🔬 mesoscale physics

Exceptional Points in a Parallel Double-Quantum-Dot Josephson Junction Coupled to a Ferromagnetic Reservoir

This paper demonstrates that while spin-independent dissipation eliminates second-order exceptional points in finite-gap superconducting double-quantum-dot Josephson junctions, the combination of spin-dependent dissipation from a ferromagnetic reservoir and magnetic flux tuning not only stabilizes these singularities but also enables the formation of third-order exceptional points with cubic-root scaling, all while maintaining continuous Josephson currents across the non-Hermitian singularities.

Original authors: Yiyan Wang, Ruixin Zhou, Bing Dong

Published 2026-07-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yiyan Wang, Ruixin Zhou, Bing Dong

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 the rules of physics get a little bit "leaky." In the standard, tidy world of quantum mechanics, energy is usually conserved, and systems are described by "Hermitian" math, which guarantees that if you measure something, you get a real, sensible number. But in the real world, nothing is perfectly isolated. Things lose energy, they interact with their surroundings, and they eventually fade away. This is the realm of "Non-Hermitian" physics. It's like studying a guitar string that isn't just vibrating in a vacuum but is also being dampened by a hand holding it, or a light bulb that is slowly running out of battery.

In this messy, open world, there are special, rare spots called "Exceptional Points" (EPs). Think of these as the quantum equivalent of a perfect storm. Usually, if you have two different states (like two different musical notes), you can tune them to get closer and closer together, but they will always stay distinct. At an Exceptional Point, however, the two notes don't just get close; they smash into each other and become one single, indistinguishable note. Even more strangely, the "direction" of the vibration (the eigenvector) also merges. It's a point where the system's behavior changes dramatically, like a switch flipping, but in a way that is much more complex and topological than a simple on/off switch. Scientists are fascinated by these points because they might help us build super-sensitive sensors or new types of quantum computers that can handle noise better than current ones.

Now, let's zoom in on a specific playground for these weird quantum effects: a tiny circuit made of two quantum dots (think of them as artificial atoms) connected in parallel between two superconducting wires. Superconductors are special materials that conduct electricity with zero resistance, and they have a "gap" in their energy spectrum—a safety zone where no electrons can exist unless they have enough energy to jump over it. The researchers in this paper wanted to see what happens when this double-dot circuit is also connected to a "reservoir" (a big bath of other electrons) that can drain energy away, and specifically, what happens if that reservoir is magnetic (ferromagnetic).

The team, led by Yiyan Wang, Ruixin Zhou, and Bing Dong, set up a theoretical experiment to map out these Exceptional Points. They started by looking at a simplified version of the system where the superconducting gap was assumed to be infinitely large. In this "infinite gap" world, the math is cleaner, and they found that if the energy drain (dissipation) is the same for all electrons (spin-independent), they could find these special merging points. However, they also found that when they made the model more realistic—by giving the superconducting gap a finite, real-world size—those specific merging points vanished. It was like finding a perfect balance point on a scale, only to realize that the scale itself was slightly bent, causing the balance to tip over.

But here is where the story gets exciting. The researchers then introduced a "ferromagnetic" reservoir. This is a magnetic bath that treats electrons with different spins (imagine them as tiny spinning tops pointing up or down) differently. It drains energy from "spin-up" electrons at a different rate than "spin-down" electrons. When they combined this spin-selective draining with a magnetic field threading through the loop of their circuit (like a magnetic wind blowing through a tunnel), something remarkable happened. Even with the realistic, finite superconducting gap, a subset of those Exceptional Points didn't disappear. They survived!

The paper shows that by carefully tuning the magnetic flux (the strength of that magnetic wind), they could even force two of these surviving second-order points to merge into a single, even more complex "third-order" point. At this third-order point, the energy levels don't just split apart in a simple square-root way; they split apart like the cube root of a number, which is a very specific and unusual mathematical behavior. The authors simulated this using a "surrogate model"—a clever, finite-sized approximation of the infinite system—and confirmed that while some of the perfect merging points from the simplified models were lost, the ones driven by the magnetic, spin-selective dissipation were robust enough to stick around.

Finally, the team checked if these weird points caused any glitches in the flow of electricity, known as the Josephson current. They calculated the current using two different mathematical methods: one based on the total energy of the system and another based on the density of particles. Both methods agreed perfectly. Most importantly, they found that the current flows smoothly and continuously right through these Exceptional Points. There is no sudden jump or break in the electricity, even though the underlying energy levels are doing something very strange.

In short, this paper suggests that if you want to create and control these robust, singular quantum points in a real-world superconducting device, you can't just rely on simple energy loss. You need to use a combination of magnetic fields and a reservoir that treats different electron spins differently. This "spin-selective dissipation" acts like a shield, protecting these delicate quantum singularities from being washed away by the finite nature of real materials. It's a recipe for making quantum systems that are not only weird but also surprisingly stable.

Technical Summary: Exceptional Points in a Parallel Double-Quantum-Dot Josephson Junction Coupled to a Ferromagnetic Reservoir

Problem Statement
The paper investigates the existence and stability of exceptional points (EPs)—non-Hermitian spectral singularities where eigenvalues and eigenvectors coalesce—in a parallel double-quantum-dot (DQD) Josephson junction coupled to a dissipative reservoir. A central challenge addressed is the validity of the "infinite superconducting gap" approximation ( $\Delta \to \infty$ ) often used in non-Hermitian treatments of Josephson transport. While previous studies predicted EPs in phase-biased junctions using static pairing models, it remains an open question whether these singularities persist when finite-gap quasiparticle degrees of freedom are restored. The study specifically contrasts spin-independent dissipation (normal reservoir) with spin-dependent dissipation (ferromagnetic reservoir) to determine how reservoir properties and magnetic flux influence the robustness of these non-Hermitian features.

Methodology
The authors employ a theoretical framework based on Grassmann path integrals to model the hybrid nanodevice. The system consists of two quantum dots connected in parallel between two superconducting leads, with one dot coupled to either a normal or ferromagnetic lead. A magnetic flux threads the loop, introducing Aharonov-Bohm phases.

Effective Action and Self-Energies: By integrating out the non-interacting leads, the authors derive an effective action. The superconducting and normal/ferromagnetic leads are characterized by self-energies ( $\Sigma_{SC}$ and $\Sigma_N$ ). In the wide-band limit, these yield frequency-dependent hybridization strengths and spin-dependent dissipation rates ( $\gamma_\uparrow, \gamma_\downarrow$ ).
Infinite-Gap Limit: To analyze the low-energy physics, the authors perform analytic continuation ( $i\omega_n \to \omega + i0^+$ ) and take the limit $\Delta \to \infty$ . This renders the superconducting self-energy frequency-independent, resulting in a non-Hermitian Bogoliubov-de Gennes (BdG) effective Hamiltonian ( $H_{eff}$ ).
Finite-Gap Surrogate Model: To test the robustness of the infinite-gap predictions, the authors construct a "surrogate" finite-level model. This involves a five-point parameter set derived from a shape-factor fit of the imaginary-time free-energy calculation, representing the superconducting leads with a finite bandwidth and gap ( $\Delta = 1$ ). This surrogate Hamiltonian is diagonalized directly in the BdG form.
Current Calculation: The Josephson current is calculated via two independent methods to ensure consistency: (i) the derivative of the free energy with respect to the phase difference ( $I_J = 2\partial F/\partial \delta\phi$ ) using Matsubara summation, and (ii) the expectation value of the current operator using a biorthogonal density matrix derived from the surrogate Hamiltonian.
Parity Analysis: A many-body Fock-space analysis is performed, separating the 16-dimensional Hilbert space into even- and odd-parity sectors to diagnose the spectral relationships governing the finite-gap behavior.

Key Results

Spin-Independent Dissipation (Normal Reservoir):
- In the infinite-gap limit, the system exhibits second-order EPs (EP2s) where eigenvalues and eigenvectors coalesce.
- However, when the finite-gap surrogate model is applied, these exact coalescences are eliminated and replaced by avoided crossings. This suggests that for spin-independent dissipation, the EPs found in the static approximation are artifacts of the infinite-gap limit for the parameters considered.
Spin-Dependent Dissipation (Ferromagnetic Reservoir):
- When coupled to a ferromagnetic reservoir (spin-selective dissipation) and subjected to magnetic flux, a subset of EP2s persists even in the finite-gap surrogate model.
- The magnetic flux acts as a continuous control parameter, shifting the positions of these EPs.
Third-Order Exceptional Points (EP3):
- In the infinite-gap model, fine-tuning the magnetic flux allows two EP2s to merge into a third-order EP (EP3). The eigenvalue splitting near this point exhibits a cubic-root scaling behavior ( $\Delta E \propto |\delta\phi - \phi_{EP}|^{1/3}$ ), confirmed by the dimension of the Jordan chain.
- Crucially, this triple coalescence does not survive in the finite-gap surrogate model; the specific parameter values that yield an EP3 in the infinite-gap limit result in lifted degeneracies in the finite-gap calculation.
Josephson Current and Continuity:
- Calculations of the Josephson current using both the free-energy derivative and the surrogate density matrix show consistent results.
- The current-phase relationship remains continuous as the system passes through the surviving second-order EPs. This confirms that the spectral non-analyticity at the EP does not manifest as a discontinuity in the thermodynamic response (current).
Parity-Sector Correlation:
- The analysis of even- and odd-parity sectors reveals a correlation between the survival of EPs in the finite-gap model and the structural matching of the real-energy spectra in these sectors. While not a definitive proof of symmetry protection, this provides a microscopic diagnostic for why certain EPs survive finite-gap corrections while others do not.

Significance and Claims
The paper establishes that spin-selective dissipation combined with interferometric flux tuning serves as an effective control mechanism for robust non-Hermitian singularities in superconducting nanostructures. The primary contribution is the distinction between predictions made in the infinite-gap limit and those that hold in a more realistic finite-gap scenario.

The authors claim that while the infinite-gap approximation is useful for identifying potential EPs, it can be misleading regarding their stability. Specifically:

Spin-independent dissipation leads to EPs that are fragile and eliminated by finite-gap effects.
Spin-dependent dissipation, in conjunction with magnetic flux, can stabilize a subset of EPs against finite-gap corrections.
The transition to higher-order EPs (EP3) is highly sensitive to the gap approximation and may not be realizable in finite-gap systems with the same parameters.

The work concludes that the Josephson current remains a continuous observable across these singularities, reinforcing the consistency of thermodynamic responses with non-Hermitian spectral features in these systems. The findings highlight the necessity of distinguishing between "directly validated continuous-model conclusions" and "parameter-specific surrogate predictions" when exploring non-Hermitian physics in superconducting devices.

Technical Summary: Exceptional Points in a Parallel Double-Quantum-Dot Josephson Junction Coupled to a Ferromagnetic Reservoir

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