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Crossed Andreev reflection in collinear $p$-wave magnet/triplet superconductor junctions

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper predicts a magnetic-field-free junction in which crossed Andreev reflection is the only subgap transport channel.

desk verdict Promising new CAR platform with a transparent scattering calculation, but the paper currently contradicts itself on the zero-bias sum rule that the central claim rests on. read the letter →

arxiv 2501.13783 v3 pith:VIJHSQCC submitted 2025-01-23 cond-mat.mes-hall cond-mat.supr-con

Abhiram Soori This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.supr-con
keywords crossedAndreevreflectionp-wavemagnettripletsuperconductornonlocalconductanceLandauer-BüttikerscatteringCooperpairsplittingspin-splitbandstransversemomentumconservation
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 predicts that a junction made of a triplet superconductor sandwiched between two $p$-wave magnets rotated by $180^\circ$ can be pushed into a pure crossed Andreev reflection (CAR) regime. In that regime an electron incident from the left magnet emerges as a same-spin hole on the right magnet, while ordinary electron tunneling and local Andreev reflection are completely suppressed, so the local and nonlocal conductivities satisfy $G_{LL}+G_{RL}=0$ with $G_{RL}\neq0$. The selection comes from transverse momentum matching: because momentum along the interface is conserved, electron and hole states with matching $k_y$ determine which processes are kinematically allowed. This matters because CAR is the basis of Cooper-pair splitting and nonlocal spin entanglement, and prior ferromagnet-based routes to enhance it require external magnetic fields; the $p$-wave magnet bands are spin-split intrinsically. The predicted pure-CAR window spans orientation angles $0.11\pi<\phi<0.89\pi$ for the representative parameters used, and the effect weakens as the superconductor length exceeds the evanescent decay length.

What carries the argument

The key machinery is the spin-selective transverse momentum matching of the constant-energy contours. A $p$-wave magnet is a collinear magnetic material whose spin splitting is linear in momentum, with no net magnetization; here its dispersion contains a term $\alpha\hbar(\hat n_\phi\cdot\mathbf{k})\tau_0\sigma_z$ that changes sign between the two sides because of the $180^\circ$ rotation. Since translational invariance along the interface conserves $k_y$, an incoming spin-up electron from the left can only scatter into a spin-up hole on the right when their $k_y$ values coincide; within an orientation window, those electron-hole contours match while no same-spin electron-electron or electron-hole contours on one side match, so CAR survives and ET and AR are kinematically forbidden. The triplet superconductor supplies the equal-spin pairing term $\Delta k_x \tau_x\sigma_\theta/k_F$ that converts the electron into a same-spin hole. A Landauer-B\"uttiker scattering calculation with current-conserving boundary conditions turns this kinematic selection into quantitative differential conductivities $G_{LL}$ and $G_{RL}$.

What would settle it

A four-terminal experiment on a short triplet superconductor contacted by two p-wave magnet leads fixed at 180 degrees relative orientation, sweeping the chemical potential or the orientation angle, would falsify the central claim if, in the predicted window $0.11\pi<\phi<0.89\pi$, one finds $G_{LL}+G_{RL}\neq0$ at zero bias or $G_{RL}=0$; equivalently, a self-consistent numerical calculation with a spatially varying $\Delta$ near the interfaces that restores finite ET or AR would falsify the complete-suppression prediction.

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Extended reading notes

Core claim

The central discovery is that the $180^\circ$ relative rotation of the two $p$-wave magnets creates a kinematic blockade of all local subgap processes at zero bias. In the model, an electron of spin $s$ incident from the left has a matching $k_y$ with a spin-$s$ hole on the right over a wide range of orientations, so equal-spin triplet pairing in the superconductor converts it into a nonlocal hole; the same matching does not exist for spin-$s$ electron states on both sides or for same-spin electron-hole pairs on the left, so ET and AR are forbidden. With $\mu=-\mu_s$, $\alpha=1.5\,\hbar/(ma)$, $\Delta=0.1\mu_s$, $\theta=0$ and $L=5a$, the paper finds $G_{LL}+G_{RL}=0$ for $\phi$ between $0.11\pi$ and $0.89\pi$ at zero bias, with $G_{RL}$ negative and nonzero, i.e. CAR is the only channel. Away from this window the constant-energy contours of the two spins overlap and both ET and AR reappear; increasing $\mu$ beyond $0.283\mu_s$ also makes ET dominant. The dependence on $\theta$ between the triplet-pairing spin and the magnet easy axis traces the pairing symmetry: at $\theta=\pi/2$ the pairing is opposite-spin and $G_{RL}$ vanishes.

Load-bearing premise

The result assumes the superconducting pairing amplitude stays uniform all the way to clean, perfectly transparent interfaces, and that the p-wave magnets have the simple parabolic dispersion used in the model; if the pairing is suppressed near the contacts or the real band structure differs, the perfect suppression window may shrink or vanish.

Editorial extensions

If this is right

  • A magnetic-field-free, all-electrical route to pure CAR becomes available; the orientation angle and chemical potential are the primary tuning knobs.
  • In the pure-CAR window, the two-terminal differential conductances obey the sum rule $G_{LL}=-G_{RL}$, a direct transport signature that can be checked without noise measurements.
  • The angle $\theta$ of the equal-spin triplet pairing relative to the magnet easy axis acts as both a control and a probe: the nonlocal conductance should drop to zero at $\theta=\pi/2$, identifying the spin structure of the triplet order parameter.
  • CAR strength is set by the superconductor length, peaking near the decay length $1/\mathrm{Im}[k_{x1}]\simeq 14a$ and showing Fabry-P\'erot oscillations with period $\pi/\mathrm{Re}[k_{x1}]\simeq 2.22a$, so length can be used to tune between nonlocal and local transport.
  • Because the suppression is momentum-based rather than spin-filtering-based, it should persist over a range of interface barrier strengths $q$, as the calculated conductance curves show.

Reading between the lines

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

  • The same transverse-momentum selection should work in any junction where the two leads have oppositely oriented spin-split Fermi surfaces; an altermagnet or spin-orbit-coupled lead with matching contours could replace the $p$-wave magnet, though the precise parameter window would shift.
  • Because the complete suppression is kinematic, it likely survives moderate interface barriers but is sensitive to Fermi-surface warping; a realistic tight-binding or band-structure calculation for a candidate $p$-wave magnet would show whether the predicted orientation window persists.
  • If realized, the junction is a natural Cooper-pair splitter: the outgoing electron and hole are spatially separated, so measuring the shot-noise cross-correlation between the left and right leads could directly confirm nonlocal pairing transport beyond the conductance sum rule.
  • The exponential suppression of CAR with superconductor length suggests that the same setup in the long-superconductor limit could act as a local probe of midgap states rather than as a nonlocal splitter.
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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 / 4 minor

Summary. The paper proposes a two-dimensional junction in which a triplet superconductor is sandwiched between two p-wave magnets whose easy axes are rotated by 180 degrees. The author formulates a lattice-consistent scattering model with explicit Hamiltonian, dispersion, boundary conditions, and Landauer-Büttiker conductance formulas, then computes local and nonlocal differential conductivities as functions of bias, magnet orientation angle, chemical potential, superconductor length, and triplet pairing direction. The central claim is that for a range of orientation angles (e.g., 0.11π < φ < 0.89π at the chosen parameters) crossed Andreev reflection is the only transport channel: ET and AR are completely suppressed and GLL + GRL = 0. The manuscript also discusses Fabry-Pérot oscillations, decay-length effects, and the role of the triplet pairing angle.

Significance. If the central claim is correct, the setup offers a magnetic-field-free platform in which crossed Andreev reflection dominates completely, which would be a practically useful step toward Cooper-pair splitting. The formalism is explicit enough to be re-implemented: the Hamiltonian, dispersion, boundary conditions, and conductance integrals are all stated with defined parameters and units, and the predictions are falsifiable through the quoted parameter windows. The paper also honestly acknowledges that proximity-induced suppression of the pairing amplitude is not modeled. However, the manuscript contains a direct internal contradiction about the zero-bias sum rule that bears on the central claim, and the claimed robustness to pairing suppression is not quantitatively supported. At present the result is not self-consistently established.

major comments (2)
  1. [Section IV (discussion of Fig. 4(b) and paragraph before Fig. 6)] The manuscript contains two mutually exclusive statements about the zero-bias sum rule. The paragraph discussing Fig. 4(b) states that in the range 0.11π < φ < 0.89π the local and nonlocal conductivities sum to zero, and that outside this range AR and ET are both allowed. The paragraph immediately before Fig. 6 states that 'at zero bias, GLL + GRL = 0 for all φ, indicating the absence of AR at the left interface.' These statements cannot both be true for the same calculation. If the sum rule holds for all φ, then the finite pure-CAR window is not special and the Fig. 4(b) discussion is incorrect; if the sum rule holds only in the window, the 'for all φ' sentence is an error. The same later paragraph adds that at nonzero bias ET is 'completely suppressed,' which also contradicts the earlier explanation that ET is allowed outside the orientation window. Since no code, raw data, or benchmark results are provided, the reader cannot determine which statement corresponds to the actual numerical solution. This contradiction must be resolved, and the precise zero-bias sum-rule statement realized by the numerics must be stated unambiguously.
  2. [Section V (first paragraph)] The paper explicitly states that suppression of the superconducting pairing amplitude near the interface is not modeled and argues that this does not qualitatively affect the results. This robustness claim is load-bearing because the pure CAR window relies on coherent subgap transport through a short superconductor (L = 5a, with the quoted decay length 14a) and on exact transverse-momentum matching in the ideal p-wave magnet dispersion. For a realistic proximity profile, the order parameter is suppressed over a finite length near each contact; if this length is a substantial fraction of L = 5a, the complete ET/AR suppression could be lifted. The manuscript provides no estimate of the suppression length or a calculation with a spatially varying Δ(x) to support the assertion that the qualitative conclusions survive. Please either model the suppression explicitly or state quantitatively the regime in which the predicted pure-CAR window remains valid.
minor comments (4)
  1. [Section IV (paragraph before Fig. 6)] The phrase 'the limit when the superconductor is very long L ≳ a' appears to be a typo; given the earlier decay length of 14a, the intended condition for the zero-bias peak in local conductivity is likely L ≳ 14a or L >> a, not L ≳ a.
  2. [Fig. 4(b)] The curves for GLL and GRL are not clearly labeled in the figure; the text mentions 'local and nonlocal conductivities' but the plot would benefit from explicit line-style labels in the caption.
  3. [Section III.E, Eqs. (7)-(9)] The current formulas include the incident current separately from the reflected-current term; it would be helpful to state explicitly that the sign convention for GRL allows negative values, since the text refers to 'nonlocal conductivity becomes positive near φ = 0' without defining the sign convention.
  4. [Section I, last paragraph] The sentence 'Within the selected range of orientations, no electron states with the same spin are available on both sides of the superconductor, preventing ET' is stated before the quantitative window is derived; please provide a cross-reference to the Fig. 4(b) discussion so the reader connects the qualitative argument to the numerical result.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the pure-CAR prediction is computed from an explicit scattering model, not fitted to or defined by the target result.

full rationale

The paper's derivation chain is self-contained. Starting from the model Hamiltonian in Eq. (1), the dispersions in Eqs. (2)-(4), the current-conserving boundary conditions in Eq. (5), and the scattering eigenfunctions in Eq. (6), the conductivities in Eqs. (7)-(9) are computed via Landauer-Büttiker scattering theory. No parameter is fitted to produce the claimed pure-CAR window; the result emerges from scanning the orientation angle ϕ and other parameters. Self-citations (e.g., Refs. [15], [22], [32]-[34]) provide methodological context or the lattice-model interpretation of the boundary parameter c, but none is load-bearing for the central prediction, and no uniqueness theorem or ansatz is imported from the author's earlier work. The explicit limitations in Sec. V (e.g., ignoring interface pairing suppression) are modeling assumptions, not circular reductions. The apparent conflict between the '0.11π<ϕ<0.89π' statement and the later 'GLL+GRL=0 for all ϕ' sentence in Sec. IV is an internal consistency or correctness issue, not a case of a prediction equaling its input by construction.

Assumptions & free parameters 8 free parameters · 6 assumptions · 0 invented entities

The model rests on the p-wave magnet Hamiltonian and the triplet pairing form from prior literature, plus standard scattering theory. The listed parameters are model inputs chosen by hand to expose the CAR regime; they are not constants fitted to experimental data. No new particles, forces, or conserved quantities are introduced; the novelty is in the junction geometry and the predicted transport regime.

free parameters (8)
  • p-wave magnet coupling alpha = 1.5 hbar/(m a)
    Chosen to make the two spin Fermi contours well separated; the central CAR regime depends on this separation.
  • chemical potentials mu and mu_s = mu = -mu_s (mu_s is the energy unit)
    Chosen to keep the spin-split contours separated; Fig. 5(a) shows the behavior changes for mu > 0.283 mu_s.
  • superconducting gap Delta = 0.1 mu_s
    Chosen to model a small gap relative to the chemical potential, typical of weak-coupling superconductors.
  • triplet spin orientation theta = 0 for the main results
    theta = 0 gives equal-spin pairing along +/- z, the case with largest CAR; theta = pi/2 gives zero CAR in the model.
  • magnet orientation angle phi = varied; pi/2 at the reference point
    The central control parameter; the pure CAR window 0.11 pi < phi < 0.89 pi is defined in terms of it.
  • superconductor length L = 5 a
    Short enough for subgap transmission; CAR decays when L exceeds the estimated decay length of about 14 a.
  • barrier parameter q = 0 in the main results
    Delta-function barrier strength at the interface set to zero for the main curves; q reduces CAR as shown in Fig. 4(a).
  • junction transparency c = 1
    Boundary condition parameter representing hopping strength; set to model transparent contacts.
assumptions (6)
  • standard math Landauer-Buttiker scattering formalism with probability current conservation determines the conductances.
    Used in Section III to convert scattering amplitudes into GLL and GRL.
  • domain assumption The p-wave magnet is described by -i alpha hbar (n_phi dot grad) tau_0 sigma_z with a spin-split, time-reversal-invariant dispersion Eq. (2).
    This is the p-wave magnet model from Refs. [5-9]; the 180 degree rotation between the two magnets enters through gamma = +/- 1 in Eq. (3).
  • domain assumption The superconductor pairing is taken as Delta k_x tau_x sigma_theta / k_F, a momentum-projected triplet pairing from Ref. [32].
    This restricts the general triplet order parameter to one momentum component; the theta dependence is the basis for probing pairing symmetry.
  • standard math ky is conserved across the junction because of translational invariance along y.
    Used to construct the scattering states in Eq. (6); the momentum-matching argument in Section II relies on it.
  • domain assumption Interfaces are modeled by Eq. (5) with parameters c and q, and no suppression of the superconducting order parameter near the boundary.
    Section V explicitly states that pairing suppression is neglected; c = 1 and q = 0 are used for the main results.
  • standard math Evanescent waves in the leads carry no current, so the beta factors in Eqs. (8)-(9) exclude them from the conductances.
    Standard transport boundary-condition treatment for decaying wavefunctions in the normal leads.

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Pith. "Pith review of Crossed Andreev reflection in collinear $p$-wave magnet/triplet superconductor junctions." pith.science (2026). https://pith.science/paper/VIJHSQCC

@misc{pith2026250113783,
  author       = {Pith},
  title        = {Pith review of: Crossed Andreev reflection in collinear $p$-wave magnet/triplet superconductor junctions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VIJHSQCC}},
  note         = {Machine review of arXiv:2501.13783}
}
abstract

Crossed Andreev reflection (CAR) is a fundamental quantum transport phenomenon that holds significant implications for spintronics and superconducting devices. However, its experimental detection and enhancement remain challenging. Recently, magnetic materials exhibiting $p$-wave magnetic ordering, distinct from conventional spin-orbit coupling, referred to as $p$-wave magnets, have attracted considerable interest. In this work, we propose a junction consisting of $p$-wave magnets and a triplet superconductor as a promising platform to enhance CAR. The setup features a triplet superconductor sandwiched between two collinear $p$-wave magnets rotated by $180^\circ$ relative to each other, allowing for precise control over transport processes. We demonstrate that CAR can dominate over electron tunneling (ET) within specific parameter regimes, such as the orientation angle of the $p$-wave magnets and their chemical potential. Enhanced CAR occurs when the constant energy contours of the two spins in the $p$-wave magnets are well-separated. Furthermore, the conductivities display Fabry-P\'erot-type oscillations due to interference effects, with CAR diminishing as the length of the superconductor exceeds the decay length of the wavefunctions. These findings underscore the potential of collinear $p$-wave magnet-superconductor junctions as a robust platform for the experimental investigation and enhancement of CAR.

Figures

Figures reproduced from arXiv: 2501.13783 by the authors.

Figure 1
Figure 1. FIG. 1. Bandstructure of a [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Setup: Two [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 4
Figure 4. (b), the conductivities are plotted as functions of ϕ with the bias fixed near zero. It is observed that in the range 0.11π < ϕ < 0.89π, the local and nonlocal conductivities sum-up to zero, indicating absence of AR. Outside this range, same-spin electron-hole pairs with -1 -0.5 0 0.5 1 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 (a) 0 0.25 0.5 0.75 1 / -1 -0.5 0 0.5 1 1.5 (b) FIG. 4. Conductivities GLL and GRL versus: (a) bias V … view at source ↗
Figures from the paper (1 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Conductivities at zero bias versus: (a) [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

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

  1. Majorana flat bands and anomalous proximity effects in $p$-wave magnet--superconductor hybrid systems

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    2023 arXiv

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.