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Spectral Bifurcation and Anomalous Supercurrent in Dissipative Topological Insulator-based Josephson Junctions

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Pith's one-line read The paper claims that a lossy metallic lead coupled to a short topological-insulator Josephson junction makes the Andreev spectrum complex and asymmetric, shifting the Majorana zero-energy crossing away from the phase difference…

desk verdict Solid spectral prediction in a non-Hermitian TI Josephson junction, but the headline anomalous supercurrent rests on an unproven use of the Furusaki-Tsukada formula. read the letter →

arxiv 2502.09397 v3 pith:H2QZQRON submitted 2025-02-13 cond-mat.supr-con cond-mat.mes-hallcond-mat.other

Ardamon Sten , Paramita Dutta , Sudeep Kumar Ghosh This is my paper · ORCID
classification cond-mat.supr-concond-mat.mes-hallcond-mat.other
keywords non-HermitianJosephsonjunctiontopologicalinsulatorAndreevboundstatesMajoranadissipationanomaloussupercurrentcurrent-phaserelationgap-exitpoints
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

The paper asks how dissipation reshapes the Andreev spectrum and Josephson current of a topological-insulator Josephson junction. Treating the loss from a metallic lead as a complex barrier in a non-Hermitian Bogoliubov–de Gennes Hamiltonian, it finds that the real part of the Andreev spectrum bifurcates beyond two gap-exit points, the zero-energy Majorana crossing moves away from the phase difference $\phi=\pi$ and acquires a finite imaginary part, and the current-phase relation becomes asymmetric with a non-vanishing current at zero phase. The authors argue this anomalous supercurrent is a genuinely non-Hermitian signature absent in an ordinary (non-topological) non-Hermitian Josephson junction, whose spectrum remains symmetric.

What carries the argument

The load-bearing object is the non-Hermitian Bogoliubov–de Gennes Hamiltonian with a complex delta-barrier $U(x)=(V_1 - iV_2)\delta(x)$ that combines a tunable real gate barrier with the dissipative lead. Solving the scattering problem in the Andreev approximation yields a quadratic equation for the complex Andreev bound-state energy whose two branches produce the asymmetric spectrum, while the Josephson current is obtained from the Furusaki–Tsukada formula in terms of the Andreev reflection coefficients. The imaginary part $V_2$ is the control parameter that moves the gap-exit points and tilts the spectrum.

What would settle it

A microscopic calculation of the Josephson current that treats the lossy lead with the full Lindblad master equation (including quantum jumps) and obtains exactly zero current at phase difference $\phi=0$ would falsify the anomalous supercurrent claim. Equivalently, an experiment on a HgTe- or Bi$_2$Te$_3$-based Josephson junction with a known dissipative lead that measures a strictly vanishing current at $\phi=0$ would contradict the paper's central prediction, even though the spectral asymmetry itself might survive.

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Extended reading notes

Core claim

The central discovery, stated on the paper's own terms, is that a small imaginary barrier $V_2$, representing a 'lossy' lead, breaks the particle-hole and Kramers protection of the helical Andreev bound states in a short topological-insulator Josephson junction. The complex barrier shifts the zero-energy crossing of the normal-incidence spectrum from $\phi=\pi$ to $\phi\neq\pi$, giving the Majorana bound states a finite lifetime; for oblique incidence the real spectrum develops two gap-exit points $\phi_1, \phi_2$, beyond which one branch pierces the continuum while the other stays subgap. Because the spectrum is no longer symmetric about $\phi=\pi$, the Furusaki–Tsukada current computed in the subgap approximation is also asymmetric and does not vanish at $\phi=0$ and $\phi=2\pi$. The authors support this by showing that the effect disappears when either the real barrier or the imaginary barrier is zero, and that an ordinary (non-topological) non-Hermitian junction with a symmetric spectrum shows no such anomalous current.

Load-bearing premise

The main load-bearing assumption is that the Furusaki–Tsukada formula, derived for Hermitian scattering, gives the correct Josephson current for the non-Hermitian Hamiltonian; the paper itself shows it differs from an ABS-based formula near the gap-exit points and offers only a qualitative reconciliation.

Editorial extensions

If this is right

  • In both the tunneling regime ($\Gamma_0 \ll \Delta$) and the strong-proximity regime, the zero-energy Majorana crossing shifts away from $\phi=\pi$ and acquires a negative imaginary energy, so the Majorana bound states have finite lifetime and lose topological protection against this type of dissipation.
  • The current-phase relation becomes $2\pi$-periodic, non-sinusoidal, asymmetric about $\phi=\pi$, and non-zero at $\phi=0$; the critical current decreases monotonically as the imaginary barrier strength $Z_2$ or the real barrier $Z_1$ increases.
  • The spectral bifurcation at the gap-exit points means that beyond a phase interval $[\phi_1,\phi_2]$, the real part of one Andreev branch lies above the superconducting gap while the other lies below it, and the imaginary part of the spectrum vanishes there.
  • The anomalous supercurrent requires both a real barrier ($Z_1 \neq 0$) and dissipation ($Z_2 \neq 0$); in an ordinary non-Hermitian planar junction the spectrum is symmetric about $\phi=\pi$ and the current at zero phase vanishes.
  • The authors propose that the effect is observable in practical devices such as Nb-on-HgTe or Nb-Bi$_2$Te$_3$ junctions with a lossy metallic lead attached, using realistic parameters ($\Delta \simeq 1$ meV, $\Gamma_0 \simeq 0.2$ meV, $\hbar v_F \simeq 250$–$350$ meV-nm).

Reading between the lines

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

  • The same asymmetry mechanism should produce a Josephson diode effect: because the current-phase relation is asymmetric about $\phi=\pi$, the critical currents for positive and negative current directions should differ, giving a dissipation-controlled diode efficiency that the paper does not explicitly compute.
  • The discrepancy between the Furusaki–Tsukada and ABS-formula currents near the gap-exit points suggests that a fully self-consistent treatment of non-Hermitian scattering, including branch-cut contributions, is needed; a Keldysh or Lindblad-with-quantum-jumps calculation would test whether the anomalous zero-phase current persists beyond the subgap approximation.
  • The shift of the Majorana zero-energy crossing with dissipation offers a route to probe non-Hermitian effects through the current-phase relation: a null measurement of current at $\phi=0$ would rule out the model's prediction, while a nonzero value with the predicted sign and magnitude would confirm the spectral asymmetry in an experiment.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper models a topological-insulator Josephson junction with a lossy metallic lead by a non-Hermitian BdG Hamiltonian with a complex delta-barrier U(x)=(V1-iV2)δ(x). Solving the scattering problem within the Andreev approximation, the authors derive a complex Andreev bound-state spectrum (Eqs. (4)-(7)) that shows an asymmetry about φ=π, a shifted zero-energy crossing, finite imaginary energies (finite lifetimes), and gap-exit points beyond which the real spectrum bifurcates. Using the Furusaki-Tsukada formula (Eq. (8)) to compute the Josephson current, they obtain a 2π-periodic, non-sinusoidal CPR with a non-zero current at zero phase difference, interpreted as an anomalous supercurrent driven by the spectral asymmetry. The paper includes a comparison with an ordinary non-Hermitian junction and with an alternative ABS-based formula (Eq. (11)), which shows a significant discrepancy near the gap-exit points.

Significance. If the central prediction survives scrutiny, this work offers a concrete model of how dissipation reshapes superconducting transport in topological junctions, with potential implications for dissipation-engineered quantum devices. The analytical derivations in Appendices A-C are transparent and the spectral features (complex ABS, GEPs, branch selection) are derived from the stated Hamiltonian. The paper honestly reports the disagreement between the two current formulas, which is a strength. However, the headline anomalous supercurrent rests on applying the Hermitian FT formula to a non-Hermitian open system; without a derivation or independent benchmark, the prediction is not yet established. The spectral-asymmetry result itself is a useful and likely correct contribution.

major comments (3)
  1. [Comparison with CPR from ABS formula (Fig. 5)] The anomalous supercurrent at zero phase is computed exclusively with the Furusaki-Tsukada formula, Eq. (8). This formula is derived for Hermitian junctions with unitary scattering, and the paper does not provide a derivation for the non-Hermitian scattering problem with the complex barrier U(x)=(V1-iV2)δ(x). The paper's own comparison with the ABS formula, Eq. (11), shows a significant discrepancy near the gap-exit points, and the offered branch-cut explanation is qualitative. Because the finite current at φ=0,2π is the headline claim, the authors must either derive Eq. (8) from the non-Hermitian Green's function of the junction or otherwise demonstrate that the discrepancy is not an artifact of applying a Hermitian formula to a non-unitary scattering problem. They should also justify the subgap approximation in the Matsubara summation, since the replacement kBT Σωn → (1/2π)∫dω over |ω|<Δ is not derived.
  2. [Complex helical ABSs and CPRs, after Eq. (7)] The selection of the negative imaginary branch as the physical spectrum and the dismissal of the positive imaginary branch as 'pumping' is an ad hoc step. In a dissipative system all physical states should have negative imaginary parts, so the appearance of a positive imaginary branch suggests that the ansatz or boundary conditions admit unphysical solutions. The authors should clarify the criterion for branch selection and verify that the scattering coefficients entering Eq. (8) are computed with the same convention. This is important because the spectral asymmetry that drives the anomalous current depends on which branches are retained.
  3. [Comparison with CPR from ABS formula] The comparison between Eq. (8) and Eq. (11) in Fig. 5 is performed only in the strong-proximity limit (Γ0=Δ), whereas the central claim of an anomalous CPR is demonstrated in the tunneling limit (Γ0≪Δ) in Figs. 2(d) and 3(b). In the tunneling limit the ABS spectrum is obtained perturbatively in γ and the effective gap is energy-dependent, so the discrepancy between the two formulas may behave differently. The authors should either extend the comparison to the tunneling regime or explain why the strong-proximity comparison is sufficient to validate the current calculation in the regime of the main results.
minor comments (6)
  1. [Eq. (10) and surrounding text] The symbol Z2 is used both for the imaginary barrier strength and for Z^2 = Z1^2+Z2^2 in the ordinary junction; this is confusing and should be fixed by using a different symbol for the sum of squares.
  2. [Abstract and Section on CPR] The phrase 'non-zero value at ϕ=0,2π' is redundant for a 2π-periodic current; consider writing 'non-zero value at ϕ=0 (equivalently 2π)'.
  3. [Main text] In the sentence 'The spectrum also exhibit an asymmetry', the verb should agree with the subject: 'exhibits'.
  4. [Fig. 5 caption] The caption does not specify the normalization of the current; please add the units (e.g., eΔ/ℏ or eNΓ0/ℏ) used in the plot.
  5. [After Eq. (8)] The subgap approximation is introduced without quantitative justification; a sentence explaining why contributions from |ω|>Δ are negligible would be helpful.
  6. [Abstract and main text] The statement that the real spectrum 'bifurcates beyond specific phase intervals' should clarify that this occurs for oblique incidence (θ≠0), since the normal-incidence spectrum remains gapless.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the complex barrier is a model input, the ABS spectrum and CPR are derived from the stated Hamiltonian, and the central claim is not built into the definitions.

full rationale

The central claim—an anomalous supercurrent that is nonzero at zero phase difference—is derived by solving the Bogoliubov-de Gennes equation with a complex delta-function barrier U(x)=(V1-iV2)δ(x). The Andreev bound-state spectrum follows from the boundary condition (Eq. 3) and secular equation (Eq. 4); the current is then computed with the Furusaki-Tsukada formula (Eq. 8) from the derived Andreev reflection coefficients and integrated over angles (Eq. 9). No parameter is fitted to the current-phase relation, and the spectral asymmetry is not assumed as an input; it emerges from the non-Hermitian barrier. The self-citations in the paper (Refs. 5–7) are contextual and not load-bearing for the main derivation, which relies on the externally established Tkachov–Hankiewicz and Fu–Kane frameworks. The discrepancy between Eq. (8) and the ABS formula (Eq. 11) near gap-exit points is a validity concern for applying the Hermitian FT formula to a non-Hermitian scattering problem, but this is a correctness risk rather than circularity: Eq. (8) is not defined in terms of the predicted nonzero current at φ=0,2π. Therefore no step in the derivation reduces by construction to its own inputs.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim relies on the Lindblad-plus-jump-neglect approximation, the specific complex barrier form, and the unproven applicability of the Furusaki-Tsukada formula to non-Hermitian systems. These are model assumptions rather than fitted parameters. No new particles, dimensions, or conserved quantities are introduced.

free parameters (3)
  • Z1 = V1/(ℏvF) = 0.4 (also 0 and 0.2 in scans)
    Real barrier strength, chosen by hand to illustrate regimes; not fitted to data.
  • Z2 = V2/(ℏvF) = 0.2 (varied 0 to 0.3 in figures)
    Imaginary barrier strength from the lossy lead; chosen to show non-Hermitian effects.
  • gamma = Γ0/Δ = 0.3 (tunneling limit) and 1.0 (strong proximity)
    Tunneling strength parameter; the perturbation expansion in Eq. (7) assumes small gamma.
assumptions (5)
  • domain assumption Lindblad formalism with neglect of quantum jumps yields an effective non-Hermitian Hamiltonian with an imaginary potential barrier.
    Invoked in Appendix A to justify the complex barrier; this is a standard but uncontrolled approximation in open quantum systems.
  • domain assumption The BdG Hamiltonian Eq. (1) from Refs. [4,8] describes the proximity-induced superconductivity on the 3DTI surface.
    The effective model is taken from earlier work and assumed valid for the tunneling and strong-proximity regimes.
  • ad hoc to paper The complex barrier couples as (V1 τz - i V2 τ0) σ0 in the BdG equation.
    This specific form is chosen to represent the gate and the lossy lead; no microscopic derivation is provided.
  • ad hoc to paper The Furusaki-Tsukada formula (Eq. 8) is applicable to a non-Hermitian scattering problem.
    The paper claims this formula resolves divergences at gap-exit points but does not derive its validity for non-Hermitian systems; this is a load-bearing assumption for the anomalous current.
  • standard math Andreev approximation and short-junction limit are valid.
    Standard assumptions in the field, stated in the model section and Appendix B.

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Cite this review

Pith. "Pith review of Spectral Bifurcation and Anomalous Supercurrent in Dissipative Topological Insulator-based Josephson Junctions." pith.science (2026). https://pith.science/paper/H2QZQRON

@misc{pith2026250209397,
  author       = {Pith},
  title        = {Pith review of: Spectral Bifurcation and Anomalous Supercurrent in Dissipative Topological Insulator-based Josephson Junctions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H2QZQRON}},
  note         = {Machine review of arXiv:2502.09397}
}
abstract

The interplay between topological protection and dissipation constitutes a critical frontier in the realization of hybrid quantum devices. Here, we investigate the transport signatures in a dissipative topological insulator-based Josephson junction, a platform that directly probes the competition between quantum coherence and loss. We model dissipation by coupling a `lossy' metallic lead to the junction, described effectively by a non-Hermitian Hamiltonian derived using the Lindblad formalism. We observe that the junction exhibits an asymmetric complex Andreev spectrum, where the imaginary energy component imposes a finite lifetime on the quasi-bound states. Furthermore, beyond specific phase intervals, the real component of the spectrum bifurcates: one branch merges with the continuum, while the other penetrates just below the superconducting gap. Crucially, the characteristic zero-energy crossing shifts away from $\phi=\pi$ and acquires a non-zero imaginary component; consequently, the associated Majorana bound states acquire a finite lifetime, signaling a loss of robustness against dissipation. Finally, this spectral asymmetry drives an anomalous supercurrent, manifested as a non-vanishing current at zero phase difference. Our results reveal how dissipation fundamentally reshapes superconducting transport in topological junctions, opening new directions for dissipation-engineered quantum devices.

Figures

Figures reproduced from arXiv: 2502.09397 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. (a). Before discussing the features introduced by non-Hermiticity, we first remind ourselves of the ABS characteristics in the Hermitian case; the spectrum for normal incidence (θ = 0) is gapless and features a zero￾energy crossing at ϕ = π which corresponds to two Ma￾jorana bound states whose spinors are related by time￾reversal symmetry and also preserve Klein tunneling [8]. The spectrum for oblique incidence (θ ̸… view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: for both the NH-JJ and the NH-TIJJ in the strong￾proximity regime. Within this subgap approximation, the ABS formula predicts ‘Josephson gaps’, intervals of vanishing supercurrent [16]. However, a significant dis￾crepancy exists between these two formulas, particularly…
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

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Forward citations

Cited by 1 Pith paper

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

  1. Anomalous supercurrents in the presence of particle losses

    cond-mat.quant-gas 2025-05 conditional novelty 7.0 of 10

    Single-particle losses in a Josephson junction can produce a reversed (pi) supercurrent, and spin-selective losses induce a dissipation-generated spin supercurrent.

Reference graph

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