🔬 mesoscale physics

Spontaneous fractional Josephson current from parafermions

This paper demonstrates that in a parafermion Josephson junction formed by proximitized counter-propagating quantum Hall edge modes, an externally controllable difference in edge lengths induces a spontaneous phase bias, enabling electrical control over Majorana or parafermion zero modes depending on the Laughlin filling fraction.

Original authors: Kishore Iyer, Amulya Ratnakar, Aabir Mukhopadyaya, Sumathi Rao, Sourin Das

Published 2026-05-01

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CC BY 4.0

Original authors: Kishore Iyer, Amulya Ratnakar, Aabir Mukhopadyaya, Sumathi Rao, Sourin Das

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 you have a very special, high-tech highway for tiny particles called electrons. Usually, on this highway, traffic flows in one direction on one side and the opposite direction on the other. This is what physicists call a "chiral edge state" in a quantum system.

Now, imagine you want to build a bridge across this highway using a superconductor (a material that conducts electricity with zero resistance). When you connect two superconductors across this highway, the electrons can "tunnel" through, creating a special kind of current called a Josephson current.

In the world of quantum physics, there are two types of exotic travelers on this highway:

Majorana modes: These are like simple, two-faced coins (heads or tails). They are already famous and being hunted for in labs.
Parafermions: These are the "super-coins." Instead of just heads or tails, they have multiple faces (like a die with many sides). They are more complex and could be the key to building much more powerful and error-proof quantum computers.

The Big Discovery
The authors of this paper found a new, simple way to control these exotic travelers.

Usually, to control the flow of current in these bridges, scientists have to change the "phase" of the superconductors (think of this as changing the timing of a wave). But this paper shows that you don't just need to change the timing; you can also change the length of the highway.

The "Treadmill" Analogy
Imagine two people running on a treadmill.

Person A is running on the left side.
Person B is running on the right side.
They are running in opposite directions.

In a standard setup, both treadmills are exactly the same length. The runners meet at the same time, and the current flows predictably.

However, the authors realized that if you make one treadmill slightly longer than the other (using a simple gate control, like adjusting the speed or position), something magical happens. Even if you don't change the timing (phase) of the runners, the difference in distance creates a "spontaneous" push. The runners feel a natural urge to move, creating a current out of nowhere.

Why This Matters for Parafermions
For the simple "Majorana" travelers, this length difference just creates a standard current. But for the complex "Parafermion" travelers (the multi-faced coins), this length difference acts like a master key.

The paper shows that by simply adjusting the length difference between the two paths, you can control the "phase" of these parafermions.
This creates a spontaneous fractional Josephson current. "Fractional" here means the current behaves in a way that is a fraction of what we usually see, which is the signature that these exotic parafermions are present.

The Bottom Line
The paper claims that you don't need complex, high-tech magnetic fields or complicated wiring to control these exotic quantum particles. You can do it with a simple "ruler." By making one path slightly longer than the other using electrical gates, you can:

Create a spontaneous current.
Control the behavior of parafermions (the complex quantum bits).
Prove that these particles exist by measuring how the current changes as you adjust the length.

The authors suggest this is a practical, "electrical" knob that scientists can turn in a lab to finally catch and control these elusive particles, potentially paving the way for better quantum computers. They even propose a specific setup using a double-layered quantum system (like a sandwich of two quantum liquids) where this length difference can be easily created and measured.

1. Problem Statement

The paper addresses the challenge of detecting and controlling parafermion zero modes, which are exotic non-Abelian anyons proposed as building blocks for topological quantum computation (offering higher fault tolerance than Majorana fermions).

Context: Parafermions are typically engineered at the interface of fractional quantum Hall (FQH) edge states and superconductors/ferromagnets.
Gap: While the fractional Josephson effect (a $4\pi m$ -periodic current) is a known signature of parafermions, existing proposals often rely on fixed geometries or complex multi-layer setups. There is a need for a mechanism to control the Josephson current and the resulting phase bias using accessible experimental parameters.
Core Question: Can the geometric asymmetry of the junction—specifically the difference in lengths of counter-propagating edge modes—act as a tunable source of spontaneous phase bias, inducing a fractional Josephson current even in the absence of an external superconducting phase difference?

2. Methodology

The authors employ a theoretical approach combining bosonization techniques for chiral Luttinger liquids and Andreev bound state (ABS) analysis.

System Model:
- A Josephson junction (JJ) formed by two counter-propagating chiral edge modes from FQH systems with filling fraction $\nu = 1/m$ (where $m$ is an odd integer).
- The edges are proximitized by two $s$ -wave superconductors ( $SC_1, SC_2$ ) and a ferromagnet ($FM$).
- Key Innovation: The lengths of the right-moving ( $L_1$ ) and left-moving ( $L_2$ ) edge modes in the ballistic region between the superconductors are allowed to be independent and gate-tunable.
Theoretical Framework:
- Majorana Case ( $m=1$ ): The Hamiltonian is treated using fermionic fields. The authors derive the ABS spectrum by applying Andreev boundary conditions at the interfaces, accounting for the phase accumulation over the distinct path lengths $L_1$ and $L_2$ .
- Parafermion Case ( $m > 1$ ): The system is modeled using bosonized chiral fields ( $\phi_R, \phi_L$ ). The superconducting and ferromagnetic regions induce pinning potentials (cosine terms) in the Hamiltonian.
- Boundary Conditions: The authors derive twisted boundary conditions for the bosonic fields, incorporating the phase difference $\phi = \phi_1 - \phi_2$ and the geometric phase shift arising from the length difference $\delta L = L_1 - L_2$ .
- Effective Hamiltonian: By taking the strong coupling limit ( $\Delta_0 \to \infty$ ), they derive an effective Hamiltonian for the ballistic region, treating the system as a ring of total length $L_1 + L_2$ with a global chiral bosonic field.

3. Key Contributions

Identification of Geometric Phase Bias: The paper demonstrates that a difference in the lengths of counter-propagating edges ( $\delta L \neq 0$ ) acts mathematically and physically equivalent to a superconducting phase bias. This creates a spontaneous Josephson current even when the external phase difference $\phi = 0$ .
Extension to Parafermions: While the spontaneous effect is known in Majorana systems ( $m=1$ ), this work generalizes the phenomenon to $Z_{2m}$ parafermions ( $m \neq 1$ ), showing that the length difference controls the $4\pi m$ -periodic fractional Josephson effect.
Experimental Proposal: The authors propose a concrete experimental setup using a bilayer FQH system (or a double quantum well) where external gates can manipulate Landau levels to displace edge states, thereby independently tuning $L_1$ and $L_2$ .

4. Key Results

A. Majorana Regime ( $m=1$ )

The Andreev Bound State (ABS) energy spectrum is derived as:
$E = \pm \Delta_0 \cos\left[ \frac{E \langle L \rangle}{\Delta_0 L_{SC}} \pm \left( \frac{\mu \delta L}{\hbar v_F} - \frac{\phi}{2} \right) \right]$
Where $\langle L \rangle = (L_1+L_2)/2$ , $\delta L = (L_1-L_2)/2$ , and $L_{SC}$ is the coherence length.
Spontaneous Current: The term $\frac{\mu \delta L}{\hbar v_F}$ is additive to the phase $\phi$ . Thus, setting $\phi=0$ but $\delta L \neq 0$ shifts the energy minimum, generating a spontaneous supercurrent.
Long Junction Limit: In long junctions, multiple ABS exist, but only one pair is topological. The energy window of this topological pair is narrowed by the factor $L_{SC}/\langle L \rangle$ .

B. Parafermion Regime ( $m > 1$ )

Ground State Degeneracy: In the limit of strong pinning (large $M_1$ ), the system exhibits a $2m$ -fold degenerate ground state manifold.
Symmetry Breaking: When the insulating gap is removed ( $M_1 \to 0$ ) to form the ballistic junction, the charge parity operators no longer commute with the Hamiltonian. The system is best described by spin operators $\hat{S}_1, \hat{S}_2$ .
Effective Hamiltonian: The low-energy effective Hamiltonian for the junction is:
$H_{\text{eff}} = \frac{m v_F}{\pi(L_1 + L_2)} \hat{\eta}^2 + \sum_{k>0} \frac{2\pi k v_F}{L_1 + L_2} \left( a_k^\dagger a_k + \frac{1}{2} \right)$
Where $\hat{\eta}$ is related to the spin of the junction and the accumulated phase.
Fractional Periodicity: The energy eigenvalues $E(\bar{s}_1)$ and the resulting Josephson current $I_\theta$ exhibit a $4\pi m$ periodicity with respect to the combined phase parameter $\theta = \frac{2\mu \delta L}{\hbar v_F} - \phi$ .
Tunability: The current can be tuned electrically by varying $\delta L$ . For typical parameters ( $v_F \sim 10^4$ m/s, $\mu \sim 10$ meV), a length difference of a few micrometers is sufficient to access the fractional effect, which is experimentally feasible.

5. Significance

New Control Knob: The paper identifies the geometric length difference ( $\delta L$ ) as a new, electrically controllable parameter for Josephson junctions, on equal footing with the superconducting phase bias.
Parafermion Detection: It provides a robust method to probe parafermions. The spontaneous fractional Josephson current serves as a direct signature of the underlying non-Abelian statistics without requiring complex interferometry.
Scalability: The proposed setup (bilayer FQH with gate-tunable edge displacement) is compatible with current semiconductor and 2DEG fabrication techniques, offering a realistic path toward realizing and manipulating topological qudits for quantum computing.
Theoretical Insight: It clarifies the role of boundary conditions and mode quantization in finite-length chiral Luttinger liquids, correcting previous assumptions about quantization lengths in such junctions.

In conclusion, the authors establish that spontaneous fractional Josephson currents can be generated and controlled purely by the geometric asymmetry of the junction edges, providing a powerful tool for the detection and manipulation of parafermion zero modes.