The Josephson effect in Fibonacci superconductors

This paper theoretically demonstrates that a quasiperiodic modulation in Fibonacci superconductors induces topological edge modes, termed Fibonacci-Andreev bound states, which can be controlled via the phason angle to dominate the Josephson current and offer new avenues for exploring exotic superconducting phenomena in quasicrystals.

The Gist

Imagine you have two superhighways for electricity, where electrons travel without any resistance. Usually, when you connect two of these highways with a small bridge, the traffic (electric current) flows smoothly based on a simple rule: the more "in sync" the two highways are, the more traffic crosses. This is the famous Josephson effect, a phenomenon that powers much of our modern quantum technology.

For decades, scientists believed that in very short bridges, this traffic was carried by a specific, predictable pair of "cars" (called Andreev bound states) that lived right inside the energy gap of the superconductor. It was a standard, well-understood rulebook.

The New Discovery: The Fibonacci Highway

This paper introduces a twist. The researchers built a bridge using a very special, unusual material called a Fibonacci quasicrystal.

To understand this, imagine a standard highway where the lanes are arranged in a perfect, repeating pattern (like A-B-A-B-A-B). Now, imagine a highway where the lanes follow the Fibonacci sequence (A, B, AB, ABA, ABAAB...). This pattern never repeats exactly; it's ordered but never periodic. It's like a musical rhythm that follows a complex, mathematical rule rather than a simple 4/4 beat.

When the scientists put superconductivity onto this weird, non-repeating highway, something surprising happened:

1. New Traffic Jams (Gaps): The strange pattern created "energy gaps" where no cars could normally drive. Think of these as invisible walls or speed bumps that appear at specific, higher energies.
2. New Types of Cars (FABSs): Inside these higher-energy gaps, new types of "cars" appeared. The authors call them Fibonacci-Andreev bound states (FABSs). These are like exotic vehicles that only exist because of the unique, non-repeating rhythm of the road.
3. The Magic Knob (Phason Angle): The researchers found a "knob" they could turn, called the phason angle. In our analogy, imagine this as a way to slightly shift the entire pattern of the highway lanes without changing the number of lanes. By turning this knob, they could move these exotic FABS cars around.

The Big Surprise: The Underdogs Take Over

In the old, standard model, the traffic on the bridge was always driven by the cars living in the main, low-energy gap. The new cars (FABSs) were just background noise.

However, the paper shows that by adjusting the "knob" (the phason angle) and the alignment of the two highways, the researchers could make the exotic FABS cars become the main drivers.

• The Shift: They found a setting where the old, standard cars stopped moving entirely (they became "dispersionless," meaning they didn't care about the phase difference anymore).
• The Takeover: At that exact moment, the exotic FABS cars, which live at higher energies, started carrying almost all the current. They became the dominant force, dictating how much electricity flows across the bridge.

Why This Matters (According to the Paper)

The paper claims this is a fundamental shift in how we understand short bridges between superconductors. It proves that in these special quasicrystal materials, the "standard rulebook" is incomplete. The traffic isn't just about the cars in the main gap; it can be entirely controlled by these new, topological cars living in the higher-energy gaps.

The researchers also showed that these FABS cars are very picky about where they sit. Depending on how you turn the "knob" (the phason angle), they can hide at the very edge of the bridge or cluster right in the middle of the junction. This gives scientists a new way to control supercurrents not just by changing voltage or magnetic fields, but by tweaking the geometric "rhythm" of the material itself.

In Summary
Think of it like a band playing music. For years, we thought the melody was always played by the lead singer (the standard Andreev states). This paper shows that if you arrange the instruments in a Fibonacci pattern and tune the "phase" just right, the backup singers (the FABSs) can suddenly take over the lead, singing the entire song while the lead singer goes silent. It's a new way to conduct electricity using the hidden geometry of quasicrystals.

Technical Summary

Technical Summary: The Josephson Effect in Fibonacci Superconductors

Problem Statement
The standard paradigm for the Josephson effect in short junctions posits that the supercurrent is carried exclusively by Andreev bound states (ABSs) residing within the superconducting energy gap ($\Delta$). These states emerge from the transfer of Cooper pairs across a normal region and are determined by the junction's transmission properties. However, this framework assumes a conventional normal state. The authors investigate whether this paradigm holds for Josephson junctions (JJs) formed by Fibonacci superconductors—one-dimensional quasicrystals with proximity-induced superconductivity. Specifically, they address how the quasiperiodic modulation of the chemical potential, which creates a complex energy spectrum with topological gaps and edge modes, alters the nature of the supercurrent carriers and the resulting current-phase relation.

Methodology
The study employs a theoretical tight-binding model for a one-dimensional JJ composed of two Fibonacci superconductors coupled via a tunnel barrier.

• Hamiltonian: The system is described by $H = H_L + H_R + H_T$, where $H_{L,R}$ represents the left and right superconducting leads with onsite quasiperiodic energies following a Fibonacci sequence, and $H_T$ models the coupling between them.
• Quasiperiodicity: The onsite energies $\varepsilon_i$ alternate between two values ($\varepsilon_a, \varepsilon_b$) determined by a generating function involving the golden ratio and a "phason angle" ($\theta$). The phason angle controls the specific arrangement of the sequence and accounts for dislocations in the quasiperiodic potential.
• Superconductivity: The leads feature proximity-induced spin-singlet $s$-wave pairing with amplitude $\Delta$ and a phase difference $\phi = \phi_R - \phi_L$.
• Analysis: The authors calculate the energy spectrum $E_n(\phi)$ as a function of the phase difference and the phason angles ($\theta_L, \theta_R$). They compute the zero-temperature supercurrent $I(\phi)$ using the derivative of the energy levels with respect to the phase. To distinguish contributions, they separate the current into components from subgap ABSs ($I_{ABS}$) and states above the gap ($I_{FABS}$).
• Validation: The study utilizes Fibonacci approximants (finite chains $S_n$) to simulate the quasicrystal, ensuring results correspond to stable topological gaps. Additionally, the authors perform self-consistent calculations of the superconducting order parameter to verify that the mean-field approximation holds and that the results are robust against spatial variations in $\Delta(x)$.

Key Contributions and Results

1. Emergence of Fibonacci-Andreev Bound States (FABSs): The primary finding is the existence of bound states located above the superconducting gap ($\Delta$). These states, termed Fibonacci-Andreev bound states (FABSs), arise from the topological edge modes of the normal-state Fibonacci quasicrystal. The quasiperiodic potential breaks the energy continuum into "Fibonacci gaps," and superconducting correlations extend into these gaps, creating FABSs.
2. Phase and Phason Dependence: Unlike conventional ABSs, which are primarily determined by the junction transmission, FABSs exhibit a strong dependence on both the superconducting phase difference ($\phi$) and the phason angle ($\theta$). The localization of these states can be tuned by $\theta$; they can be localized at the outer edges of the superconductors or, crucially, at the junction interface.
3. Dominance of FABSs in Short Junctions: The authors demonstrate regimes where the conventional subgap ABSs become dispersionless (independent of $\phi$) and contribute negligibly to the supercurrent. In these regimes, the FABSs localized at the junction interface acquire the strongest phase dependence and dominate the total supercurrent. This occurs even in short junctions, challenging the standard view that short JJs are governed solely by subgap states.
4. Current-Phase Characteristics: When FABSs dominate, the current-phase relation ($I$-$\phi$) becomes more sinusoidal, resembling that of a tunnel junction, whereas ABS-dominated currents often exhibit non-sinusoidal behavior characteristic of highly transmitting junctions.
5. Control via Phason Angle: The contribution of FABSs versus ABSs can be actively controlled by adjusting the phason angle, which effectively tunes the transparency of the junction and the alignment of the quasiperiodic potentials at the interface.

Significance
The paper claims to reveal a new mechanism for the Josephson effect in quasicrystalline systems. It establishes that the topological properties of the parent quasicrystal (specifically the Fibonacci gaps and edge modes) can induce superconducting correlations at energies above the gap, leading to a supercurrent dominated by these high-energy states. This interplay between the Josephson effect and nontrivial edge modes in quasiperiodic systems offers a new avenue for exploring exotic superconducting phenomena. The work suggests that the "phason degree of freedom" serves as a control parameter to modulate the effective transparency of the junction and the relative weight of subgap versus over-gap Andreev states, potentially enabling the engineering of supercurrents in hybrid superconductor-quasicrystal structures.