Fibonacci-Andreev Bound States

• Fibonacci-Andreev bound states are discrete, topologically hybridized levels emerging in Josephson junctions formed by Fibonacci quasicrystals with quasiperiodic chemical potentials.
• The contribution of these states to the Josephson current is controlled by the phason angle, which tunes the spatial placement of edge modes at the junction interface.
• In the short-junction limit, Fibonacci-Andreev bound states can dominate the supercurrent, yielding a nearly sinusoidal current-phase relation distinct from conventional subgap Andreev states.

Searching arXiv for the specified paper and closely related work on Fibonacci superconductors and Josephson effects. Fibonacci-Andreev bound states are discrete Andreev-like levels that arise in a short Josephson junction formed by two proximitized Fibonacci quasicrystals, where a quasiperiodic modulation of the chemical potential on a superconducting substrate induces topological gaps and edge modes with energies above the superconducting gap. When superconducting pairing is present, these normal-state edge modes develop superconducting correlations and contribute directly to the Josephson current; this contribution is controlled by the Fibonacci sequence arrangement, or phason angle, and can dominate over the conventional subgap Andreev bound states in short junctions (Sardinero et al., 25 Jul 2025).

1. Microscopic formulation

The theoretical setting is a one-dimensional chain of lattice sites $i=1,\dots,N$ with nearest-neighbor hopping $t$, chemical potential $\mu$, an on-site quasiperiodic potential $\varepsilon_i$, and proximity-induced $s$-wave pairing $\Delta e^{i\phi_\alpha}$ in each superconducting lead. The total Hamiltonian is written as

$H = H_L + H_R + H_T,$

with

$$H_\alpha = \sum_{i=1}^{N_\alpha}\Bigl[(\varepsilon_i-\mu)\,c^\dagger_{i\sigma}c_{i\sigma} -t\,(c^\dagger_{i+1,\sigma}c_{i\sigma}+\mathrm{H.c.}) +\Delta\,e^{i\phi_\alpha}\,c^\dagger_{i\uparrow}c^\dagger_{i\downarrow}+\mathrm{H.c.}\Bigr],$$

and

$$H_T = -\,t_0\;\bigl(c^\dagger_{N_L+1,\sigma}c_{N_L,\sigma}+\mathrm{H.c.}\bigr),$$

where $t_0$ is the tunnel coupling across the junction and $t$0 is the superconducting phase difference (Sardinero et al., 25 Jul 2025).

In Nambu space, with $t$1, the Bogoliubov-de Gennes Hamiltonian is

$t$2

This formulation makes explicit that the superconducting and quasiperiodic structures are not added phenomenologically after the fact; they are encoded directly in the microscopic BdG problem. A plausible implication is that the resulting bound states inherit both the phase sensitivity of Andreev physics and the gap hierarchy of the Fibonacci quasicrystal.

2. Quasiperiodic modulation and phason control

The quasiperiodic structure is implemented through the “diagonal” Fibonacci chain, in which each on-site energy $t$3 takes one of two values, $t$4 or $t$5, according to

$t$6

and

$t$7

The control parameter $t$8 is the phason angle. Changing $t$9 rigidly shifts the quasiperiodic pattern along the chain and is the analog of sliding the “cut” in higher-dimensional projections of a quasicrystal (Sardinero et al., 25 Jul 2025).

This phason degree of freedom is central because it tunes the spatial placement of edge modes relative to the Josephson interface. The paper states that, by tuning the phason angles $\mu$0, one can pin particular edge modes right at the junction interface, thereby maximizing the corresponding effective transmission while suppressing the conventional subgap channel. This is the specific mechanism by which a geometric degree of freedom of the quasicrystal enters the current-phase relation.

3. Emergence of Fibonacci-Andreev bound states

In the normal, non-superconducting Fibonacci chain there exists a hierarchy of gaps, referred to as Fibonacci gaps, in the spectrum. Each finite, topologically stable gap carries a nonzero Chern number when $\mu$1 is viewed as a synthetic dimension, and therefore hosts localized edge modes at energies $\mu$2 above or below the band. When superconducting pairing is switched on, each normal-state edge mode $\mu$3 hybridizes electron-like and hole-like components and is pulled into the full BdG spectrum as a discrete subgap-like state inside the broadened Fibonacci gap. These hybridized states are the Fibonacci-Andreev bound states, or FABS (Sardinero et al., 25 Jul 2025).

In the short-junction limit, each localized pair of edge modes from the left and right sides can be treated analogously to a single-channel SNS junction with an effective normal-state transmission $\mu$4. The resulting Andreev-level dispersion is

$\mu$5

with $\mu$6 determined by the overlap of the left and right Fibonacci edge wavefunctions at the junction, and

$\mu$7

For high-energy gaps, one often linearizes near small $\mu$8 and writes $\mu$9.

A common source of confusion is the relation between FABS and ordinary Andreev bound states. In the notation of the model, the familiar conventional subgap Andreev states correspond to $\varepsilon_i$0, associated with the smallest normal gap at $\varepsilon_i$1, whereas $\varepsilon_i$2 labels discrete FABS inside higher Fibonacci gaps at $\varepsilon_i$3. Thus FABS are neither purely normal edge modes nor merely a relabeling of conventional subgap ABS; they are Andreev-hybridized descendants of topological Fibonacci edge modes.

4. Bound-state dispersion and Josephson current

At zero temperature, the Josephson supercurrent is given by the sum over occupied Andreev levels,

$\varepsilon_i$4

The decomposition is

$\varepsilon_i$5

for the conventional subgap contribution, and

$\varepsilon_i$6

for the Fibonacci-gap channels (Sardinero et al., 25 Jul 2025).

In the short-junction regime, where the separation between the two superconductors is much smaller than the coherence length $\varepsilon_i$7, each channel obeys the universal Beenakker expression

$\varepsilon_i$8

and therefore

$\varepsilon_i$9

Here $s$0 or $s$1 for the conventional and Fibonacci gaps respectively, and $s$2 depends sensitively on $s$3.

Because the FABS disperse with $s$4 inside each Fibonacci gap, they carry a sizable current, often with a nearly sinusoidal $s$5 shape, whereas the conventional ABS often have non-sinusoidal current-phase relations in the high-$s$6 regime. This distinction in the CPR is one of the principal operational consequences of FABS in the Josephson problem.

5. Dominance in the short-junction limit

The defining regime emphasized in the paper is the short junction, where the FABS contribution can dominate the Josephson effect. The stated criterion is explicit: if $s$7 while $s$8 for some Fibonacci channel, then $s$9 (Sardinero et al., 25 Jul 2025).

Numerically, this regime is reached by tuning the phason angles so that particular edge modes are pinned at the junction interface, which maximizes $\Delta e^{i\phi_\alpha}$0 and suppresses $\Delta e^{i\phi_\alpha}$1. In that limit the Josephson current is carried almost entirely by FABS and is nearly sinusoidal, with a reduced amplitude $\Delta e^{i\phi_\alpha}$2.

This directly qualifies a common intuition imported from conventional SNS junctions. In ordinary settings, the dominant supercurrent channel is expected to be the lowest-energy subgap ABS. Here, by contrast, the hierarchy of quasiperiodic topological gaps provides higher-$\Delta e^{i\phi_\alpha}$3 channels whose phase-dispersing Andreev character can outweigh the conventional contribution. This suggests that the spectral location of a bound state relative to the superconducting gap is not, by itself, a reliable indicator of its importance for the Josephson response in quasiperiodic superconductors.

6. Topological interpretation and experimental signatures

The topological origin of FABS is traced to the Fibonacci gaps of the diagonal Fibonacci model. Each stable gap is characterized by an integer Chern number $\Delta e^{i\phi_\alpha}$4 when $\Delta e^{i\phi_\alpha}$5 is regarded as a synthetic momentum. As $\Delta e^{i\phi_\alpha}$6 is adiabatically cycled from $\Delta e^{i\phi_\alpha}$7 to $\Delta e^{i\phi_\alpha}$8, each gap pumps $\Delta e^{i\phi_\alpha}$9 electrons from one end of the chain to the other, so the bulk gap cannot close without changing $H = H_L + H_R + H_T,$0. By bulk-boundary correspondence, each gap hosts $H = H_L + H_R + H_T,$1 chiral edge modes whose energies disperse in $H = H_L + H_R + H_T,$2; these are the normal-state edge modes that become FABS when pairing is included (Sardinero et al., 25 Jul 2025).

The paper identifies several measurable signatures. First, the current-phase relation differs qualitatively between conventional-ABS-dominated and FABS-dominated regimes. A junction dominated by conventional ABS at high $H = H_L + H_R + H_T,$3 exhibits a forward-skewed, non-sinusoidal CPR, whereas a FABS-dominated regime yields an almost perfect $H = H_L + H_R + H_T,$4 CPR while retaining a critical current that depends strongly on the phason angle. Second, by gating the quasicrystal region one effectively changes $H = H_L + H_R + H_T,$5, and the prediction is that $H = H_L + H_R + H_T,$6 shows sharp peaks whenever a new FABS channel becomes resonant at the junction interface; notably, $H = H_L + H_R + H_T,$7 can remain large even when the subgap channel is suppressed. Third, microwave spectroscopy, photon absorption, or Shapiro-step spectroscopy can probe the FABS spectrum at energies $H = H_L + H_R + H_T,$8, where one would observe phase-dispersing Andreev-level transitions above the gap.

Taken together, these signatures establish the intended meaning of the term. Fibonacci-Andreev bound states are a hierarchy of over-gap Andreev states rooted in the topological edge structure of a Fibonacci quasicrystal, with existence, phase dispersion, and transport weight controlled by phason degrees of freedom and the associated topological invariants. In short Josephson junctions, they provide a mechanism by which quasiperiodic topology can determine both the spectral content of the BdG problem and the dominant contribution to the supercurrent.