🔬 mesoscale physics

Phase-controlled quasi-bound states in the continuum and thermoelectric enhancement in Majorana-quantum-dot nanostructures

This study demonstrates that phase-controlled symmetry breaking of bound states in the continuum within a Majorana-quantum-dot nanostructure significantly enhances electronic thermoelectric performance, achieving a figure of merit of approximately 0.75 through superconducting-phase tuning and a pronounced violation of the Wiedemann-Franz law.

Original authors: Alejandro Garrido, David Zambrano, Hishan Farfán-Bachiloglu, Juan Pablo Ramos-Andrade, Vladimir Juričić, Pedro Orellana

Published 2026-07-20

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CC BY 4.0

Original authors: Alejandro Garrido, David Zambrano, Hishan Farfán-Bachiloglu, Juan Pablo Ramos-Andrade, Vladimir Juričić, Pedro Orellana

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

Imagine a world where electricity doesn't just flow like water in a pipe, but dances like a troupe of acrobats on a tightrope. This is the realm of quantum physics, specifically the study of "topological superconductors." Think of these materials as special highways for electrons where the particles behave in a very strange way: they can act as their own opposites, like a shadow that is also the person casting it. These mysterious particles are called "Majorana zero modes." Scientists are obsessed with them because they might be the key to building super-powerful, unbreakable computers that don't crash easily. But there's a catch: these particles are incredibly shy. They hide at the very ends of tiny wires and are notoriously hard to spot because they look a lot like other, less interesting things. To find them, researchers need to build tiny, intricate traps and see how electricity and heat move through them. If they can find a unique "signature" in how heat flows, they might finally prove these ghostly particles are real.

This paper is about building one of those tiny traps and discovering a clever way to make the hidden particles show off. The researchers designed a microscopic device shaped like a crossbar: a central dot connected to two normal wires and two special "topological" wires that host the Majorana particles. In this setup, the electrons can get stuck in a special state called a "Bound State in the Continuum" (BIC). Imagine a song that is so perfectly tuned to a room's acoustics that it never leaks out; the sound stays trapped inside forever. In physics, these are states that exist inside the flow of energy but don't actually let any energy pass through. They are "dark" states—silent and invisible to the flow of electricity.

The team used complex math (Green's functions) to simulate how this system behaves under different conditions. They tested three ways to break the perfect symmetry of the device: making the two special wires different lengths, shifting the energy level of the central dot, and changing the "phase" (a kind of timing or rhythm) of the superconducting wires. They found that simply making the wires different lengths or shifting the dot's energy only made a small, modest improvement in how well the device could convert heat into electricity. However, when they tuned the superconducting phase, something magical happened. This tuning didn't just make the dark states visible; it turned them into "quasi-bound states" that acted like a perfect filter.

The results were striking. By adjusting the phase, the researchers created a "quadratic transmission zero," which is a fancy way of saying they created a perfect, sharp dip where electricity is blocked, but right next to it, the flow changes very rapidly. This sharp change acts like a super-efficient sieve, sorting electrons by their energy. In these simulations, this method boosted the device's ability to turn heat into useful power (measured by a number called the thermoelectric figure of merit, $ZT_{el}$ ) by about sixty times compared to just making the wires different lengths. The best configuration reached a value of $ZT_{el} \simeq 0.75$ , and it broke a fundamental rule of physics known as the Wiedemann–Franz law, which usually says heat and electricity travel together in a fixed ratio. Here, the ratio was violated, reaching a value of $21/5$ (or 4.2), proving that the system was behaving in a unique, quantum-mechanical way.

The paper concludes that while the other methods worked a little bit, the "phase control" is the real star. It suggests that by simply twisting the rhythm of the superconducting wires, scientists can engineer these nanostructures to be incredibly efficient at harvesting energy from heat. This doesn't mean we have a working computer or a new power plant yet; it means that in these simulated environments, the recipe for finding and using these elusive Majorana particles has been found. The authors propose that this phase-tunable signature is a reliable way to identify these particles and control how energy moves through future high-tech devices.

Technical Summary: Phase-controlled quasi-bound states in the continuum and thermoelectric enhancement in Majorana–quantum-dot nanostructures

Problem Statement
The identification of Majorana zero modes (MZMs) in topological superconductor nanowires (TSCNs) remains a significant challenge, as conventional zero-bias conductance anomalies can be mimicked by trivial effects such as disorder or Andreev bound states. While thermoelectric measurements offer a complementary probe through thermal conductance, Seebeck coefficients, and violations of the Wiedemann–Franz (WF) law, the mechanisms governing these responses in hybrid nanostructures require further elucidation. Specifically, the interplay between MZMs and bound states in the continuum (BICs)—states that do not decay despite residing within the energy continuum—has not been fully explored in the context of thermoelectric transport. This paper investigates how symmetry breaking converts these symmetry-protected BICs into transport-active quasi-BICs and how this conversion influences the electronic thermoelectric response in a crossbar-shaped quantum dot (QD) system coupled to two TSCNs.

Methodology
The authors model a crossbar geometry consisting of a central QD coupled to two normal leads (source and drain) and two TSCNs, each hosting a pair of MZMs. The system is described by an effective low-energy Hamiltonian incorporating:

Lead and Dot Terms: Standard electronic contributions for the normal leads and the QD (energy level $\epsilon_0$ ).
Coupling Terms: Tunneling between the QD and the leads, and coupling between the QD and the MZMs ( $\gamma_{1A}, \gamma_{1B}$ ).
Majorana Hybridization: Finite-length coupling between MZM pairs at the ends of the wires ( $\epsilon_{MA}, \epsilon_{MB}$ ), which decays exponentially with wire length.

The analysis employs the Green-function formalism within the Nambu basis to account for anomalous terms arising from the dot–Majorana coupling. Transport properties are derived using exact linear-response energy integrals for electrical conductance ( $G$ ), Seebeck coefficient ( $S$ ), and electronic thermal conductance ( $\kappa_{el}$ ). The electronic figure of merit ( $ZT_{el}$ ) and the Lorenz ratio ( $L/L_0$ ) are calculated. To analyze low-temperature behavior, the authors utilize a fourth-order Sommerfeld expansion, comparing these analytic approximations against exact numerical integrations.

Key Contributions and Results
The study systematically compares three symmetry-breaking mechanisms to determine their impact on converting BICs into quasi-BICs and enhancing thermoelectric performance:

Wire-Length Asymmetry (Unequal Hybridization):
Introducing unequal wire lengths ( $\epsilon_{MA} \neq \epsilon_{MB}$ ) breaks the symmetry that protects BICs. This converts the zero-linewidth BICs into finite-linewidth quasi-BICs, manifesting as narrow resonances in the QD spectral function. While this generates a finite thermoelectric response, the enhancement is modest, yielding a maximum electronic figure of merit of $ZT_{el}^{max} \simeq 0.012$ .
Quantum Dot Level Detuning:
Detuning the QD energy level ( $\epsilon_0$ ) away from the Fermi energy significantly alters the transport landscape. This mechanism produces Fano-like structures and thermoelectric doublets on the flanks of the resonances. Detuning increases the maximum response by approximately one order of magnitude compared to the wire-length asymmetry case, reaching $ZT_{el}^{max} \simeq 0.105$ .
Superconducting Phase Control:
The most significant enhancement arises from tuning the superconducting phase difference ( $\theta$ ) between the two TSCNs. In symmetric configurations, specific phase values convert BICs into quasi-BICs and, crucially, generate quadratic transmission zeros at the resonance energy.
- At these quadratic zeros, the transmission function scales as $T(\omega) \propto (\omega - \omega_0)^2$ .
- This power-law behavior leads to a strong violation of the Wiedemann–Franz law. The authors derive universal low-temperature values for this regime: a maximum electronic figure of merit of $ZT_{el}^{max} \simeq 0.75$ (specifically $0.75489$ analytically and $\approx 0.752$ via exact integration) and a Lorenz ratio of $L/L_0 = 21/5 = 4.2$ .
- Compared to wire-length asymmetry, phase tuning enhances the thermoelectric response by approximately sixtyfold; compared to QD detuning, the enhancement is roughly sevenfold.

Significance and Claims
The paper establishes that superconducting-phase control is an efficient mechanism for engineering the electronic response of topological hybrid nanostructures. The primary significance lies in demonstrating that phase tuning can transform symmetry-protected BICs into transport-active quasi-BICs, thereby generating a robust thermoelectric signature.

The authors claim that the observation of a quadratic transmission zero, accompanied by a specific violation of the WF law ( $L/L_0 = 21/5$ ) and a high $ZT_{el}$ , serves as a distinct, phase-tunable thermoelectric signature of the Majorana-coupled interference structure. These results provide a theoretical basis for distinguishing true MZM physics from trivial subgap states using thermoelectric measurements, offering a route to control electronic energy filtering in topological systems. The study remains focused on the electronic contribution to the figure of merit, noting that phonon heat transport would reduce the total $ZT$ in a physical device.

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