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arxiv: 2308.01980 · v2 · submitted 2023-08-03 · ❄️ cond-mat.mes-hall

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

Kishore Iyer , Amulya Ratnakar , Sumathi Rao , Sourin Das This is my paper

Pith reviewed 2026-05-24 07:02 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall
keywords parafermionsedge reconstructionquantum Hall edgesMajorana zero modesJosephson periodicitytopological sectorsfractional quantum HallZ2 degeneracy
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The pith

Edge reconstruction of a ν=1 quantum Hall edge adds a ν=1/3 side strip that turns parafermion modes into Z₂ × Z₂ degenerate states with 4π Josephson periodicity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper asks whether Majorana zero modes trapped at domain walls of proximitized quantum Hall edges can survive when the edge reconstructs. It shows that reconstruction creates a stable ν=1/3 fractional side strip alongside the original ν=1 edge. This strip doubles the number of topological sectors and imposes an extra Z₂ factor, producing overall Z₂ × Z₂ degeneracy for each sector. The resulting many-body spectrum exhibits 4π periodicity in the Josephson effect, and the two Z₂ copies remain energetically decoupled. Signatures of these new states appear in the fractional Josephson current once the two edge velocities are allowed to differ.

Core claim

The ν = 1/3 fractional quantum Hall side strip arising due to edge reconstruction of a ν = 1 edge doubles the number of topological sectors such that each of them is Z₂ × Z₂ degenerate. The many-body spectrum displays a 4π Josephson periodicity, with the states in each Z₂ being energetically decoupled.

What carries the argument

The ν=1/3 side strip produced by edge reconstruction, which enforces an additional Z₂ factor and produces Z₂ × Z₂ degeneracy on the parafermion zero modes.

If this is right

  • The original topological sectors double in number because of the side strip.
  • Each new sector carries an extra Z₂ degeneracy that remains energetically decoupled from its partner.
  • The Josephson spectrum acquires 4π periodicity instead of the usual 2π periodicity.
  • When the two edge velocities differ, the fractional Josephson current carries clear signatures of the doubled degeneracy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Reconstruction modifies the degeneracy pattern but does not remove the topological protection of the zero modes.
  • The same doubling mechanism could appear at other fractional fillings that produce stable side strips.
  • Velocity tuning offers a practical experimental knob to isolate the new periodicity without requiring changes to the superconducting or magnetic gaps.

Load-bearing premise

Proximitization by superconductors and ferromagnets leaves the Z₂ × Z₂ structure intact and the ν=1/3 side strip stable without extra mixing or gap-opening terms that would lift the degeneracy.

What would settle it

Observation of 4π (rather than 2π) periodicity in the Josephson current through a reconstructed ν=1 edge, together with two energetically decoupled sets of states, when the two edge velocities are tuned to be unequal.

Figures

Figures reproduced from arXiv: 2308.01980 by Amulya Ratnakar, Kishore Iyer, Sourin Das, Sumathi Rao.

Figure 1
Figure 1. Figure 1: The system is then described by the bosonized Hamiltonian, H = H0 + HSC + HFM where H0 = X 2 α=1 ℏvα να Z dx h (∂xφα(x))2 + (∂xθα(x))2 i HSC = − Z dx{∆A cos [2(φ1 + φ2)] + ∆B cos [6φ1] +∆C cos [3φ2]} HFM = − Z dx{MA cos [2(θ1 + θ2)] + MB cos [6θ1] +MC cos [3θ2]} (1) where να denotes the conductance of α th edge, with ν1 = 1/3, ν2 = 2/3, and vα denotes the velocity of the α-th bosonic mode. ∆γ and Mγ are th… view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The energy spectrum ( [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Schematic shows two quantum Hall systems (shown in light red and light yellow) with different spin ( [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
read the original abstract

Parafermion zero modes can be trapped in the domain walls of quantum Hall edges proximitized by superconductors and ferromagnets. The $\nu = 1/3$ fractional quantum Hall side strip arising due to edge reconstruction of a $\nu = 1$ edge doubles the number of topological sectors such that each of them is $Z_{2} \times Z_{2}$ degenerate. The many-body spectrum displays a $4\pi$ Josephson periodicity, with the states in each $Z_{2}$ being energetically decoupled. Signatures of the new states appear in the fractional Josephson current when the edge velocities are taken to be different.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript examines the survival of parafermion zero modes (arising from Majorana modes in proximitized quantum Hall edges) under edge reconstruction. It claims that reconstruction of a ν=1 edge produces a stable ν=1/3 fractional quantum Hall side strip; the combined topological sectors yield Z₂ × Z₂ degeneracy per sector. The many-body spectrum then exhibits 4π Josephson periodicity with energetic decoupling of states within each Z₂ copy. Observable signatures appear in the fractional Josephson current when the two edge velocities are unequal. The derivation employs bosonized chiral Luttinger liquid theory for the reconstructed edge together with a proximitized Hamiltonian that introduces no explicit inter-copy mixing terms.

Significance. If the central counting and periodicity results hold, the work is significant because it directly addresses a practical obstacle (edge reconstruction) to realizing protected parafermion modes in quantum Hall–superconductor–ferromagnet heterostructures. The derivation from combined chiral Luttinger liquids supplies a parameter-free sector-counting argument and yields falsifiable predictions for the Josephson current; these features strengthen the manuscript’s contribution to the literature on topological superconductivity.

minor comments (2)
  1. [Abstract] The abstract states that the ν=1/3 strip is 'arising due to edge reconstruction' but does not indicate the section in which the stability of this strip against additional gap-opening operators is demonstrated.
  2. [Results] Notation for the two distinct edge velocities (v1 and v2) is introduced without an explicit equation reference when the fractional Josephson current is computed; a cross-reference to the relevant Hamiltonian term would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and significance assessment of our work on the survival of parafermion zero modes under edge reconstruction. We note the recommendation for minor revision and will address any specific suggestions in a revised manuscript. Since no major comments were raised, we provide no point-by-point responses below.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained in bosonized theory

full rationale

The paper derives the Z₂ × Z₂ degeneracy and 4π Josephson periodicity from the combined chiral Luttinger liquid edge modes of the ν=1 edge plus the reconstructed ν=1/3 strip, using standard bosonization plus explicit proximitized Hamiltonian terms. Sector counting follows directly from the absence of mixing/gap-opening operators between copies, with no reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. Assumptions about SC/FM preservation are stated explicitly rather than smuggled. This is the normal case of an independent theoretical derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard domain assumptions about edge reconstruction and proximitization; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Edge reconstruction of a ν=1 quantum Hall edge produces a stable ν=1/3 side strip that interacts with the proximitizing superconductor and ferromagnet without destroying the topological sectors.
    Directly stated in the abstract as the origin of the doubled degeneracy.

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