🔬 mesoscale physics

Interplay of dimerization and quasiperiodicity in the superconducting proximity effect of a one-dimensional hybrid ring

This study utilizes the self-consistent Bogoliubov-de Gennes formalism to demonstrate that in a one-dimensional hybrid ring, quasiperiodicity (via the AAH model) suppresses the magnitude of proximity-induced superconductivity and enhances spatial fluctuations, while hopping dimerization (via the SSH model) governs the oscillatory behavior and penetration depth of the induced pairing, with their combined interplay offering a versatile mechanism for controlling superconducting correlations in quasiperiodic systems.

Original authors: Sourav Karmakar, Santanu K. Maiti

Published 2026-09-16

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CC BY 4.0

Original authors: Sourav Karmakar, Santanu K. Maiti

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

Superconductivity is a state of matter where electricity flows without any resistance, a phenomenon that usually requires extremely cold temperatures. In a standard superconductor, electrons pair up and move through the material in perfect unison. However, scientists have long been fascinated by what happens when a superconductor touches a normal, non-superconducting material. This contact creates a "proximity effect," where the superconducting properties leak across the boundary and induce a similar pairing in the neighboring material. The depth and strength of this leak depend entirely on the internal structure of the normal material. If the material is a perfect crystal, the superconducting influence travels far. If the material is disordered or random, the influence stops quickly. But there is a middle ground between perfect order and total chaos: quasicrystals. These are materials that have a repeating pattern but lack the simple, shifting symmetry of a standard crystal. They possess a unique, complex order that sits somewhere between the predictable and the random. Understanding how superconductivity behaves in these strange, in-between structures is a key question for modern physics, as it could reveal new ways to control electronic properties in future devices.

In a recent study, researchers investigated this question by simulating a one-dimensional ring made of two parts: a superconducting segment and a normal segment. They wanted to see how the superconducting influence would travel through the normal part if that part was built according to three different rules. The first rule involved a quasiperiodic pattern, where the energy levels of the atoms vary in a complex, non-repeating way. The second rule involved a dimerized pattern, where the connections between atoms alternate between strong and weak links, a structure known to create topological phases. The third rule combined both of these patterns. Using a powerful computational method that accounts for how electrons interact with each other, the team mapped out exactly how the superconducting pairing changed as they adjusted the strength of these patterns.

The researchers first looked at the quasiperiodic pattern alone. They found that as the complexity of the pattern increased, the superconducting influence became more erratic. Instead of a smooth, steady flow of induced pairing, the strength of the effect began to fluctuate wildly from one atom to the next. More importantly, the overall strength of the superconducting signal dropped significantly. Once the pattern became strong enough to trap the electrons in specific locations, the superconducting influence essentially vanished from the center of the normal region. This confirmed that the ability of the material to support the leak of superconductivity is directly tied to whether its electrons are free to move or are stuck in place.

Next, the team examined the alternating strong-and-weak connection pattern. Here, the behavior was different. When the connections were nearly equal, the superconducting influence spread smoothly through the material. However, as the difference between the strong and weak connections grew, the smooth flow turned into a series of ripples. These ripples, or oscillations, were a direct signature of the alternating structure. If the difference between the strong and weak links was small, these ripples traveled deep into the material. But if the difference was large, the ripples were confined to the very edge where the superconductor touched the normal material, dying out almost immediately. This showed that the structural rhythm of the material acts as a gatekeeper, determining how far the superconducting influence can penetrate.

Finally, the researchers combined both the complex energy pattern and the alternating connections to see how they would interact. The results revealed a clear division of labor between the two effects. The alternating connections continued to dictate the shape of the ripples and how far they could travel, while the complex energy pattern acted as a dampener, making the ripples more irregular and further reducing their overall strength. In the most extreme cases, where the connections were very uneven and the energy pattern was very complex, the superconducting influence was almost entirely suppressed, leaving the center of the ring with almost no induced pairing at all.

These findings suggest that scientists can use two distinct knobs to control superconductivity in these hybrid systems. One knob, the alternating connections, controls the reach and the rhythmic pattern of the effect. The other knob, the complex energy pattern, controls the uniformity and the total magnitude of the effect. By tuning these two factors, it may be possible to engineer materials where superconducting properties are precisely localized or extended as needed. While these results come from computer simulations rather than physical experiments, they provide a clear roadmap for what to expect when such hybrid rings are built in the lab. Given recent advances in creating artificial quasicrystals and superconducting nanostructures, testing these predictions in real materials is now a realistic possibility, potentially opening the door to new types of electronic devices that rely on the unique interplay of order and complexity.

Technical Summary: Interplay of Dimerization and Quasiperiodicity in the Superconducting Proximity Effect of a One-Dimensional Hybrid Ring

Problem Statement
The superconducting proximity effect, where Cooper pairs penetrate a normal material from a superconductor, is a sensitive probe of the electronic states in the normal region. While the effect is well-understood in periodic crystals (extended states) and random disordered systems (localized states), its behavior in quasicrystalline systems remains an active area of research. Quasicrystals possess long-range order without translational symmetry, often hosting critical, multifractal electronic states that are neither fully extended nor exponentially localized. Furthermore, the interplay between topological band structures (specifically dimerized hopping) and quasiperiodicity in the context of proximity-induced superconductivity has not been systematically explored. This work addresses the spatial variation of the proximity-induced pairing amplitude in a one-dimensional hybrid ring where the normal region is modeled by three distinct lattice scenarios: the diagonal Aubry-André-Harper (AAH) model, the Su-Schrieffer-Heeger (SSH) chain, and a combined SSH-AAH model.

Methodology
The authors employ a self-consistent Bogoliubov-de Gennes (BdG) formalism at zero temperature to study a hybrid ring of total length $L$ , comprising a superconducting (SC) segment and a normal (N) segment of length $L_N$ .

Hamiltonian: The normal region is described by a tight-binding Hamiltonian incorporating nearest-neighbor hopping amplitudes ( $t_A, t_B$ ) and a quasiperiodic on-site potential ( $\lambda_{AAH} \cos(2\pi\beta i + \phi)$ ). The superconducting region features uniform hopping and an attractive on-site interaction ( $V_{SC}$ ) driving Cooper pairing. The two regions are coupled via interface hopping ( $t_{NS}$ ).
Models Investigated:
1. AAH Model: Uniform hopping ( $t_A = t_B$ ) with varying quasiperiodic potential strength $\lambda_{AAH}$ .
2. SSH Model: No on-site potential ( $\lambda_{AAH} = 0$ ) with alternating hopping ( $t_A \neq t_B$ ), allowing investigation of the topological phase transition at $t_A = t_B$ .
3. SSH-AAH Model: A combination of both dimerized hopping and quasiperiodic potential.
Numerical Procedure: The BdG equations are solved iteratively. The superconducting order parameter (OP) $\Delta_i = \langle c_{i\downarrow} c_{i\uparrow} \rangle$ and the Hartree shift are updated until convergence is achieved (relative change $< 10^{-3}$ ). The study analyzes the spatial profile of $\Delta_i$ across the ring and its dependence on system parameters, including the quasiperiodic phase $\phi$ to assess sample-to-sample fluctuations.

Key Results

AAH Normal Region (Quasiperiodicity):
- Spatial Fluctuations: Increasing the quasiperiodic potential strength $\lambda_{AAH}$ enhances spatial fluctuations in the induced order parameter within the normal region.
- Magnitude Suppression: As $\lambda_{AAH}$ increases, the overall magnitude of the induced pairing decreases.
- Localization Transition: Beyond the critical strength ( $\lambda_{AAH} > 2t_N$ ), where the AAH model undergoes a localization transition, the proximity-induced pairing is strongly diminished. This suppression is attributed to the exponential localization of electronic states, which reduces their spatial overlap and ability to support pairing correlations, rather than a depletion of states near the Fermi energy. In fact, the number of low-energy states increases near the critical point, yet the proximity effect vanishes due to localization.
- Phase Sensitivity: For finite systems, varying the AAH phase $\phi$ induces sample-to-sample fluctuations in the order parameter, which grow in amplitude as $\lambda_{AAH}$ increases.
SSH Normal Region (Dimerization):
- Oscillatory Modulations: Weak dimerization ( $t_A/t_B$ close to 1) introduces oscillatory modulations in the induced pairing that extend deep into the normal region, distinct from the smooth decay seen in uniform chains.
- Confinement by Strong Dimerization: Strong dimerization ( $t_A/t_B$ far from 1) confines these oscillations to the vicinity of the normal-superconductor interfaces. The penetration depth of superconducting correlations is significantly reduced, leaving negligible pairing in the center of the chain.
- Topological Signature: The induced order parameter peaks sharply at the topological phase transition point ( $t_A = t_B$ ), where the bulk energy gap closes. As the gap opens (moving away from unity), the number of available low-energy states to mediate pairing drops, suppressing the proximity effect.
SSH-AAH Normal Region (Interplay):
- Distinct Roles: The two mechanisms play complementary but distinct roles. The SSH dimerization primarily determines the oscillatory behavior and the penetration depth of the induced pairing. The AAH potential primarily enhances spatial inhomogeneity (fluctuations) and further suppresses the overall magnitude of the pairing.
- Combined Effect: In weakly dimerized chains, the oscillatory profile survives deep into the bulk but becomes increasingly irregular and suppressed as $\lambda_{AAH}$ increases. In strongly dimerized chains, where pairing is already confined to the interfaces, the AAH potential acts to further suppress the residual correlations without significantly altering the confinement length scale set by the dimerization.

Significance and Claims
The paper claims to provide a systematic investigation of how quasiperiodicity and hopping dimerization influence proximity-induced superconductivity in a 1D hybrid ring. The authors assert that their results offer a "versatile means of controlling proximity-induced superconductivity" in quasiperiodic hybrid systems. Specifically, they demonstrate that:

The proximity effect serves as a sensitive real-space indicator of the AAH localization transition and the SSH topological phase transition.
Dimerization and quasiperiodicity act as complementary control knobs: dimerization governs the penetration depth and oscillatory nature, while quasiperiodicity controls spatial inhomogeneity and magnitude.
The findings suggest that engineering these parameters could allow for the tuning of superconducting correlations in low-dimensional quasiperiodic structures.

The authors note that experimental realizations of such systems are becoming feasible through advances in quasicrystal growth, synthetic electronic quasicrystals, and moiré quasicrystals, providing a potential platform for testing these theoretical predictions. Future work is suggested to explore finite temperature effects, longer-range hopping, and the impact of magnetic flux on charge and spin currents.

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