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arxiv: 2510.18963 · v1 · submitted 2025-10-21 · ❄️ cond-mat.supr-con

Helical phases and Bogoliubov Fermi surfaces probed by superconducting diode effects

Zekun Zhuang , Daniel Shaffer , Jaglul Hasan , Alex Levchenko This is my paper

Pith reviewed 2026-05-18 04:47 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con
keywords superconducting diode effectJosephson diode effectnoncentrosymmetric superconductorRashba spin-orbit couplinghelical phaseBogoliubov Fermi surfaceLifshitz transitionfinite-momentum pairing
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The pith

Superconducting diode efficiency approaches its maximum at the Lifshitz transition where Bogoliubov Fermi surfaces emerge in helical phases.

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

Noncentrosymmetric superconductors with Rashba spin-orbit coupling under in-plane magnetic fields support helical phases with finite-momentum Cooper pairs. The analysis finds that bulk diode efficiency can reach near its highest possible value exactly at the endpoint of a first-order Lifshitz transition between weak and strong helical phases, where Bogoliubov Fermi surfaces first appear. In Josephson junctions, finite-momentum pairing in the leads controls the diode effect for short junctions while the Zeeman field dominates in long junctions, producing field oscillations at low strength and strong directional suppression once the Fermi surfaces form. These transport signatures offer a route to detect the surfaces through current measurements.

Core claim

For the bulk system, the diode efficiency can nominally approach its maximal value at the critical endpoint of the first-order Lifshitz transition between weak and strong helical phases featuring finite-momentum Cooper pairs, the latter marked by the emergence of Bogoliubov Fermi surfaces. In a Josephson junction, finite-momentum pairing in the superconducting leads is the dominant mechanism behind the Josephson diode effect in short junctions, whereas in long junctions it is primarily governed by the Zeeman field in the normal region, with the onset of Bogoliubov Fermi surfaces producing sharp suppression and anisotropy in the current.

What carries the argument

the critical endpoint of the first-order Lifshitz transition between weak and strong helical phases, where finite-momentum Cooper pairs produce Bogoliubov Fermi surfaces that control diode efficiency and junction anisotropy

If this is right

  • Bulk diode efficiency reaches near its theoretical maximum at the specific magnetic field marking the Lifshitz transition endpoint.
  • Short Josephson junctions show a diode effect driven primarily by finite-momentum pairing in the leads.
  • Long junctions exhibit oscillations of diode efficiency sign with magnetic field at low fields, enabling tunability.
  • At higher fields the Josephson current develops strong anisotropy and suppression when flow is aligned with Bogoliubov Fermi surface momenta.

Where Pith is reading between the lines

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

  • The predicted anisotropy could detect Bogoliubov Fermi surfaces in materials that lack a measurable diode effect.
  • Sweeping field while monitoring diode efficiency might locate the helical phase boundary more directly than spectroscopic probes.
  • The same Lifshitz endpoint behavior could appear in other noncentrosymmetric setups with different spin-orbit forms.

Load-bearing premise

The system is assumed to be perfectly clean with no impurities, and the quasiclassical Eilenberger formalism is assumed to remain valid through both helical phases and at their transition point.

What would settle it

An experiment that sweeps in-plane magnetic field and finds no sharp peak in diode efficiency near the predicted Lifshitz transition field, or that measures Josephson current without the expected directional suppression aligned to Bogoliubov Fermi surface momenta, would contradict the central claim.

Figures

Figures reproduced from arXiv: 2510.18963 by Alex Levchenko, Daniel Shaffer, Jaglul Hasan, Zekun Zhuang.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) The schematic setup of the S-N-S junction, where [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) The supercurrent [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The Cooper pair momentum [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The phase diagram of bulk helical superconduc [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The single-band energy spectrum in the normal region as a function of [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The single-band CPR [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The angular dependence of the root-mean-square [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: (d). When the magnetic field further increases and the su￾perconductor enters the strong helical phase, the diode efficiency is greatly suppressed. This is because the cur￾rent contribution from the lower DOS λ = − band de￾creases significantly due to the emergence of BFS, as dis￾cussed earlier. The total current hence originates only from that of the higher DOS λ = + band, resulting in a strong reduction … view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. The polar plot of [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The diode efficiency as a function of [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
read the original abstract

Noncentrosymmetric superconductors (NCSs) with Rashba spin-orbit coupling (SOC) and in-plane magnetic fields have emerged as natural platforms for realizing both the bulk superconducting diode effect (SDE) and the Josephson diode effect (JDE) - phenomena characterized by unequal critical currents in opposite directions due to the simultaneous breaking of time-reversal and inversion symmetries. Using the quasiclassical Eilenberger formalism, we systematically investigate both the bulk SDE and the JDE in a clean NCS with Rashba SOC and in-plane magnetic fields. For the bulk system, we find that the diode efficiency can nominally approach its maximal value at the critical endpoint of the first-order Lifshitz transition between weak and strong helical phases featuring finite-momentum Cooper pairs, the latter marked by the emergence of Bogolyubov Fermi surfaces (BFSs). In a Josephson junction, we show that finite-momentum pairing in the superconducting leads is the dominant mechanism behind the JDE in short junctions, whereas in long junctions it is primarily governed by the Zeeman field in the normal region. In the long-junction regime, the diode efficiency additionally oscillates between positive and negative values as a function of magnetic field at low fields, providing a route toward a highly tunable Josephson diode. At higher fields, the onset of BFSs in the strong helical phase leads to a sharp suppression of both the JDE and the Josephson current when the current direction is aligned with momenta along the BFS, resulting in strong anisotropy. We propose that this anisotropy in the Josephson current offers an alternative method for detecting BFSs, applicable to systems with or without a JDE.

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.

Referee Report

1 major / 2 minor

Summary. The manuscript uses the quasiclassical Eilenberger formalism to study the bulk superconducting diode effect (SDE) and the Josephson diode effect (JDE) in clean noncentrosymmetric superconductors with Rashba SOC and in-plane Zeeman fields. It claims that the bulk diode efficiency approaches its maximal value at the critical endpoint of a first-order Lifshitz transition separating weak and strong helical phases with finite-momentum pairing, the latter signaled by the appearance of Bogoliubov Fermi surfaces. In Josephson junctions the work distinguishes dominant mechanisms (finite-momentum pairing in short junctions versus Zeeman field in long junctions), reports low-field oscillations of the diode efficiency, and identifies strong anisotropy and suppression of the Josephson current once BFSs appear, proposing the anisotropy as an alternative probe for BFSs.

Significance. If the central claims hold, the work would be significant for providing concrete, parameter-based predictions that connect helical phases and Bogoliubov Fermi surfaces to measurable diode efficiencies and anisotropies. The systematic treatment with Rashba strength and Zeeman field as free parameters, together with the distinction between short- and long-junction regimes, offers clear experimental routes for detecting exotic pairing states and for realizing tunable Josephson diodes. The clean-limit Eilenberger approach is a reproducible starting point, though its quantitative reliability at the Lifshitz endpoint remains to be verified.

major comments (1)
  1. [Abstract and bulk SDE section] Abstract and the section presenting the bulk SDE results: the claim that diode efficiency nominally reaches its maximum at the critical endpoint of the first-order Lifshitz transition (marked by emergence of BFSs) is load-bearing for the paper's main result. At this point the gap closes along nodal directions, the spectrum becomes gapless, and the density of states diverges; the standard clean-limit Eilenberger treatment (linearization around the Fermi surface and Matsubara integration) can miss non-analytic contributions or require additional regularization that is not shown or justified in the derivation.
minor comments (2)
  1. The abstract uses the qualifier 'nominally approach its maximal value'; an explicit definition of diode efficiency (including its formula) and the numerical values obtained near the transition would improve clarity and allow readers to assess how close to the theoretical maximum the computed efficiency actually gets.
  2. Notation for the 'weak' and 'strong' helical phases is introduced without a concise reminder of their defining characteristics (e.g., momentum of Cooper pairs or presence/absence of BFSs); adding a short definitional sentence in the introduction or model section would aid accessibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We are grateful to the referee for the detailed review and valuable feedback on our manuscript. We address the major comment below and have revised the manuscript accordingly to improve clarity and address concerns about the formalism at the critical point.

read point-by-point responses

  1. Referee: [Abstract and bulk SDE section] Abstract and the section presenting the bulk SDE results: the claim that diode efficiency nominally reaches its maximum at the critical endpoint of the first-order Lifshitz transition (marked by emergence of BFSs) is load-bearing for the paper's main result. At this point the gap closes along nodal directions, the spectrum becomes gapless, and the density of states diverges; the standard clean-limit Eilenberger treatment (linearization around the Fermi surface and Matsubara integration) can miss non-analytic contributions or require additional regularization that is not shown or justified in the derivation.

    Authors: We appreciate the referee's concern regarding the applicability of the quasiclassical Eilenberger formalism precisely at the Lifshitz transition endpoint. In our calculations, the diode efficiency is computed for parameters where the superconducting gap remains finite, and we find that it approaches its theoretical maximum value of 1 as the system approaches the critical endpoint from the weak helical phase. At the exact endpoint, the gap closes along certain directions, rendering the system gapless with a diverging density of states. We acknowledge that the standard Eilenberger approach, based on linearization around the Fermi surface and Matsubara summation, assumes a gapped spectrum and may require additional regularization or careful treatment of non-analytic terms in the gapless regime, which we have not explicitly derived in the manuscript. To address this, we will revise the abstract and the relevant section to emphasize that the maximum efficiency is approached asymptotically near the transition rather than exactly attained at the endpoint. We will also add a discussion paragraph on the limitations of the formalism at the critical point and suggest that future work could employ full microscopic calculations to verify the behavior exactly at the transition. This clarification strengthens the presentation without altering the core findings. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from standard Eilenberger setup

full rationale

The paper applies the standard quasiclassical Eilenberger formalism to a clean NCS with Rashba SOC and in-plane Zeeman field as external inputs. The central result on diode efficiency reaching a nominal maximum at the Lifshitz endpoint is obtained by solving the resulting equations for the current-phase relation and critical currents across the weak-to-strong helical transition; no parameter is fitted to the diode efficiency itself, no prediction is statistically forced by a subset of the same data, and no load-bearing uniqueness theorem or ansatz is imported via self-citation. The derivation therefore remains independent of its target observables.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard quasiclassical treatment of Rashba superconductors plus the clean-limit assumption; no new entities are postulated and the parameters are physical inputs rather than ad-hoc fits.

free parameters (2)
  • Rashba SOC strength
    Standard input parameter controlling band splitting; value not fitted to diode data in the reported results.
  • Zeeman field strength
    External control parameter varied to map helical phases and diode response.
axioms (2)
  • domain assumption Quasiclassical Eilenberger formalism remains valid across the weak-to-strong helical transition and at the Lifshitz endpoint.
    Invoked to derive both bulk SDE and JDE results.
  • domain assumption Clean limit (no impurity scattering) applies to the modeled NCS.
    Stated as the regime for the systematic investigation.

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Reference graph

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