Researchers are increasingly investigating the interplay between topological quantum matter and aperiodic order. William Caiger, Felix Flicker, and Miguel-Ángel Sánchez-Martínez, all from the School of Physics, Bristol, demonstrate a fractal character in the topological phase transitions associated with Majorana bound states within superconducting quasicrystals. This work is significant because it reveals a novel ‘Kitaev Butterfly’ , a spectral fractal analogous to the well-known Hofstadter butterfly, yet uniquely defined by a central superconducting gap. Their findings establish a fractal topological phase diagram governed by the competition between quasicrystalline order and superconducting pairing, ultimately dictating the stability of the topological phase against fragmentation.

These states are considered promising candidates for building fault-tolerant quantum computers, requiring robust topological protection against environmental disturbances.

This work demonstrates that the transitions between topological and trivial phases, characterised by the emergence of Majorana Bound States, exhibit a fractal nature. By analysing a family of models generated by Sturmian words, the study reveals a spectral fractal dubbed “Kitaev’s Butterfly”, analogous to the well-known Hofstadter’s butterfly but distinguished by a central superconducting gap.
The research establishes that the competition between quasicrystalline order and superconducting pairing dictates a hierarchy of stability for these Majorana Bound States. Specifically, the survival of the topological phase against fractal fragmentation is determined by the relative strength of the quasicrystalline and superconducting energy scales.

A key finding is that only quasicrystalline gaps exceeding a certain threshold, defined by the superconducting gap size, contribute to maintaining the topological phase. Smaller gaps do not eliminate the topological phase entirely, but instead induce a hierarchy of finite hybridisations of the Majorana Bound States.

This investigation generalises Quasicrystal Kitaev Chains to a broader family of models generated by Sturmian words, revealing a connection to the Z-invariant topology of Hofstadter’s butterfly. The resulting fractal topological phase diagram, termed “Majorana’s Butterfly”, is tunable and controlled by the balance between quasicrystalline and superconducting effects.

The model generates 1D quasicrystals by projecting an irrational slope in 2D Euclidean space, defining a flow over a two-torus and subsequently projecting it into a sequence of hopping terms for the Kitaev chain. The bandwidth of the periodic model is defined as 4t, with a critical on-site potential of 2t, establishing the parameters for the topological phase hosting Majorana Bound States.

Majorana polarisation calculation and identification of bound states in quasicrystalline Kitaev chains reveal novel topological properties

A Majorana Polarisation (MP) measurement forms the core of this study, enabling differentiation between Majorana Bound States (MBS) and trivial zero-energy modes. The research investigates the interplay between quasicrystallinity and superconductivity in a single quasicrystalline Kitaev chain (QKC), utilising a 200-site chain with a density of ρ = 1.5 and a superconducting gap of ∆ = 0.05.

The MP is calculated as M = PL · P ∗ R, where PL and PR represent projections onto the left and right halves of the chain, and P ∗ denotes the complex conjugate. This quantity assesses the particle-antiparticle equivalence of Majorana states via the particle-hole operator C, constructed as C = eiζτx K, with τx being the Pauli-x operator and K the complex conjugate operator.

Energy eigenstates are ordered by their eigenvalues, and the local action of C is then calculated for each state to compare candidate MBS eigenstates. A value of M = −1 signifies non-overlapping MBS localised at the chain’s ends, with zero energy. Due to the continuous nature of M(μ), a tolerance of ε is introduced, requiring M ∆ESC, where ∆EQC represents the size of quasicrystalline energy gaps and ∆ESC denotes the superconducting gap.

Regions satisfying this criterion indicate a breakdown of the MBS phase, as the quasicrystalline gap strength overpowers the superconducting gap. This competition dictates a hierarchy of stability, with the survival of the topological phase against fractal fragmentation determined by the relative strength of these competing energy scales, ultimately defining a finite-scale fractal topological phase diagram controlled by ρ/∆′.

Topological phase stability correlates with competing quasicrystalline and superconducting gap sizes in these materials

Kitaev’s butterfly, a spectral fractal analogous to Hofstadter’s butterfly, emerges from the analysis of topological phase transitions in quasicrystalline systems. This work demonstrates that these transitions, characterised by the appearance of bound states, exhibit a fractal character intrinsically linked to the competition between quasicrystallinity and superconductivity.

The research establishes a hierarchy of stability where the survival of the topological phase is determined by the relative strength of these competing energy scales. Analysis reveals a direct competition between quasicrystalline and superconducting effects, captured by a figure of merit representing the difference between the quasicrystalline gap size (∆EQC) and the superconducting gap size (∆ESC).

Results indicate that only quasicrystalline gaps satisfying ∆EQC ∆ESC survive projection to zero energy and sustain the topological phase hosting Majorana bound states. Smaller quasicrystalline gaps do not eliminate the topological phase but instead induce a hierarchy of finite hybridisations of the Majorana bound states.

Generalisation to a family of two-hopping models generated by Sturmian words reveals a similarity to the fractal Z-invariant topology of Hofstadter’s butterfly, distinguished by a topologically nontrivial superconducting gap in the central region. Characterisation of the collection of bulk gap spectra defines Kitaev’s butterfly, with the resulting fractal topological phase diagram containing Majorana bound states termed Majorana’s butterfly.

This fractal structure is tunable and controlled by the ratio of quasicrystalline to superconducting competition. The bandwidth of the periodic model is established as 4t, with a critical on-site potential of μc = 2t defining the phase transition for Majorana bound state hosting. In finite-length quasicrystals, the bandwidth expands to approximately 4tγ, where tγ = (1−γ)+γρ, leading to a modified critical chemical potential of |μ′ c| ≈2 tγ.

Persistent energy gaps, characteristic of quasicrystalline energy spectra, are identified and classified using the integrated density of states N(E) = p + γq, where integers (p, q) uniquely label each energy gap and q represents the mid-gap state winding number. The central superconducting gap receives a trivial Z label of q = 0, while simultaneously exhibiting a distinct Z2 topology indicative of Majorana bound state existence.

Fractal topology and Majorana phase stability in quasicrystalline superconductors offer novel routes to topological quantum computation

Quasicrystalline order induces a fractal energy spectrum and this work demonstrates its impact on topological protection within one-dimensional superconductors. Researchers have shown that topological phase transitions, specifically the appearance of bound states, exhibit a fractal character when induced by quasicrystalline order.

Analysis extending to a family of Sturmian words revealed a spectral fractal termed Kitaev’s Butterfly, analogous to the well-known Hofstadter’s Butterfly but uniquely distinguished by a central superconducting gap. This Butterfly diagram represents a fractal topological phase diagram governed by the interplay between quasicrystallinity and superconducting pairing.

The stability of the topological phase against fragmentation is determined by the relative strengths of these competing energy scales, establishing a clear hierarchy. A key finding is an energetic criterion predicting which quasicrystalline gaps interrupt the Majorana phase, dependent on whether the quasicrystalline gap exceeds the local superconducting gap.

The resulting hierarchy of finite Majorana hybridisations allows for precise classification of phases using Majorana polarisation in finite systems. The authors acknowledge a limitation in that the current analysis focuses on two-hopping Quasicrystal Kitaev Chains, and suggest future research could explore the behaviour in more complex systems.

Experimentally, these fractal topological transitions are expected to be observable through electrostatic gating, allowing for mapping of the sequence of transitions by tuning carrier density. This topological fractality provides a means to differentiate Majorana bound states from other zero-energy modes present in quasicrystals by correlating zero-energy response with the underlying quasicrystalline spectrum.