze the pinning of these cosine terms in the limit of strong coupling ( $\Delta, M \to \infty$ ) to determine the ground state degeneracy.
Ground State Analysis:
- They define charge and spin parity operators for the SC and FM regions.
- They apply bulk-edge correspondence constraints to determine allowed integer vs. fractional charge/spin configurations.
- They construct the ground state manifold by counting the distinct ways to distribute spin/charge parity across the interfaces.
Josephson Junction Simulation:
- To probe the physics, they model a Josephson junction between two SC regions.
- They calculate the energy spectrum and the resulting Josephson current ( $J$ ) as a function of the SC phase bias ( $\phi_{SC}$ ), considering cases where the velocities of the two bosonic modes ( $v_1, v_2$ ) are equal and unequal.

3. Key Contributions and Results

A. Survival of Majorana Modes (No Parafermions)

Contrary to the intuition that a $\nu=1/3$ strip might host Parafermions ( $Z_3$ symmetry), the authors prove that only Majorana Zero Modes ( $Z_2$ symmetry) survive.

Mechanism: While the $\nu=1/3$ bulk allows for fractional excitations, the global constraints imposed by the $\nu=1$ bulk and the requirement for the total charge/spin on the SC/FM regions to be integers (to remain at the minima of the cosine potentials) forbid fractional occupation of the edge states.
Result: The system effectively "demotes" the potential Parafermions to Majoranas. The ground state degeneracy is doubled compared to a sharp edge (4 states vs. 2 states), but they decouple into two independent sets of Majoranas.

B. Decoupled Majorana Pairs

At each SC-FM interface, the system hosts two decoupled Majorana operators:

$\chi$ -type: Associated with the $\psi_A$ electron (involving both $\phi_1$ and $\phi_2$ modes, linked to the $\nu=1$ bulk).
$\alpha$ -type: Associated with the $\psi_B$ electron (involving only $\phi_1$ , linked to the $\nu=1/3$ bulk).

These two Majoranas are localized at the same physical interface but are topologically distinct and decoupled because $\psi_A$ and $\psi_B$ cannot transform into one another without violating energy constraints.

C. Signatures in the Josephson Current

The paper proposes a specific experimental signature to detect these two decoupled Majoranas via the fractional Josephson effect:

Case 1: Equal Velocities ( $v_1 = v_2$ ): The Josephson current depends only on the total spin parity of the junction. The signatures of the two distinct Majorana types overlap, making them indistinguishable in the current.
Case 2: Unequal Velocities ( $v_1 \neq v_2$ ): The energy spectrum splits. The Josephson current $J(\phi_{SC})$ $J (ϕ_{S C})$ displays distinct signatures for the different states within the ground-state manifold.
- The current retains a $4\pi$ periodicity (characteristic of Majoranas) regardless of the velocity difference.
- However, the amplitude and shape of the current differ for states with different combinations of $\psi_A$ and $\psi_B$ spin parities. This allows experimentalists to distinguish the presence of the second Majorana mode induced by edge reconstruction.

4. Significance

Robustness of Topological Phases: The work demonstrates that edge reconstruction, often considered a nuisance in FQH experiments, does not necessarily destroy the topological nature of engineered Majorana modes. Instead, it enriches the system by adding a second, decoupled Majorana mode.
Clarification of Parafermion Engineering: It clarifies that simply having a $\nu=1/3$ strip is insufficient to generate Parafermions; the interplay with the $\nu=1$ bulk and the specific gapping mechanisms (SC/FM) enforces $Z_2$ symmetry.
Experimental Guidance: The paper provides a concrete protocol for experimental verification: measuring the fractional Josephson current with tunable edge velocities (or in systems where velocities naturally differ) can reveal the multiplicity of the zero modes. This distinguishes the reconstructed edge scenario from both the ideal sharp edge and potential Parafermion scenarios.
Theoretical Framework: It establishes a rigorous bosonization framework for analyzing reconstructed edges with multiple gapping terms, resolving ambiguities regarding charge quantization and ground state degeneracy in complex FQH edge structures.

In summary, the paper concludes that edge reconstruction in $\nu=1$ QH systems leads to a $Z_2 \times Z_2$ symmetric ground state with two decoupled Majorana zero modes, rather than Parafermions, and that these modes can be distinguished via the velocity-dependent signatures in the fractional Josephson current.