🔬 mesoscale physics

Fractal Topology of Majorana Bound States in Superconducting Quasicrystals

This paper reveals that topological phase transitions in superconducting quasicrystals exhibit a fractal structure known as "Majorana's Butterfly," where the stability of Majorana Bound States is governed by a hierarchical competition between quasicrystalline order and superconducting pairing.

Original authors: William Caiger, Felix Flicker, Miguel-Ángel Sánchez-Martínez

Published 2026-02-04

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CC BY 4.0

Original authors: William Caiger, Felix Flicker, Miguel-Ángel Sánchez-Martínez

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

Imagine you are trying to build a very special kind of bridge made of quantum materials. This bridge is designed to hold a very fragile, magical object called a Majorana Bound State (MBS). In a perfect, orderly world (a regular crystal), this bridge is stable, and the magical object sits safely at the ends.

However, this paper asks: What happens if we build the bridge on a "quasicrystal"?

A quasicrystal is like a pattern that repeats, but never quite the same way twice. It's like a musical rhythm that follows a complex, non-repeating rule (think of the Fibonacci sequence: 1, 1, 2, 3, 5, 8...). The authors discovered that this irregularity doesn't just make the bridge wobbly; it turns the entire map of where the bridge is stable into a fractal.

Here is the breakdown of their discovery using simple analogies:

1. The Two Competing Forces

The stability of this quantum bridge depends on a tug-of-war between two forces:

The Quasicrystal Force (QC): This is the irregular, fractal pattern of the bridge itself. It tries to break the bridge into tiny, disconnected pieces.
The Superconducting Force (SC): This is the "glue" holding the bridge together, trying to keep it as one solid, stable unit.

2. The "Butterfly" Discovery

In the world of physics, there is a famous fractal shape called Hofstadter's Butterfly. It looks like a butterfly with wings made of infinitely many smaller wings, representing energy gaps in a magnetic field.

The authors found something similar, but for their superconducting bridges. They call it Kitaev's Butterfly.

The Difference: In the original butterfly, the center is empty. In this new "Kitaev's Butterfly," the center is filled with a special "Superconducting Gap." This is the safe zone where our magical Majorana objects live.

3. The Rule of Survival (The "Big Gap" Rule)

The most important finding is a simple rule for when the magical object survives: Size matters.

The quasicrystal pattern creates many "gaps" (weak spots) of different sizes.
If a weak spot (a quasicrystal gap) is bigger than the strength of the glue (the superconducting gap), it wins. It breaks the bridge, and the magical object disappears.
If a weak spot is smaller than the glue, the glue wins. The bridge stays intact, but the magical object gets slightly "jiggled" or hybridized. It doesn't break, but it's not perfectly still.

This creates a hierarchy of stability. The biggest weak spots break the bridge first. As you tune the materials, smaller and smaller weak spots start to win, breaking the bridge in more and more places.

4. Majorana's Butterfly

When the authors mapped out exactly where the magical objects survive across all these different patterns, they got a new shape they call Majorana's Butterfly.

This shape is a "subset" of the bigger Kitaev's Butterfly.
It looks like a fractal map where the "safe zones" (where the Majorana objects exist) are chopped up into a complex, self-similar pattern.
The more you tune the competition between the irregular pattern and the glue, the more detailed and "fractal" this map becomes.

5. Why This Matters (According to the Paper)

The paper suggests that this fractal pattern is a unique "fingerprint."

If you see a zero-energy signal (a sign of a Majorana object) that follows this specific fractal pattern, you know it's the real deal.
If the signal doesn't follow this pattern, it might just be a "fake" zero-energy state caused by the irregularity of the material.

In summary: The paper shows that when you mix superconductors with quasicrystals, the stability of the quantum states doesn't just break randomly. It breaks in a beautiful, mathematical, fractal pattern (a butterfly shape), governed by a simple rule: the irregular pattern only wins if its "weak spots" are stronger than the glue holding the system together.

Technical Summary: Fractal Topology of Majorana Bound States in Superconducting Quasicrystals

Problem Statement
Majorana Bound States (MBS) localized at the ends of one-dimensional (1D) topological superconductors are a primary candidate for fault-tolerant quantum computing. In periodic systems, such as the standard Kitaev chain, the topological phase hosting MBS is characterized by a single connected interval of chemical potential, bounded by a topological phase transition where the bulk energy gap closes. However, introducing quasiperiodicity (quasicrystalline order) into the bulk spectrum generically fragments the energy spectrum into a Cantor-set-like hierarchy of gaps. A fundamental open question remains: how does this spectral fractality impact the stability of the topological phase and the transitions characterizing the emergence of MBS? While previous works have noted the self-similarity of the order parameter, a comprehensive understanding of the fractality of the topological phases themselves and the mechanisms governing their stability has been lacking.

Methodology
The authors investigate Quasicrystal Kitaev Chains (QKCs), 1D models where the hopping potential $t_j$ varies according to a quasiperiodic sequence generated by the projection of an irrational slope in 2D Euclidean space. The study utilizes the following approach:

Model Construction: The system is defined by the Kitaev Hamiltonian with fixed on-site potential $\mu$ and superconducting pairing $\Delta$ , while hopping terms $t_j$ alternate between two values ( $t_0, t_1$ ) based on Sturmian words generated by an irrational slope $\gamma$ and a phase offset $\phi$ .
Topological Indicator: Due to the lack of translational symmetry, reciprocal space methods are insufficient. The authors employ Majorana Polarisation (MP), a real-space quantity defined by the particle-hole equivalence of eigenstates in the left and right halves of the chain. MP effectively distinguishes true MBS from trivial zero-energy states.
Competition Analysis: The study analyzes the competition between the energy scales of quasicrystallinity (QC) and superconductivity (SC). Specifically, it compares the size of quasicrystalline energy gaps ( $\Delta_{EQC}$ ) against the local superconducting gap ( $\Delta_{ESC}$ ) projected onto the chemical potential axis.
Generalization: The analysis extends from a specific Fibonacci case ( $\gamma = \phi^{-1}$ ) to the full family of Sturmian words (all irrational $\gamma \in (0, 1)$ ), mapping the resulting spectral and topological structures.

Key Contributions and Results

Fractal Topological Phase Transitions:
The authors demonstrate that the sequence of transitions between topological and trivial phases in QKCs is itself fractal. The bulk energy spectrum's fractal gap structure is projected onto the chemical potential dependence, creating a hierarchy of gaps in the topological phase diagram.
The QC-SC Competition Criterion:
A central finding is a simple energetic criterion determining the survival of the topological phase: a quasicrystalline gap breaks the MBS phase only if its size exceeds the local superconducting gap ( $\Delta_{EQC} > \Delta_{ESC}$ ).
- Surviving Gaps: When $\Delta_{EQC} > \Delta_{ESC}$ , the QC gap "wins," projecting a gap down to zero energy and interrupting the topological phase.
- Non-Surviving Gaps: When $\Delta_{EQC} < \Delta_{ESC}$ , the topological phase is not destroyed. Instead, these smaller QC gaps induce a structured hierarchy of finite hybridizations of the MBS. The magnitude of this hybridization is captured by the Majorana Polarisation, revealing a "height hierarchy" of domes in the MP data.
Kitaev's Butterfly (KB):
By generalizing to all Sturmian words, the authors identify a spectral fractal analogous to Hofstadter's butterfly, termed Kitaev's Butterfly.
- Similarities: KB shares the antisymmetric labeling structure ( $q$ -labels) and behavior around rational values of $\gamma$ (Farey sequence) with Hofstadter's butterfly.
- Distinctions: Unlike Hofstadter's butterfly, KB features a central superconducting gap at zero energy ( $E=0$ ) with a trivial $Z$ winding number ( $q=0$ ) but a non-trivial $Z_2$ topology hosting MBS. This central gap is absent in the standard Hofstadter model.
Majorana's Butterfly (MB):
Projecting the KB structure onto the chemical potential axis yields Majorana's Butterfly, a fractal topological phase diagram.
- MB is a tunable fractal subset of KB, controlled by the ratio of the QC-SC competition ( $\rho/\Delta'$ ).
- Only the subset of gaps in KB satisfying the competition criterion ( $\Delta_{EQC} > \Delta_{ESC}$ ) appears in MB.
- As the ratio $\rho/\Delta'$ increases, finer fractal structures from KB are realized in MB, following a predictable order dictated by the MBS hybridization hierarchy.

Significance and Claims
The paper claims to establish that quasicrystalline order renders the topological phase structure fractal in 1D superconductors in a non-trivial manner. The fractality is not a direct inheritance of the bulk spectrum but is governed by the direct competition between quasicrystallinity and superconductivity.

The authors assert that their results provide a clear framework for understanding the stability of MBS in quasiperiodic systems. Specifically:

The QC-SC competition criterion allows for the prediction of which spectral gaps will interrupt the topological phase.
The hierarchy of finite hybridizations explains the behavior of MBS in regimes where the topological phase is not fully broken, offering a method to fix the tolerance parameters ( $\epsilon$ ) needed for classifying phases in finite systems.
The identification of "Majorana's Butterfly" provides a distinct topological fingerprint. This structure can be used to distinguish true Majorana Bound States from other trivial zero-energy modes (such as mid-gap states arising from phason variations) by correlating the zero-energy response with the underlying fractal structure of the quasicrystalline spectrum.
Experimentally, the authors suggest that these fractal topological transitions should be accessible via electrostatic gating, which tunes the effective chemical potential to map the sequence of transitions.