arxiv: 2604.08231 · v1 · submitted 2026-04-09 · ❄️ cond-mat.supr-con

Recognition: unknown

Exploring the conventional and anomalous Josephson effects at arbitrary disorder strength in systems with spin-dependent fields

Maryam Darvishi , F. Sebasti\'an Bergeret , Stefan Ili\'c

Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:23 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con

keywords Josephson currentarbitrary disorderspin-orbit couplinganomalous Josephson effectaltermagnetismEilenberger equationSNS junction0-pi transition

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The pith

A compact expression for the Josephson current in SNS junctions holds for arbitrary disorder when spin-dependent fields are present.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives a single formula for the supercurrent through a normal-metal link between two superconductors that incorporates spin-orbit coupling, Zeeman fields, or altermagnetism without restricting the amount of disorder. The derivation starts from the linearized quasiclassical Eilenberger equation and produces an expression that can be evaluated for any scattering strength between the ballistic and diffusive extremes. The authors then apply the formula to three concrete cases: the magnetic-field dependence of the critical current with Rashba versus Dresselhaus spin-orbit coupling, the persistence of the anomalous phi-zero shift under disorder, and the rounding of the zero-pi transition in altermagnetic junctions. A reader who works with real high-mobility samples would care because those samples routinely sit in the intermediate-disorder regime where standard ballistic or diffusive limits fail to give quantitative predictions.

Core claim

Using the linearized quasiclassical Eilenberger equation we obtain a compact expression for the Josephson current that remains valid at arbitrary disorder strength in the presence of generic spin-dependent fields. The same expression is evaluated for an applied magnetic field together with Rashba or Dresselhaus spin-orbit coupling, for the anomalous phi-zero effect under Rashba coupling, and for the 0-pi transition in junctions containing an altermagnet.

What carries the argument

The linearized quasiclassical Eilenberger equation, which yields a closed-form supercurrent once the Green's functions are integrated along trajectories at any disorder level.

If this is right

The magnetic-field dependence of the critical current can be used to extract the relative strength of Rashba versus Dresselhaus spin-orbit coupling in a given junction.
The anomalous Josephson shift survives moderate disorder and can even grow with increasing scattering in sufficiently long junctions.
Disorder rounds and ultimately suppresses the sharp 0-pi transition that would otherwise appear in clean altermagnetic junctions.
The same compact expression supplies quantitative predictions for high-mobility samples that contain residual disorder and therefore fall outside the usual limiting cases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same trajectory-integration technique could be reused to compute differential conductance or noise in the same class of junctions.
Disorder might be deliberately introduced in device design to stabilize or amplify the anomalous phase shift for superconducting electronics.
The approach provides a practical bridge between microscopic calculations and mesoscopic experiments on hybrid structures that contain both spin-orbit and magnetic order.

Load-bearing premise

The linearized quasiclassical Eilenberger equation remains accurate enough to give the Josephson current when disorder is neither weak nor strong and spin-dependent fields are also present.

What would settle it

Measure the critical current versus magnetic field in an SNS wire whose mean free path is known to lie between the ballistic and diffusive limits; the data should match the new formula but deviate systematically from both the pure-ballistic and pure-diffusive limits.

Figures

Figures reproduced from arXiv: 2604.08231 by F. Sebasti\'an Bergeret, Maryam Darvishi, Stefan Ili\'c.

**Figure 2.** Figure 2: FIG. 2. Josephson critical current as a function of junction length [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Normalized critical current [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. (a–c) Anomalous phase shift [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Slope of the anomalous phase shift at low Zeeman fields, ∂ [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. Josephson critical current in an altermagnet junction as a [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10. Bulk anomalous current [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9. Homogeneous bilayer of a normal metal with Rashba SOC [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗

read the original abstract

We present a theory of the Josephson current in superconductor-normal metal-superconductor (SNS) junctions in the presence of generic spin-dependent fields, such as spin-orbit coupling (SOC), Zeeman fields, and altermagnetism. We consider systems with arbitrary disorder strength, going beyond the usual diffusive and ballistic approximations. Using the linearized quasiclassical Eilenberger equation, we derive a compact expression for the Josephson current, which is then applied to various situations of experimental interest. First, we investigate the evolution of the Josephson critical current in an applied magnetic field in the presence of Rashba and Dresselhaus SOC, and discuss how this dependence can be used to probe SOC in the junction. We then study the anomalous Josephson ($\varphi_0$) effect in systems with Rashba SOC and show that it remains robust over a wide range of disorder strength, and can even be enhanced by moderate disorder in sufficiently long junctions. Finally, we investigate the Josephson current in disordered junctions with altermagnets, and show how the $0$-$\pi$ transition in such systems is suppressed by disorder. Our results may be useful for describing experimental setups with high-mobility samples, which nevertheless always contain some amount of disorder, and where neither purely ballistic nor diffusive approximations are adequate.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper's main contribution is a compact Josephson-current formula from the linearized Eilenberger equation that works across all disorder strengths in SNS junctions with SOC, Zeeman, or altermagnetic fields.

read the letter

The useful new element is the closed-form expression that interpolates between ballistic and diffusive limits without additional approximations. They obtain it by linearizing the quasiclassical Eilenberger equation in the normal region, where the gap is zero, and the resulting kernel directly gives the current-phase relation for generic spin-dependent fields. This is applied to three cases: the magnetic-field dependence of the critical current with Rashba or Dresselhaus SOC, the robustness of the anomalous phi0 effect (which can even strengthen with moderate disorder in long junctions), and the suppression of the 0-pi transition in disordered altermagnetic junctions. These applications follow straightforwardly from the formula and address setups with real samples that sit between the clean and dirty limits.

Referee Report

0 major / 0 minor

Summary. The manuscript develops a theory of the Josephson current in SNS junctions subject to generic spin-dependent fields (SOC, Zeeman, altermagnetism) at arbitrary disorder strength. It employs the linearized quasiclassical Eilenberger equation to obtain a compact expression for the current that interpolates between ballistic and diffusive regimes, then applies the result to the magnetic-field dependence of the critical current in the presence of Rashba and Dresselhaus SOC, the robustness of the anomalous φ₀ effect, and the disorder-induced suppression of the 0-π transition in altermagnetic junctions.

Significance. If the central derivation holds, the work supplies a practical, parameter-free interpolation formula for Josephson currents in moderately disordered systems that are neither purely ballistic nor diffusive; this is directly relevant to high-mobility experimental samples. The applications to SOC probing via critical-current magnetometry and to the stability of anomalous Josephson effects under disorder constitute concrete, falsifiable predictions that can guide experiments in superconducting spintronics. The derivation from the Eilenberger equation without additional fitting parameters is a clear technical strength.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive overall assessment, including the recommendation for minor revision. The referee summary accurately captures our derivation of a compact Josephson-current expression from the linearized Eilenberger equation that interpolates between ballistic and diffusive limits, together with the applications to magnetic-field dependence in the presence of Rashba/Dresselhaus SOC, robustness of the anomalous φ₀ effect, and disorder suppression of the 0-π transition in altermagnetic junctions.

Circularity Check

0 steps flagged

No significant circularity: derivation from standard Eilenberger equation is self-contained

full rationale

The central step is the use of the linearized quasiclassical Eilenberger equation (a well-established, externally validated framework in quasiclassical superconductivity theory) to obtain a compact integral expression for the Josephson current that interpolates between ballistic and diffusive limits. No parameter is fitted to the target observable and then re-labeled as a prediction; no uniqueness theorem or ansatz is imported via self-citation; the subsequent applications (magnetic-field dependence, φ₀ effect robustness, 0-π suppression in altermagnets) are direct evaluations of the derived kernel. The derivation chain therefore does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of the linearized quasiclassical Eilenberger equation to systems with arbitrary disorder and generic spin-dependent fields; no free parameters or new entities are mentioned in the abstract.

axioms (1)

domain assumption The linearized quasiclassical Eilenberger equation is valid for deriving the Josephson current at arbitrary disorder strength
Invoked to obtain the compact expression for the current in the presence of spin-dependent fields.

pith-pipeline@v0.9.0 · 5545 in / 1226 out tokens · 47531 ms · 2026-05-10T17:23:58.494000+00:00 · methodology

discussion (0)

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Spin-polarized Josephson current induced by inhomogeneous altermagnetic interlayers
cond-mat.supr-con 2026-05 unverdicted novelty 7.0

An inhomogeneous altermagnetic interlayer in a Josephson junction induces a net spin-polarized Josephson current at π misorientation of Néel vectors, enhancing the critical current while suppressing 0-π transitions.
Spin-polarized Josephson current induced by inhomogeneous altermagnetic interlayers
cond-mat.supr-con 2026-05 unverdicted novelty 7.0

An inhomogeneous altermagnetic interlayer in a Josephson junction produces enhanced critical current and spin-polarized supercurrent at π misorientation of Néel vectors through cancellation of pair-breaking oscillations.

Reference graph

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and the Research Council of Finland through the Finnish Quantum Flagship project 359240. S.I. is supported by the Research Council of Finland (Grant Number 355056). F. S. B. thanks financial support from the Spanish MCIN/AEI/10.13039/501100011033 through the grant PID2023-148225NB-C31, and the Eu- ropean Union’s Horizon Europe research and innova- tion pr...

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