Fractal Topology of Majorana Bound States in Superconducting Quasicrystals

TL;DR

Quasicrystalline order fragments the bulk spectrum of a 1D superconductor and raises the question of how fractality affects the topological protection of Majorana Bound States (MBS). The authors construct Quasicrystal Kitaev Chains (QKCs) by modulating hopping with Sturmian words defined by an irrational slope $γ$ and phason $φ$, and analyze the system with Majorana Polarisation (MP) together with gap labeling $N(E) = p + γ q$, extending to all Sturmian words to reveal Kitaev's butterfly (KB) and Majorana's butterfly (MB). The findings show a fractal, topological phase diagram in MB governed by the QC–SC competition; a simple criterion $ΔE_{QC} > ΔE_{SC}$ selects QC gaps that survive projection to zero energy and explains a hierarchy of MBS stability and finite-hybridisation, while phason winding produces trivial mid-gap states that do not signal MBS. The results provide a fractal fingerprint for Majorana physics, offer experimentally accessible routes to map fractal transitions via gating, and help distinguish true MBS from trivial zero-energy modes in quasicrystals.

Abstract

Quasicrystalline order induces a fractal energy spectrum, yet its impact on topological protection remains an open fundamental question. Here, we demonstrate that the topological phase transitions characterised by the appearance of Majorana Bound States themselves have a fractal character. By extending this analysis to the full family of Sturmian words, we uncover Kitaev's Butterfly $-$ a spectral fractal analogous to Hofstadter's butterfly, but fundamentally distinguished by a central superconducting gap. Within this framework, we identify Majorana's Butterfly as a fractal topological phase diagram governed by the competition between quasicrystallinity and superconducting pairing. We show that this competition dictates a hierarchy of Majorana stability, where the survival of the topological phase against fractal fragmentation is determined by the relative strength of these competing energy scales.

Figures (6)

• Figure 1: (a) A line of irrational slope $\gamma\in(0,1)$ and real $y$-intercept $\phi\in[0,1)$, modulo the periodic 2D integer lattice, defines a flow over a two-torus (b). We project this into two intervals $(0;1-\gamma)\to t_0$ (blue) and $(1-\gamma;1)\to t_1$ (red) and use $t_0$, $t_1$ as hoppings in the Kitaev chain, shown in (c) for $L=11$. Each $\gamma$ defines a local isomorphism class of Sturmian words distinguished by $\phi$.
• Figure 2: Results for the Fibonacci QKC $\gamma = \varphi^{-1}$ with $L=200$, $\rho=1.5$, $\Delta=0.05$, over $0 \leq \mu' \leq 3$ including $\mu'_c\approx2.61$. (b) shows the full fractal energy spectrum. Along $\mu'=0$ the QC energy gaps satisfying the QC-SC condition from Eq. \ref{['eq:QC-SCCompCriterion']} are coloured by the topological winding number $q$ from Eq. \ref{['eq:energy_gap_labelling']}. (a) shows QC wins this criterion in the five shaded regions; these gaps survive projection down to zero energy. They correspond to regions of $\mu'$ where $\mathcal{M}>-1+\epsilon_c$ defines trivial MBS phase, shown by the shaded (yellow) regions between (b) & (c). (d) shades topological regions (purple) and trivial regions (yellow) defined by $\mathcal{M}$ as $\log\epsilon$ is varied; at $\epsilon=\epsilon_c$ the expected five MBS phase gaps are realised, below this, further structure in the MP contains information about the hierarchy of finite MBS hybridisation induced by weaker QC gaps.
• Figure 3: (a) Hofstadter's butterfly (HB) and (b) Kitaev's butterfly (KB) both normalised to their bandwidth $(W_\gamma)$, and coloured by $q$-label values, fixing $|q_\text{max}|=20$ for visual clarity. KB here is plotted for $L=500$, $\mu'=0$, $\rho=2.0$ and $\Delta'=0.05$ for clarity. Notably, HB and KB exhibit the same antisymmetric labelling structure and behaviour around rational values of $\gamma$, except for the superconducting gap (SC) along $E=0$ (gray) in KB which is not labelled by $q$. (c) Majorana's butterfly (MB) is composed of regions in the MBS topological phase using the Majorana polarisation $\mathcal{M}$ for the same parameters as KB and with $\mu'$ normalised to the same bandwidth as KB. The fractal structure of the MBS phase transition in MB is a subset of KB with finite-scale controlled by the QC-SC competition ratio $\rho/\Delta'$.
• Figure A.1: Energy spectrum (a) plotted for fixed $\gamma = \varphi^{-1}$ over a full phason period $0<\phi\leq1$ showing the how individual mid-gap states exhibit integer winding labelled by $q$, taking positive (red) or negative (blue) values. (b) shows the minimum positive energy $(|E_\text{min}|)$ as a function of $\mu'$ and $\phi$, revealing the winding behaviour of the trivial zero-energy states induced by their energy mid-gap-state counterparts shown in the top panel. The MBS topology is undisturbed by $\phi$ for $\epsilon_c=10^{-3}$(c).
• Figure B.1: (a) The relative height of the MP domes remains constant whilst varying $L$ for a fixed ratio between QC and SC strength; shown for $\rho=1.5$ and $\Delta'=0.05$ for $L=400$ (left), 600 (middle), and 800 (right). Meanwhile, (b) shows how $\rho$ and $\Delta'$ both change the relative height of domes. The $\log \epsilon$ scales are fixed in each column to illustrate the emergence of new domes and MBS phase gaps as $\rho'/\Delta'$ is increased.