🔬 mesoscale physics

Vortex-core Majorana coupling to a chiral edge in a $p_x+ip_y$ superconductor: Nonmonotonic spectral reorganization and coherent fermion-parity dynamics

This study demonstrates that while a vortex-core Majorana wave packet maintains a high retained norm when projected onto the low-energy subspace of a finite $p_x+ip_y$ superconducting disk, its coupling to a chiral edge induces nonmonotonic spectral reorganization that leads to diverse dynamical behaviors—including rapid dephasing and sign-changing oscillations—thereby showing that a large retained norm does not guarantee spectral concentration or persistent fermion-parity memory.

Original authors: Peiyao Liu, Yi Zhou

Published 2026-09-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Peiyao Liu, Yi Zhou

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

In the quest to build a quantum computer that can withstand the noise of the real world, physicists are hunting for a very specific kind of particle. They are looking for Majorana fermions, exotic states of matter that behave like their own antiparticles. These particles are special because they can store information in a way that is naturally protected from errors, a property known as "topological protection." Imagine a knot in a string; you can shake the string, but the knot remains until you actively untie it. In a similar way, information stored in these particles is robust against local disturbances. To create them, scientists use a special type of superconductor, a material that conducts electricity without resistance, where the electrons pair up in a unique, spinning configuration. In these materials, if you create a tiny whirlpool, or vortex, in the flow of electrons, a Majorana particle can get trapped right in the center of that swirl. At the same time, the edge of the material itself hosts a stream of similar particles. The big question for researchers is what happens when you bring the trapped particle in the center close to the stream of particles at the edge. Do they stay separate and keep their memory safe, or do they mix up and lose the information they were meant to hold?

A team of researchers at the Institute of Physics in Beijing has simulated this exact scenario to see how the two types of particles interact. They modeled a circular disk of this special superconductor and placed a single vortex inside it, then moved the vortex closer and closer to the edge of the disk. Their goal was to watch how the energy levels of the system changed and to see if a specific packet of quantum information, prepared at the center, would remain stable as the vortex moved. They found that the relationship between the center and the edge is far more complicated than a simple, smooth transition. As the vortex approaches the boundary, the system does not just gradually mix the two states; instead, it undergoes a series of sudden, non-linear reorganizations. The researchers discovered that the distance between the vortex and the edge matters immensely, with a particularly dramatic shift occurring when the separation is roughly seven times the size of the fundamental quantum length scale of the material.

The study reveals that simply keeping the vortex physically close to the center of the disk is not enough to guarantee that the information remains safe. The team tracked a specific quantum state, which they prepared as a localized packet of energy at the vortex core, and watched how it evolved as the vortex was moved to different positions. They found that even when the packet remained largely intact—retaining more than 98 percent of its original form—the way its energy was distributed changed drastically depending on the distance. In some positions, the energy stayed concentrated in a single, nearly zero-energy state, which is ideal for keeping the information stable over time. In other positions, the energy scattered across several different levels. This scattering causes the quantum information to lose its coherence rapidly, a process the researchers call "coherent dephasing," where the different energy components cancel each other out, leading to a loss of the stored memory.

Perhaps the most surprising finding is that the system behaves in a non-monotonic way, meaning that moving the vortex closer does not always make the situation worse in a predictable, steady line. Instead, the quality of the memory storage fluctuates wildly as the distance changes. At certain specific separations, the system reorganizes itself so that the information remains stable, while at others, it fragments. The researchers identified a region around a separation of seven times the coherence length where these changes happen most rapidly. They also observed that the system can exhibit "recurrences," where the information, after appearing to be lost, suddenly reappears at later times due to the finite size of the disk. This suggests that in a real, finite-sized device, the timing of operations would be critical, as the stability of the qubit depends on the precise moment you measure it.

The work challenges the idea that spatial separation alone is the key to protecting quantum information. The researchers showed that a state can be highly localized in space, sitting firmly at the vortex core, yet still fail to maintain its quantum memory if its energy is spread across multiple levels. This distinction is vital for future quantum computing designs, as it implies that engineers cannot rely solely on keeping vortices far apart to ensure stability. Instead, they must account for the complex, oscillating nature of the interaction between the core and the edge. The simulations suggest that the behavior of these particles is governed by the intricate interference of their wave-like tails, which extend far beyond the visible core of the vortex. When these tails overlap with the edge states, they create a complex pattern of energy levels that can either preserve or destroy the quantum state depending on the exact geometry.

By mapping out these energy landscapes, the study provides a detailed guide for how these systems behave under realistic conditions. The researchers used a model that mimics the behavior of electrons in iron-based superconductors, a class of materials where these effects are being actively investigated. Their calculations show that the transition from a stable, protected state to a chaotic, dephasing one is not a smooth slide but a jagged landscape of peaks and valleys. This means that for a quantum computer based on these principles to work, the placement of the vortices must be controlled with extreme precision. A small shift in position could move the system from a safe zone into a region where the information is scrambled. The findings highlight that the path to reliable quantum computing involves not just finding the right materials, but mastering the subtle, non-linear dance of energy levels that emerges when these exotic particles are brought together.

Technical Summary: Vortex-core Majorana coupling to a chiral edge in a $p_x + ip_y$ superconductor

Problem Statement
This study investigates the low-energy spectral reorganization of a finite two-dimensional $p_x + ip_y$ superconducting disk as a function of the separation $d$ between a unit-winding vortex and the disk boundary. While isolated vortices in the weak-pairing phase bind Majorana zero modes (MZMs) and open boundaries support chiral Majorana edge modes, the coupling between these two entities in a finite geometry remains a complex problem. Specifically, the authors address two distinct questions: (1) How do individual finite-disk eigenstates redistribute between the vortex core and the boundary as $d$ varies? and (2) Does a prescribed, core-localized, self-conjugate wave packet remain spectrally concentrated and retain fermion-parity memory when coupled to the edge, or does it suffer from coherent dephasing due to spectral fragmentation?

Methodology
The authors employ a Bogoliubov–de Gennes (BdG) numerical framework to solve the mean-field Hamiltonian for a spinless $p_x + ip_y$ superconductor on a disk of radius $R = 30\xi$ (where $\xi$ is the coherence length).

Model: The system is defined by a continuum Hamiltonian with a spatially varying pairing field $\Delta(\mathbf{r})$ . To simulate a single vortex near a smooth boundary without periodic artifacts, the authors utilize the method of images, placing an antivortex outside the disk to enforce a vanishing normal phase gradient ( $\partial_r \phi|_{r=R} = 0$ ) and hard-wall boundary conditions ( $u=v=0$ ) for quasiparticles.
Parameters: Calculations are performed for two coupling regimes: $\Delta_0/E_F = 0.08$ (weak coupling) and $\Delta_0/E_F = 0.36$ (motivated by iron-based superconductors).
Eigenstate Analysis: The authors compute the lowest 50 positive-energy eigenstates. To diagnose the spatial nature of these states, they introduce "regional diagnostics":
- Core Weight ( $W_{core}$ ): The probability weight of the state within a region defined by proximity to the vortex versus the boundary.
- Core-Edge Polarization ( $M_{ce}$ ): A measure of complementary self-conjugacy between the core and edge regions. $M_{ce} \approx -1$ indicates well-separated Majorana components, while $M_{ce} \approx 0$ indicates a lack of such separation.
Wave Packet Construction: To address the second question, the authors construct a "translated core-reference packet." They take the lowest particle-hole pair of a centered vortex (where symmetry ensures a well-defined MZM), rigidly translate this profile to various off-center positions, restrict it to the disk, and project it onto the retained low-energy subspace without intermediate normalization.
Parity Dynamics: The spectral measure of this projected packet is used to calculate the signed parity correlator $C(t)$ , which describes the time evolution of fermion parity when the packet is paired with an ideal remote Majorana mode.

Key Results

Nonmonotonic Spectral Reorganization: The coupling between the vortex core and the chiral edge exhibits strong nonmonotonic behavior as a function of separation $d$ . While the lowest eigenstate generally shows separated Majorana components at large $d$ , the system undergoes rapid and complex reorganization near $d \simeq 7\xi$ . At this scale, the energy, core weight, and polarization of the lowest state vary significantly and non-monotonically.
Distinction Between Localization and Separation: The study demonstrates that strong spatial localization of a state in the vortex core (high $W_{core}$ ) does not guarantee the presence of separated Majorana components (high $|M_{ce}|$ ). For instance, Caroli–de Gennes–Matricon (CdGM) states can be highly core-localized ( $W_{core} \approx 1$ ) yet possess negligible core-edge polarization ( $M_{ce} \approx 0$ ).
Spectral Fragmentation vs. Retained Norm: A critical finding is that a high "retained norm" (the projection of the translated packet onto the low-energy subspace, which exceeds 0.98 for all sampled separations) does not imply spectral concentration or persistent parity memory.
- Depending on $d$ , the spectral weight of the packet can be concentrated near zero energy (yielding slow temporal variation in parity), fragmented over several low-energy levels (yielding rapid coherent dephasing), or dominated by finite-energy weight (yielding sign-changing oscillations).
- Specifically, at $d \simeq 5.75\xi$ and $7.5\xi$ , the packet is dominated by a near-zero mode, preserving parity memory. Conversely, at $d \simeq 7\xi$ and $8.25\xi$ , the weight fragments, causing rapid dephasing.
Scale of Interaction: The strongest reorganization occurs near $d \simeq 7\xi$ , which is consistent with an order-of-magnitude estimate of the combined spatial extent of the vortex core and the chiral edge mode ( $\sim (2\pi + 1)\xi$ ).

Significance and Claims
The paper claims to resolve a fundamental ambiguity in finite-geometry Majorana systems: spatial localization is insufficient to guarantee the topological protection or stability of parity memory. The authors establish that the dynamical behavior of a Majorana-based qubit (specifically its parity memory) is controlled by the detailed spectral weight distribution of the wave packet across the low-energy eigenstates, rather than by the packet's spatial overlap with the core alone.

The study highlights that vortex-edge coupling can induce "coherent dephasing" and "sign-changing oscillations" in parity dynamics even when the wave packet remains largely within the low-energy subspace. This implies that for Majorana-based quantum operations, simply ensuring a vortex is far from the boundary is not the only criterion; the specific separation distance relative to the coherence length critically determines whether the system exhibits stable parity memory or suffers from coherent errors due to spectral fragmentation. The authors note that these findings are derived from a finite-disk model and that the discrete spectrum leads to finite-size recurrences, which would evolve into a smooth resonance in the thermodynamic limit.

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