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REVIEW 2 major objections 5 minor 36 references

Probing Cooper pair momentum by quasiparticle steering with planar Josephson junctions

T0 review · 2 major / 5 minor · reviewed 2026-07-13 · grok-4.5

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Pith's one-line read Andreev modes in ballistic planar Josephson junctions eject into a normal region at a phase-controlled angle that scales as √(Δ/μ), larger than ordinary Cooper-pair momentum.

desk verdict Clean theory paper: phase-steered quasiparticle ejection from planar JJs at √(Δ/μ), larger than the usual Doppler scale, with analytics, numerics, and open code. read the letter →

arxiv 2607.08845 v1 pith:DMODCJAE submitted 2026-07-09 cond-mat.mes-hall cond-mat.supr-con

Isidora Araya Day , Anton R. Akhmerov , Antonio R. L. Manesco This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.supr-con
keywords CooperpairmomentumAndreevboundstatesplanarJosephsonjunctionquasiparticleejectionDopplershiftballistictransportelectronopticsphasecontrol
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

Cooper-pair momentum is usually tiny next to the Fermi momentum, so it is hard to see in the direction quasiparticles move. This paper shows that Andreev bound states traveling along a short, clean planar Josephson junction leave the junction into an adjacent normal metal at a finite angle set by the superconducting phase difference. That angle grows as the square root of the gap over the chemical potential, so it is parametrically larger than the usual Doppler scale Δ/μ and remains measurable even deep in the Andreev regime. The effect arises because repeated Andreev reflections from a phase-biased condensate give the bound state a transverse momentum that pure electrons then inherit at a transparent interface. If confirmed, phase-controlled ejection becomes a kinematic, momentum-space probe of condensate motion rather than an energy-shift measurement.

What carries the argument

Phase-controlled quasiparticle ejection: the next-to-leading-order transverse ABS momentum ⟨ky⟩_ABS ∼ √(Δ/a) obtained from a two-state Andreev-approximation basis, which is then matched at a transparent SNS–normal interface so that the transmitted electron current carries the same angular scale.

What would settle it

In a short epitaxial planar SNS device with a transparent normal lead, measure the transmission-weighted average emission angle of subgap quasiparticles versus phase and bias (via quantum point contacts or scanning gate microscopy). If the angle fails to track the predicted √(Δ/μ) phase and energy dependence, or remains finite at time-reversal-symmetric phase, the claim is false.

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Extended reading notes

Core claim

Subgap Andreev bound states that propagate along a short ballistic planar Josephson junction are transmitted into pure electron modes in an adjacent normal region at a phase- and energy-dependent average angle Θ that scales as √(Δ/μ). This scale exceeds the conventional Cooper-pair momentum ratio Δ/μ because the bound state accumulates transverse momentum through repeated Andreev reflections, and that momentum is inherited by the ejected electron current.

Load-bearing premise

The junction must be fully ballistic and the SNS–normal interface fully transparent, so Andreev modes convert into pure electron modes without ordinary band-mismatch reflection that would scramble the angle.

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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

2 major / 5 minor

Summary. The manuscript shows that Andreev bound states propagating along a short ballistic planar Josephson junction eject into an adjacent normal region at a phase- and bias-controlled angle that scales as Θ ∼ √(Δ/μ). This is parametrically larger than the conventional Cooper-pair momentum scale Δ/μ. The result is obtained from a controlled expansion of the linearized BdG Hamiltonian beyond the Andreev approximation (Eqs. 2–10), a continuum scattering solution that matches the SNS half-plane modes to pure electron modes in the normal lead (Appendix B), and supporting tight-binding Kwant simulations at Δ/μ up to 0.1–0.2 (Fig. 2). The authors propose detection via quantum point contacts or scanning gate microscopy and estimate material parameters for InAs–Al and graphene platforms.

Significance. If the result holds, the work supplies a kinematic probe of condensate momentum transfer that is complementary to existing Doppler-shift spectroscopies: the signal is a momentum-space deflection of emitted quasiparticles rather than an energy shift. The √(Δ/μ) enhancement makes the effect sizeable even deep in the Andreev regime, which is a genuine conceptual advance over single-reflection Doppler estimates. Strengths that raise confidence include a parameter-free scaling argument from Hamiltonian rescaling, an explicit continuum interface solution, quantitative agreement with open tight-binding code (Zenodo), and concrete experimental geometries with material estimates. The paper is a natural extension of recent work on ballistic Andreev-mode transport and is of clear interest to the mesoscopic superconductivity community.

major comments (2)
  1. The central theoretical claim is well supported inside the stated regime (ballistic junction, transparent SNS–normal interface, short junction). The only load-bearing experimental caveat is that the inheritance of the ABS transverse-momentum scale by the ejected electron current relies on the absence of ordinary band-mismatch reflection and on clean conversion of Andreev modes into pure electron modes (Sec. 2, interface paragraphs; Appendix B matching conditions ψ_h(0,y)=0 and continuous wavefunction). A short quantitative estimate of how weak interface mismatch or residual normal reflection would degrade ⟨Θ⟩ would strengthen the experimental section without changing the theory result.
  2. Near the gap edge the perturbative transverse momentum (Eq. 10) has a slow fourth-root divergence that is cut off by the finite normal-state bandwidth. Figure 2(c) shows good continuum–tight-binding agreement away from this cutoff, but the experimental proposals (Sec. 3) do not specify how close to the minigap or gap edge the bias should be kept for the √(Δ/μ) scaling to remain reliable. A brief statement of the usable energy window for the quoted material parameters would make the detection claims more falsifiable.
minor comments (5)
  1. Eq. (1) writes k = |k| while later using separate kx, ky components; a short clarification that the kinetic term is (ℏ²/2m)(kx² + ky²) would avoid momentary confusion.
  2. Figure 2(c) collapses data using a nontrivial combination of E and ϕ. Adding one sentence in the caption that states the expected collapse variable from Eq. (10) would help readers interpret the plot without flipping back to the text.
  3. In Appendix B the residual minimization for Ae(k), Be(k) is described at a high level; a pointer to the specific grid sizes and regularization used in the Zenodo code (already cited) would improve reproducibility of the continuum curves.
  4. The self-field estimate in Sec. 3 is useful; stating the assumed junction length and number of modes more explicitly would make the <10⁻⁴ rad bound easier to recompute.
  5. A few recent arXiv references (e.g. 2025–2026) are cited as published Phys. Rev. entries; verify final bibliographic details before production.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: ejection angle follows from BdG rescaling, perturbation, and independent interface matching.

full rationale

The claimed scaling Θ∼√(Δ/μ) is obtained from dimensional analysis of the linearized, rescaled BdG Hamiltonian (Eqs. 2–4), a next-to-leading-order perturbative evaluation of the ABS transverse momentum (Eq. 10), and a self-contained continuum scattering construction at the SNS–normal interface (Appendix B) that enforces wavefunction continuity and ψ_h(0,y)=0. These steps are checked against tight-binding scattering (Fig. 2c) with open code. Self-citations (notably [11]) supply background on Andreev-mode transport and the standard Andreev-approximation basis states used as a starting point for the perturbation; they are not used as a uniqueness theorem, fitted input, or definitional stand-in for the ejection angle. No prediction reduces by construction to its inputs, and the central kinematic claim is independently derived within the paper.

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

The claim rests on standard BdG modeling of a short planar Josephson junction, the Andreev approximation regime, ballistic propagation, and a transparent SNS–normal interface. No free parameters are fitted to data; numerical values of Δ/μ are illustrative. No new particles or forces are introduced.

assumptions (5)
  • domain assumption Bogoliubov–de Gennes description of a short planar SNS junction with step-like phase profile (Eq. 1).
    Standard continuum model for proximity-induced planar junctions; invoked throughout Sec. 2 and Appendix B.
  • domain assumption Andreev approximation Δ/μ ≪ 1 with controlled next-to-leading corrections for transverse momentum.
    Used to linearize longitudinal dispersion and obtain the √(Δ/a) transverse scale (Eqs. 2–10).
  • domain assumption Ballistic Andreev-mode transport along the junction with suppressed backscattering.
    Taken from Tam–Kane and prior author work; required for a well-defined propagating ABS that reaches the normal interface.
  • domain assumption Transparent SNS–normal interface: no ordinary band-mismatch reflection; Andreev modes convert to pure electron modes.
    Stated in Sec. 2; matching conditions in Appendix B set hole component to zero at the interface.
  • domain assumption Parabolic normal-state dispersion used to convert transverse momentum into ejection angle Θ ≈ ⟨ky⟩/k_F.
    Enters the scaling Θ ∼ √(Δ/μ) and the continuum angle extraction; graphene warping is noted as a possible correction.

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Pith. "Pith review of Probing Cooper pair momentum by quasiparticle steering with planar Josephson junctions." pith.science (2026). https://pith.science/paper/DMODCJAE

@misc{pith2026260708845,
  author       = {Pith},
  title        = {Pith review of: Probing Cooper pair momentum by quasiparticle steering with planar Josephson junctions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DMODCJAE}},
  note         = {Machine review of arXiv:2607.08845}
}
abstract

The Cooper pair momentum in a superconductor is associated with a phase gradient of the superconducting order parameter. In general, this momentum is small compared to the Fermi momentum, which makes it challenging to measure. Josephson junctions, however, enable the creation of large phase gradients and transfer of the Cooper pair momentum to quasiparticles via Andreev reflection. In this work we demonstrate that Andreev bound states propagating along ballistic planar Josephson junctions eject into an adjacent normal region at a phase-controlled angle that scales as $\Theta \sim \sqrt{\Delta/ \mu}$, where $\Delta$ is the superconducting gap and $\mu$ is the chemical potential. This angle parametrically exceeds the conventional Cooper pair momentum scale $\Delta/ \mu$, and thus this phenomenon is sizeable even within the Andreev approximation regime $\Delta / \mu \ll 1$. Our results establish phase-controlled quasiparticle ejection as a kinematic probe of condensate momentum transfer: unlike existing probes that detect the Doppler energy shift, the signal appears as a momentum-space deflection of emitted quasiparticles.

Figures

Figures reproduced from arXiv: 2607.08845 by the authors.

Figure 1
Figure 1. Electron current density ejected from a planar Josephson junction into [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Current angle distribution of the ejected electron current density and [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Proposed experimental setup to detect the ejection angle of quasiparticles [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Root structure of the bounded subgap solutions of Eq. [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: Real-space profiles of the bounded subgap solution families shown in [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]
Figure 6
Figure 6. Figure 6: Transmitted intensity profiles |t(y)| 2 at the SNS–normal interface, obtained from the dense linear-k solve of the half-line integral equations. Each panel corresponds to a different phase difference ϕ, and each curve corresponds to one of five evenly spaced energies b…

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Works this paper leans on

36 extracted references · 27 canonical work pages

  1. [1]

    Bardeen, L

    J. Bardeen, L. N. Cooper and J. R. Schrieffer,Theory of superconductivity, Phys. Rev. 108(5), 1175 (1957), doi:10.1103/physrev.108.1175

    doi:10.1103/physrev.108.1175 1957

  2. [2]

    Tkachov and V

    G. Tkachov and V. I. Fal’ko,Magnetic field influence on the proximity effect in semiconductor-superconductor hybrid structures and their thermal conductance, Physi- cal Review B69(9) (2004), doi:10.1103/physrevb.69.092503

    work page doi:10.1103/physrevb.69.092503 2004

  3. [3]

    Tunneling into Current-Carrying Surface States of High T$_c$ Superconductors

    M. Fogelström, D. Rainer and J. A. Sauls,Tunneling into current-carrying sur- face states of high-t _csuperconductors, Phys. Rev. Lett.79(2), 281 (1997), doi:10.1103/physrevlett.79.281, arXiv:cond-mat/9705260

    work page Pith review arXiv doi:10.1103/physrevlett.79.281 1997

  4. [4]

    Doppler Shift in Andreev Reflection from a Moving Superconducting Condensate in Nb/InAs Josephson Junctions

    F. Rohlfing, G. Tkachov, F. Otto, K. Richter, D. Weiss, G. Borghs and C. Strunk,Doppler shift in andreev reflection from a moving superconduct- ing condensate in nb/inas josephson junctions, Phys. Rev. B80(22) (2009), doi:10.1103/physrevb.80.220507, arXiv:0903.0321

    work page Pith review arXiv doi:10.1103/physrevb.80.220507 2009

  5. [5]

    M. R. Lykkegaard, A. E. Dahl, T. Lindemann, M. J. Manfra, K. Flensberg and C. M. Marcus,Phase-biased andreev diffraction grating, Phys. Rev. Lett.135(6) (2025), doi:10.1103/d4kk-sgmh, arXiv:2411.13355

    work page Pith review arXiv doi:10.1103/d4kk-sgmh 2025

  6. [6]

    A. E. Dahl, A. Maiani, M. Geier, J. Shabani and K. Flensberg,Directional conductance of andreev crystals in hybrid josephson junction arrays, Phys. Rev. B113(18) (2026), doi:10.1103/zrt6-971d, arXiv:2508.11768

    work page Pith review arXiv doi:10.1103/zrt6-971d 2026

  7. [7]

    Universal Josephson diode effect

    M. Davydova, S. Prembabu and L. Fu,Universal josephson diode effect, Sci. Adv. 8(23) (2022), doi:10.1126/sciadv.abo0309, arXiv:2201.00831

    work page Pith review arXiv doi:10.1126/sciadv.abo0309 2022

  8. [8]

    Nonreciprocal superconductivity

    M. Davydova, M. Geier and L. Fu,Nonreciprocal superconductivity, Sci. Adv.10(48) (2024), doi:10.1126/sciadv.adr4817, arXiv:2407.01681

    work page Pith review arXiv doi:10.1126/sciadv.adr4817 2024

Show all 36 references
  1. [9]

    P. M. Tam and C. L. Kane,Probing fermi sea topology by andreev state trans- port, Phys. Rev. Lett.130, 096301 (2023), doi:10.1103/PhysRevLett.130.096301, arXiv:2210.08048

    2023 doi arXiv

  2. [10]

    P. M. Tam, C. De Beule and C. L. Kane,Topological andreev rectification, Phys. Rev. B107, 245422 (2023), doi:10.1103/PhysRevB.107.245422, arXiv:2302.14050

    2023 doi arXiv

  3. [11]

    Araya Day, K

    I. Araya Day, K. Vilkelis, A. L. R. Manesco, A. M. Bozkurt, V. Fatemi and A. R. Akhmerov,Chiral adiabatic transmission protected by Fermi surface topology, SciPost Phys.18, 098 (2025), doi:10.21468/SciPostPhys.18.3.098

    2025 doi

  4. [12]

    C. W. Groth, M. Wimmer, A. R. Akhmerov and X. Waintal,Kwant: a software package for quantum transport, New Journal of Physics16(6), 063065 (2014), doi:10.1088/1367- 2630/16/6/063065

    2014 doi

  5. [13]

    L. W. Molenkamp, A. A. M. Staring, C. W. J. Beenakker, R. Eppenga, C. E. Timmering, J. G. Williamson, C. J. P. M. Harmans and C. T. Foxon,Electron- beam collimation with a quantum point contact, Phys. Rev. B41(2), 1274 (1990), doi:10.1103/physrevb.41.1274

    1990 doi

  6. [14]

    K. L. Shepard, M. L. Roukes and B. P. Van der Gaag,Direct measurement of the transmission matrix of a mesoscopic conductor, Phys. Rev. Lett.68(17), 2660 (1992), doi:10.1103/physrevlett.68.2660. 8 SciPost Physics Submission

    1992 doi

  7. [15]

    M. A. Topinka, B. J. LeRoy, S. E. J. Shaw, E. J. Heller, R. M. Westervelt, K. D. Maranowski and A. C. Gossard,Imaging coherent electron flow from a quantum point contact, Science289(5488), 2323 (2000), doi:10.1126/science.289.5488.2323

    2000 doi

  8. [16]

    M. A. Topinka, B. J. LeRoy, R. M. Westervelt, S. E. J. Shaw, R. Fleischmann, E. J. Heller, K. D. Maranowski and A. C. Gossard,Coherent branched flow in a two-dimensional electron gas, Nature410(6825), 183 (2001), doi:10.1038/35065553

    2001 doi

  9. [17]

    M. P. Jura, M. A. Topinka, M. Grobis, L. N. Pfeiffer, K. W. West and D. Goldhaber- Gordon,Electron interferometer formed with a scanning probe tip and quantum point contact, Phys. Rev. B80(4) (2009), doi:10.1103/physrevb.80.041303

    2009 doi

  10. [18]

    Bhandari, G.-H

    S. Bhandari, G.-H. Lee, K. Watanabe, T. Taniguchi, P. Kim and R. M. West- ervelt,Imaging andreev reflection in graphene, Nano Lett.20(7), 4890 (2020), doi:10.1021/acs.nanolett.0c00903, arXiv:2003.00367

    2020 doi arXiv

  11. [19]

    M. D. Bachmann, A. L. Sharpe, A. W. Barnard, C. Putzke, M. König, S. Khim, D. Goldhaber-Gordon, A. P. Mackenzie and P. J. W. Moll,Super-geometric electron focusing on the hexagonal fermi surface of pdcoo2, Nat Commun10(1) (2019), doi:10.1038/s41467-019-13020-9

    2019 doi

  12. [20]

    C. Gold, A. Knothe, A. Kurzmann, A. Garcia-Ruiz, K. Watanabe, T. Taniguchi, V. Fal’ko, K. Ensslin and T. Ihn,Coherent jetting from a gate-defined channel in bilayer graphene, Phys. Rev. Lett.127(4) (2021), doi:10.1103/physrevlett.127.046801, arXiv:2008.09145

    2021 doi arXiv

  13. [21]

    Ingla-Aynés, A

    J. Ingla-Aynés, A. L. R. Manesco, T. S. Ghiasi, S. Volosheniuk, K. Watanabe, T. Taniguchi and H. S. J. van der Zant,Specular electron focusing between gate- defined quantum point contacts in bilayer graphene, Nano Lett.23(12), 5453 (2023), doi:10.1021/acs.nanolett.3c00499, arX...

    2023 doi arXiv

  14. [22]

    Ingla-Aynés, A

    J. Ingla-Aynés, A. L. R. Manesco, T. S. Ghiasi, K. Watanabe, T. Taniguchi and H. S. J. vanderZant,Ballistic electron source with magnetically controlled valley polarization in bilayer graphene, Phys. Rev. Lett.133(15) (2024), doi:10.1103/physrevlett.133.156301, arXiv:2310.15293

    2024 doi arXiv

  15. [23]

    Sivan, M

    U. Sivan, M. Heiblum, C. P. Umbach and H. Shtrikman,Electrostatic elec- tron lens in the ballistic regime, Physical Review B41(11), 7937 (1990), doi:10.1103/physrevb.41.7937

    1990 doi

  16. [24]

    Noguchi, H

    M. Noguchi, H. S. Hideki Sakakibara and T. I. Toshiaki Ikoma,Collimation of two-dimensional ballistic electrons using equivalent snell’s law, Japanese Journal of Applied Physics32(11R), 5014 (1993), doi:10.1143/jjap.32.5014

    1993 doi

  17. [25]

    C. Yan, M. Pepper, P. See, I. Farrer, D. Ritchie and J. Griffiths,Demonstration of electron focusing using electronic lenses in low-dimensional system, Scientific Reports 10(1) (2020), doi:10.1038/s41598-020-59453-x, arXiv:2002.04307

    2020 doi arXiv

  18. [26]

    Shabani, M

    J. Shabani, M. Kjaergaard, H. J. Suominen, Y. Kim, F. Nichele, K. Pakrouski, T. Stankevic, R. M. Lutchyn, P. Krogstrup, R. Feidenhans’l, S. Kraemer, C. Nayak et al.,Two-dimensional epitaxial superconductor-semiconductor heterostructures: A platform for topological superconduct...

    2016 doi arXiv

  19. [27]

    Aghaee, A

    M. Aghaee, A. Akkala, Z. Alam, R. Ali, A. Alcaraz Ramirez, M. Andrzejczuk, A. E. Antipov, P. Aseev, M. Astafev, B. Bauer, J. Becker, S. Boddapatiet al.,Inas-al hybrid devices passing the topological gap protocol, Phys. Rev. B107, 245423 (2023), doi:10.1103/PhysRevB.107.245423,...

    2023 doi arXiv

  20. [28]

    F. D. Natterer, J. Ha, H. Baek, D. Zhang, W. G. Cullen, N. B. Zhitenev, Y. Kuk and J. A. Stroscio,Scanning tunneling spectroscopy of proximity superconductivity in epi- taxial multilayer graphene, Phys. Rev. B93(4) (2016), doi:10.1103/physrevb.93.045406, arXiv:1602.06831

    2016 doi arXiv

  21. [29]

    Cortés-del Río, S

    E. Cortés-del Río, S. Trivini, J. I. Pascual, V. Cherkez, P. Mallet, J.-Y. Veuillen, J. C. Cuevas and I. Brihuega,Shaping graphene superconductivity with nanometer precision, Small20(20) (2024), doi:10.1002/smll.202308439, arXiv:2401.09288

    2024 doi arXiv

  22. [30]

    Trivini, T

    S. Trivini, T. Kokkeler, J. Ortuzar, E. Cortés-del Río, B. Viña Bausá, P. Mallet, J.-Y. Veuillen, J. C. Cuevas, I. Brihuega, F. S. Bergeret and J. I. Pascual,From local to collective superconductivity in proximitized graphene(2025), arXiv:2504.19620

    2025 arXiv

  23. [31]

    F. K. de Vries, E. Portolés, G. Zheng, T. Taniguchi, K. Watanabe, T. Ihn, K. Ensslin and P. Rickhaus,Gate-defined Josephson junctions in magic-angle twisted bilayer graphene, Nature Nanotechnology16(7), 760 (2021), doi:10.1038/s41565-021-00896-2, arXiv:2011.00011

    2021 doi arXiv

  24. [32]

    Araya Day, A

    I. Araya Day, A. R. Akhmerov and A. R. L. Manesco,Code and data for Probing Cooper pair momentum by quasiparticle steering with planar Josephson junctions, doi:10.5281/zenodo.20763591 (2026)

    2026 doi

  25. [33]

    J. D. Hunter,Matplotlib: A 2d graphics environment, Computing in Science & Engineering9(3), 90 (2007), doi:10.1109/MCSE.2007.55

    2007 doi

  26. [34]

    C. R. Harris, K. J. Millman, S. J. van der Walt, R. Gommers, D. Cournapeau, E. Wieser, J. Taylor, S. Berg, N. J. Smith, R. Kern, M. Picus, S. Hoyeret al.,Array programming with NumPy, Nature585, 357 (2020), doi:10.1038/s41586-020-2649-2

    2020 doi

  27. [35]

    Virtanen, R

    P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett et al.,SciPy 1.0: Fundamental algorithms for scientific computing in python, Nature Methods17, 261 (2020), doi:10.103...

    2020 doi

  28. [36]

    Meurer, C

    A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Viget al.,Sympy: symbolic computing in python, PeerJ Computer Science3, e103 (2017), doi:10.7717/peerj-cs.103. 10 SciPost Physics Submissi...

    2017 doi

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