Field-free Josephson diode effect in interacting chiral quantum dot junctions

Debika Debnath; Paramita Dutta

REVIEW 3 major objections 5 minor 123 references

Field-free Josephson diode effect in interacting chiral quantum dot junctions

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read An interacting chiral quantum dot between two superconductors passes more supercurrent in one direction than the other with no magnetic field, reaching about 72 percent rectification.

desk verdict A coherent mean-field study of a field-free Josephson diode in an interacting chiral QD, but the central spin-polarization mechanism is an unchecked Hartree-Fock artifact, so the 72% rectification is not yet established. read the letter →

arxiv 2411.18325 v2 pith:E5L5XRWH submitted 2024-11-27 cond-mat.supr-con cond-mat.mes-hall

Debika Debnath , Paramita Dutta This is my paper · ORCID

classification cond-mat.supr-concond-mat.mes-hall

keywords JosephsondiodeeffectquantumdotchiralityCoulombinteractionKeldyshnon-equilibriumGreen'sfunctionHartree-Fockmeanfieldsupercurrentrectificationspinimbalance

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that a single chiral quantum dot placed between two superconducting leads can act as a Josephson diode without any external magnetic field or magnetic impurity. The proposed mechanism is that electron-electron repulsion $U$ drives the dot into a spin-asymmetric state during transport, $\langle n_{d\uparrow}\rangle \neq \langle n_{d\downarrow}\rangle$, and this spontaneous spin imbalance, combined with the chirality-induced inversion asymmetry, makes the critical Josephson current different for the forward and reverse directions. Using the Keldysh non-equilibrium Green's function technique with a Hartree-Fock decoupling of the interaction, the authors report a rectification coefficient that changes sign with the Coulomb interaction and the lead-to-dot coupling and reaches about 72 percent at moderate $U/\Delta$ in the symmetric-coupling case. A field-free, gate-tunable supercurrent rectifier at the single-dot scale would matter because it removes the need for Zeeman fields and magnetic elements in superconducting diode devices.

What carries the argument

The load-bearing object is the self-consistent spin polarization of the dot produced by the Hartree-Fock decoupling of the interaction, $U n_{d\uparrow}n_{d\downarrow} \to U(\langle n_{d\downarrow}\rangle n_{d\uparrow} + \langle n_{d\uparrow}\rangle n_{d\downarrow} - \langle n_{d\downarrow}\rangle\langle n_{d\uparrow}\rangle)$. That decoupling generates a finite density difference $\delta n = |\langle n_{d\uparrow}\rangle - \langle n_{d\downarrow}\rangle|$, which enters the retarded dot Green's function and becomes intertwined with the superconducting phase difference. The chirality term $\sigma\alpha I$ shifts the dot level oppositely for the two spins and reverses with the direction of the current, providing the inversion asymmetry. The relative sizes of $U$, the lead-to-dot coupling $v$, and the superconducting gap $\Delta$ decide whether forward or reverse critical current dominates, which is why the rectification coefficient changes sign.

What would settle it

Compute the same junction with a method that treats the Coulomb interaction exactly at $U/\Delta = 2$ and $v/\Delta = 0.8$: if no up- versus down-spin density difference appears, or if the forward and reverse critical currents come out equal, the claimed diode effect is a mean-field artifact. A direct experimental check is to measure the zero-field current-phase relation of a chiral quantum dot Josephson junction and look for $I_c^+ \neq |I_c^-|$.

Watch

Extended reading notes

Core claim

The central claim is that time-reversal symmetry need not be broken explicitly in the Hamiltonian for a Josephson diode: the Coulomb repulsion on the dot itself creates an imbalance between up- and down-spin electron densities during nonequilibrium transport, turning the dot effectively magnetic. In a chiral dot, the level shifts by $\sigma\alpha I$ with the sign of the current, so forward and reverse transport see opposite helicity, and once the correlation-generated spin imbalance is coupled to the superconducting phase, the maximum Josephson current in one direction no longer matches the other, $I_c^+ \neq I_c^-$. The authors solve the Anderson impurity model in the symmetric limit $\varepsilon_d = -U/2$ with a Hartree-Fock mean-field decoupling and Keldysh Green's functions, keeping parameters below the Kondo scale. They find a current-phase relation with asymmetric extrema and a rectification coefficient $R = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$ that oscillates in sign as $U$ and $v$ are varied, reaching $R \sim 72\%$ at $U/\Delta = 2$ for symmetric coupling and $R \sim 60\%$ for strongly asymmetric couplings at weak correlation.

Load-bearing premise

The load-bearing premise is that electron repulsion on the dot genuinely makes more electrons of one spin occupy it than the other; if that spin imbalance is only an artifact of the mean-field approximation, the diode effect is not real.

Editorial extensions

If this is right

A single chiral quantum dot junction can rectify supercurrent with no external magnetic field and no magnetic impurity, so superconducting diode functionality can be built into a minimal weak link.
The rectification direction and magnitude are controllable through the gate-tuned dot level and through the lead-to-dot coupling, allowing the same junction to be switched between forward-favoring and reverse-favoring states.
At $U/\Delta = 2$ the computed rectification coefficient of about 72 percent is higher than the values the authors compare against for other quantum-dot Josephson diodes, suggesting this geometry is competitive for superconductor-based switching.
The diode effect persists for asymmetric lead couplings, where the ratio $v_L/v_R$ also controls the sign and size of the rectification.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same mechanism should appear in any correlated weak link that can spontaneously spin-polarize under current and lacks inversion symmetry, so the design principle is not limited to chiral dots.
Because the calculation is at zero temperature, the reported rectification is likely an upper bound; a finite-temperature version should show the spin imbalance and $R$ being progressively washed out as the junction approaches its critical temperature.
The sign oscillation of $R$ with $U/v$ implies a single gate-controlled device could pass through diode-on, off, and reversed-diode regimes, a switching protocol the paper mentions but does not spell out.
Whether the spin imbalance is genuine or a mean-field artifact is the key open question; an exact treatment of the same Anderson model would isolate the true origin of the effect.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Gerda,

Here's my read on 2411.18325. The genuinely new thing is a field-free Josephson diode in a single interacting chiral quantum dot, with the nonreciprocity driven by a correlation-induced spin imbalance rather than by an external field or magnetic impurity. The mechanism is distinct from the earlier QD JDE designs (Cheng & Sun, Sun et al.), and the parameter study is systematic: current-phase relations, rectification coefficient versus v and U, symmetric and asymmetric couplings, sign changes, and a peak RC of about 72%. The Keldysh/Hartree-Fock machinery is standard, the appendices give the derivations, and the authors are careful to claim validity only in the regime U/v >> 1, T_K << Delta, citing Karrasch and Yoshioka for HF-vs-NRG agreement on the Josephson current.

The soft spot is exactly the one the stress test flags, and I think it is load-bearing. The diode effect requires the HF spin polarization delta_n ~ 0.74 (Fig. 9). For a single Anderson impurity at particle-hole symmetry, the exact ground state is a spin singlet in the screened regime or a degenerate doublet with zero average spin; spin-rotation invariance forbids a spontaneous moment in a finite dot. HF admits broken-symmetry solutions, and the cited NRG checks cover only the total Josephson current in the nonmagnetic regime, not the spin-resolved occupations or the rectification coefficient. If delta_n is an artifact, the 72% rectification collapses. The central quantitative claim is therefore not established.

Two smaller issues. First, the chiral term sigma alpha I is a phenomenological input, alpha = 0.2 is uncalibrated, and the magnitude of the effect depends on it. Second, the claim to have achieved the highest JDE among QD-based junctions overreaches, since the comparison mixes different geometries and RC definitions.

Is it worth engaging? Yes. This is a coherent mean-field calculation of a plausible mechanism, and the question of whether correlations plus chirality can produce a field-free JDE in a QD is worth settling. But it should not be accepted on the strength of the HF polarization alone. I would send it to a referee and ask for an exact-method check of delta_n and the RC (NRG or QMC, or at least a compelling argument that the broken-symmetry solution survives in the exact ground state). Without that, the paper is a suggestion rather than a result.

I would not cite it in my own work yet, though I might bring it to a reading group: the symmetry discussion is instructive.

Referee Report

3 major / 5 minor

Summary. The paper studies a single chiral quantum dot Josephson junction with local Coulomb interaction U, modeled by an Anderson impurity coupled to two BCS leads. Using a Hartree-Fock decoupling of the Coulomb term and Keldysh Green's functions, the authors compute the Josephson current as a function of superconducting phase difference and define a rectification coefficient R=(I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|). They report unequal forward and reverse critical currents for both symmetric and asymmetric lead couplings, a sign-changing R as a function of U and v, and a maximum |R| of about 72% at moderate interaction strength. The proposed mechanism is that the interaction generates a spin polarization on the dot, which, combined with the chiral current-induced term σαI, breaks the symmetry required for a field-free Josephson diode effect.

Significance. If the central claim survives scrutiny, the result would be a useful addition to the field-free Josephson diode literature: a single quantum dot with no magnetic impurity or external field, showing a large and gate-tunable rectification coefficient. The paper is systematic in its parameter scans, gives explicit definitions of the current and rectification coefficient, and provides appendices documenting the Hartree-Fock treatment, the Kondo-temperature estimate, and the spin-density imbalance. The comparison with existing QD-based diode proposals, including the claimed improvement over Refs. [25,44,77], is useful. However, the significance is currently conditional: the diode effect rests on a Hartree-Fock spin polarization that is not validated against an exact method, and part of the nonreciprocity is built into the model through the free chiral parameter α.

major comments (3)

[Appendix D, Eq. (D1), Fig. 9] The time-reversal-symmetry-breaking ingredient of the diode effect is the spin imbalance δn = |<n_up> - <n_down>|, shown in Fig. 9 to be about 0.74. This quantity is obtained entirely from the Hartree-Fock decoupling of Eq. (A1). At the particle-hole symmetric point εd = -U/2 used throughout the paper, the exact ground state of a single-orbital Anderson impurity is a spin singlet in the screened regime or a degenerate doublet with <S_z> = 0 in the local-moment regime; spin-rotation invariance forbids a spontaneous magnetization in a finite dot. The chiral term σαI can orient a pre-existing moment, but it is itself proportional to the self-consistent current, so it cannot be invoked as an external field that justifies the large HF polarization. The cited NRG comparisons (Refs. [52,69]) validate only the total Josephson current in the nonmagnetic regime and do not test the spin-resolved occupations or the rectification coefficient. Since the claimed RC up to 72% (Fig. 3) and its sign changes are direct consequences of δn, the central claim requires an exact check of δn and RC, for example with NRG or DMRG, or a clear demonstration that the HF broken-symmetry solution is not an artifact.
[Eq. (2), Appendix B, Figs. 6 and 7] The diode effect is not purely correlation-induced: Eq. (2) contains the chiral term σαI with α a free input parameter. Figure 6 shows that α = 0 gives no forward/reverse asymmetry, while increasing α increases the asymmetry, and Fig. 7 shows that the magnitude of the RC grows with α. Thus a substantial part of the nonreciprocity is baked into the model through the assumed current-induced spin-dependent field. The abstract and Sec. III.A.1 overstate the role of U by calling the effect 'correlation-induced' and claiming that no TRS breaking is present in the microscopic Hamiltonian. The paper should explicitly separate the correlation-induced polarization from the current-induced chiral field and quantify how the RC depends on the minimal α required for the effect.
[Appendix C, Eq. (C1), Figs. 3 and 8] The stated validity condition TK/Δ << 1 is not satisfied over the full parameter range used in the main results. Using Eq. (C1) with εd = -U/2 gives TK = (1/2)√(Uv) exp(-πU/(8v)) in units of Δ. For U/Δ = 2 and v/Δ = 0.8, this is TK/Δ ≈ 0.24, and for v/Δ = 0.9 it is about 0.29, which is not much smaller than unity. Since Fig. 3 emphasizes RC for U/Δ = 2 over v/Δ up to about 0.9, the claim that the calculation is confined to the Kondo-free regime is quantitatively incorrect. This weakens the justification for using Hartree-Fock in exactly the parameter region where the largest rectification is reported. The authors should either restrict the parameter scans to the regime where TK/Δ is genuinely small or provide HF results with an explicit discussion of the expected Kondo corrections.

minor comments (5)

[Sec. II and Sec. IV] The manuscript repeatedly describes the calculation as 'non-equilibrium transport', but the Josephson current is computed at zero bias as a function of the superconducting phase difference, with the equilibrium fluctuation-dissipation relation G< = -f(E)(Gr - Ga). The authors should use consistent terminology, e.g., 'phase-biased equilibrium supercurrent', unless a genuine out-of-equilibrium bias is introduced.
[General] There are numerous typos and grammatical errors, including 'reults', 'interaciton', 'tempereatures', 'sign-chaning', and 'arte fact'. The manuscript should be carefully proofread before resubmission.
[Eq. (13)] The notation G<_{dL,11} and G<_{dL,33} in the current formula should be defined explicitly in terms of the spin and Nambu indices introduced in Eqs. (10) and (11), since the current expression is central to the paper and the index convention is not transparent.
[Appendix D, Fig. 9] The vertical axis of Fig. 9 spans only 0.7405–0.7410, making the phase dependence invisible to the eye. The authors should either use a scale that shows the variation of δn with ϕ or state the numerical magnitude and its phase dependence explicitly in the text.
[Sec. IV] The statement that 'RC ∼ 60%' can be extracted from Ref. [77] is not clearly supported, since the text acknowledges that Ref. [77] does not report RC explicitly. The extraction procedure should be described or this comparison should be softened.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the field-free Josephson diode effect is a genuine self-consistent consequence of the input chiral coupling and the interaction-induced spin imbalance; no fitted parameter is relabeled as a prediction.

full rationale

The paper's derivation is self-contained given its model. The chiral term σαI in Eq. (2) is an explicit model input, introduced as a known current-induced chiral effect and cited to prior work; it is not a parameter fitted to the target result. The interaction-induced spin imbalance δn = |⟨n↑⟩ − ⟨n↓⟩| is obtained self-consistently from Eq. (D1) using the Keldysh Green's function, not imposed by hand or fitted to data. The diode effect is then a nontrivial consequence of the combination of this self-consistent spin polarization with the chiral coupling: with either ingredient absent, the calculation yields no nonreciprocity, as stated in Sec. III.A and Appendix B. The Hartree-Fock approximation is a methodological limitation, and the paper explicitly restricts parameters to a regime where HF agrees with external NRG results (Refs. [52,69]); this is an external validity check, not a circular justification. The self-citation Ref. [26] for the chiral quantum dot model is not load-bearing because the model is also attributed to Ref. [44] and the underlying concept to Ref. [25]. The magnitude of the rectification coefficient (up to 72%) is a numerical output of the model, not a quantity that was fed into the Hamiltonian. No equation is equivalent to its own input by construction, and no fitted parameter is renamed as a prediction. The main scientific risk—whether the HF spin polarization survives an exact treatment—is a correctness/approximation concern, not circularity.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central prediction rests on three main inputs the reader must grant: the phenomenological chiral current-spin coupling with adjustable strength alpha, the validity of the Hartree-Fock approximation in producing the spin imbalance that drives the diode, and the restriction to the local-moment, Kondo-suppressed parameter regime. There are no fitted experimental constants and no invented physical entities.

free parameters (2)

chirality coefficient alpha = 0.2 (in units of Delta)
Strength of the current-induced spin splitting sigma alpha I in Eq. (2). The rectification coefficient grows with alpha (Appendix B, Fig. 7), so the predicted RC magnitude depends on this ad hoc input.
dot level epsilon_d = -U/2
Chosen to enforce particle-hole symmetry, the symmetric Anderson point. The paper claims results are qualitatively unchanged for other epsilon_d but does not demonstrate this.

assumptions (5)

domain assumption The chiral quantum dot is modeled by a current-induced spin splitting sigma alpha I in the dot level.
Eq. (2) postulates a linear coupling between spin, current direction, and a phenomenological coefficient alpha; this form is inherited from Refs [25,26] and is not derived from a microscopic chiral structure in this paper.
domain assumption The Hartree-Fock mean-field decoupling of the Coulomb interaction is quantitatively reliable in the regime U/v >> 1 and T_K << Delta.
Invoked in Sec. II and Appendix A based on agreement with NRG for the Josephson current in Refs [52,69]; the same HF solution supplies the spin imbalance that drives the diode effect, so the diode prediction inherits this assumption.
standard math The Keldysh lesser Green's function follows from the equilibrium fluctuation-dissipation relation G< = -f(E)(Gr - Ga).
Used in Sec. II for a zero-voltage, phase-biased junction at zero temperature; standard within the non-equilibrium Green's function formalism.
standard math The superconducting phase can be rotated into the tunnel amplitudes through a canonical transformation.
Eq. (5) decouples phase and gap, following Refs [26,83,84]; standard gauge manipulation.
domain assumption The system is kept below the Kondo temperature, T_K << Delta, so Kondo correlations do not invalidate the HF treatment.
The Kondo temperature in Eq. (C1) is used to restrict U and v such that the local-moment regime holds; this is necessary for the HF approximation to be trusted.

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Cite this review

Pith. "Pith review of Field-free Josephson diode effect in interacting chiral quantum dot junctions." pith.science (2026). https://pith.science/paper/E5L5XRWH

@misc{pith2026241118325,
  author       = {Pith},
  title        = {Pith review of: Field-free Josephson diode effect in interacting chiral quantum dot junctions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E5L5XRWH}},
  note         = {Machine review of arXiv:2411.18325}
}

abstract

We investigate chiral quantum dot (QD)-based Josephson junction and show the correlation-induced Josephson diode effect (JDE) in it. The presence of electron-electron interaction spontaneously creates an imbalance between up- and down-spin electrons during the non-equilibrium transport making the QD effectively magnetic. The simultaneous presence of the chirality and the interaction eventually results in the field-free JDE in our chiral QD junction. We employ the Keldysh non-equilibrium Green's function technique to study the behavior of the Josephson current (JC) and the rectification coefficient (RC) of our Josephson diode (JD). We show a sign-changing behavior of the RC with the Coulomb correlation and the lead-to-dot coupling strength and find the maximum magnitude of the RC $\sim 72\%$ for moderate interaction strength. Our proposed field-free JD based on interacting chiral QD may be a potential switching component in superconductor based devices.

Figures

Figures reproduced from arXiv: 2411.18325 by the authors.

**Figure 2.** FIG. 2. Josephson current [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. Rectification coefficient [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 5.** Figure 5: FIG. 5. Rectification coefficient [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: FIG. 6. Josephson current [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: FIG. 7. RC as a function of correlation strength [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

**Figure 9.** Figure 9: FIG. 9. Difference in up-spin and down-spin electron number [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]

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Reference graph

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Current-phase relation (CPR) We start by referring to Fig. 2, where we plot the JC as a function of the superconducting phase difference ϕ for two different lead-to-QD coupling coefficients, weak and moderate by setting v/∆= 0.3 and v/∆= 0.8 in Fig. 2(a) and (b), respectively, considering various Coulomb in- teractions. The competition among the lead-to-Q...
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2, we now investigate the rectifica- tion property of our QD-based JD

Diode rectification property With the discussion of the nonreciprocal behavior of the current in Fig. 2, we now investigate the rectifica- tion property of our QD-based JD. Using the definition of Eq. (14) in terms of the forward and reverse current, we plot the RC in Fig. 3 with respect to the lead-to- QD coupling strength v for various Coulomb interacti...
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Diode rectification property In order to investigate the effect of the asymmetry of the coupling constants on the rectification, we now present the behavior of the RC as a function of the ratio of the lead-to-QD coupling vL/vR for various correlation strength in Fig. 5. We find that the RC changes sign as the asymmetry vL/vR increases. The change in the s...

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Current-phase relation (CPR) We start by referring to Fig. 2, where we plot the JC as a function of the superconducting phase difference ϕ for two different lead-to-QD coupling coefficients, weak and moderate by setting v/∆= 0.3 and v/∆= 0.8 in Fig. 2(a) and (b), respectively, considering various Coulomb in- teractions. The competition among the lead-to-Q...

[2] [2]

2, we now investigate the rectifica- tion property of our QD-based JD

Diode rectification property With the discussion of the nonreciprocal behavior of the current in Fig. 2, we now investigate the rectifica- tion property of our QD-based JD. Using the definition of Eq. (14) in terms of the forward and reverse current, we plot the RC in Fig. 3 with respect to the lead-to- QD coupling strength v for various Coulomb interacti...

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4, we show the effect of asymmetric coupling on the JC for various Coulomb correlation strengths

Current-phase relation (CPR) In Fig. 4, we show the effect of asymmetric coupling on the JC for various Coulomb correlation strengths. With the increase in the Coulomb correlation, the JC is re- duced due to the Coulomb repulsion. We find that the asymmetry in the coupling coefficients in the two leads enhances the asymmetry in the forward and reverse cur...

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Diode rectification property In order to investigate the effect of the asymmetry of the coupling constants on the rectification, we now present the behavior of the RC as a function of the ratio of the lead-to-QD coupling vL/vR for various correlation strength in Fig. 5. We find that the RC changes sign as the asymmetry vL/vR increases. The change in the s...

2022

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