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Persistent spin currents in superconducting altermagnets

T0 review · 2 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper argues that superconducting altermagnets support persistent spin currents because their leading superconducting state consists of two independent equal-spin triplet condensates, one for spin-up and one for spin-down electrons.

desk verdict Solid, internally consistent blueprint for persistent spin currents in superconducting altermagnets, but the 'almost certainly' claim about triplet pairing overreaches the two toy models tested. read the letter →

arxiv 2507.22139 v1 pith:2DDWNCKT submitted 2025-07-29 cond-mat.supr-con cond-mat.str-el

Kyle Monkman , Joan Weng , Niclas Heinsdorf , Alberto Nocera , Marcel Franz This is my paper
classification cond-mat.supr-concond-mat.str-el
keywords altermagnetismspinsupercurrentsuperconductivityspin-tripletpairingchiralp-wavepersistentcurrentsspintronicsGinzburg-Landautheory
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

Superconducting altermagnets, metals with zero net magnetization but spin-split Fermi surfaces, can support persistent spin currents, the paper argues. The reason is that their leading superconducting instability is a pair of independent equal-spin triplet condensates, one formed from spin-up and one from spin-down electrons, which remain decoupled in the absence of spin-orbit coupling. Because each condensate carries its own supercurrent, the two can flow in opposite directions and produce a pure spin supercurrent with no net charge transport; in geometries like the half-flux ring this state is the global free-energy minimum, so the spin current is persistent in the same sense as a charge supercurrent. The paper also shows that spin-orbit coupling and magnetic disorder, which normally destroy spin transport, only make the spin current oscillate in space without dissipating it, and it proposes a 'spin-current dynamo' geometry in which an ordinary charge current generates a pure spin supercurrent.

What carries the argument

The load-bearing object is the two-condensate order parameter $(\psi_\uparrow,\psi_\downarrow)$ with phases $\varphi_\uparrow,\varphi_\downarrow$, governed by a Ginzburg-Landau free energy whose kinetic terms are $\gamma \psi_0^2 (\nabla\varphi_\sigma - (2e/\hbar c)\mathbf{A})^2$ and whose spin-orbit coupling produces a Josephson term $-J\cos(\varphi_\uparrow-\varphi_\downarrow)$ with $J \propto \lambda_R^2$. The spin current $j_s = j_\uparrow - j_\downarrow$ is set by the relative phase gradient, while the charge current is set by the average phase gradient; the Josephson coupling converts the relative-phase equation into an undamped pendulum equation $\varphi'' + k_J^2 \sin\varphi = 0$, whose non-decaying oscillatory solutions are the mathematical reason spin currents persist under spin-orbit coupling. In d-wave altermagnets the superfluid density tensor $\hat{\rho}_\sigma = \rho_0 + \sigma \rho_\eta \operatorname{diag}(1,-1)$ has opposite anisotropy for the two spins, producing $j_s = (\rho_\eta/\rho_0)\, j_e$ along principal axes and a transverse spin current for diagonal drive.

What would settle it

A gap-structure measurement on a superconducting altermagnet that detects singlet pairing, nodal gap, or finite-momentum FFLO order would falsify the two-condensate basis. More directly, a thin-ring experiment that scans flux near $\Phi_0/2$ and finds no near-zero-magnetization plateau, or a long-wire experiment that sees the spin supercurrent decay with distance, would falsify the persistent-spin-current claim.

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Extended reading notes

Core claim

The paper's central claim is that a superconducting altermagnet naturally realizes two coexisting equal-spin triplet $p_x \pm i p_y$ condensates, $\Delta_\uparrow$ and $\Delta_\downarrow$, whose phases are independent degrees of freedom in the absence of spin-orbit coupling. Because the altermagnetic band splitting is pair-breaking for conventional spin-singlet pairing, the leading instability is odd-parity equal-spin triplet order even when the attraction is a weak conventional phonon-mediated interaction; the gap equations show triplet $T_c$ survives while singlet $T_c$ is rapidly suppressed once the splitting $\eta$ exceeds the singlet gap scale. From this the paper derives persistent spin-polarized supercurrents, pure spin supercurrents in the charge counterflow regime, a spin-current dynamo effect in d-wave altermagnets, and a thin-ring ground state with pure spin current near half a flux quantum. Away from the non-relativistic limit, spin-orbit coupling introduces a Josephson coupling between the two condensates that makes the spin current oscillate spatially with constant amplitude; the current does not decay, in contrast to spin currents in normal metals.

Load-bearing premise

The argument depends on the claim that the leading superconducting instability of every altermagnet is equal-spin triplet $p_x \pm i p_y$ pairing, but that claim is demonstrated only for two model Hamiltonians with nearest-neighbor attraction before being generalized to any altermagnet; if a real material instead pairs in the singlet channel or in a finite-momentum FFLO state, the two-condensate description and all persistent spin-current results built on it collapse.

Editorial extensions

If this is right

  • Any spin polarization injected into a superconducting altermagnet wire persists over arbitrarily long distances in the non-relativistic limit, because the two condensates conserve spin separately.
  • Driving a charge current along the (1,1) direction of a d-wave altermagnet generates a pure spin supercurrent along (1,-1), with magnitude set by $\rho_\eta/\rho_0 \sim \eta/t$, a mechanism the paper calls the spin-current dynamo effect.
  • A thin altermagnet ring threaded by half a flux quantum selects the charge counterflow state $(n_\uparrow,n_\downarrow)=(1,0)$ or $(0,1)$ as its global free-energy minimum, so a pure spin supercurrent flows while the measured magnetization nearly vanishes.
  • With spin-orbit coupling, spin supercurrents acquire spatial oscillations of period $\sim 2\pi \xi (\Delta_0/\lambda_R) k_F^{-2}$ but no decay; the critical injected spin current separating oscillating and rotating regimes is $j_s^c \simeq j_e^c (\lambda_R/\Delta_0)\sqrt{3} k_F^2$.
  • g-wave altermagnets cannot generate spin current by the dynamo effect because their symmetry enforces $\rho_\eta=0$, but they still support persistent spin currents; strain can restore the generation asymmetry.

Reading between the lines

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

  • The same two-condensate phase dynamics imply a spin analogue of the Josephson effect: relative-phase winding between $\Delta_\uparrow$ and $\Delta_\downarrow$ should be controllable and observable in ring or junction geometries, a testable extension the paper does not develop.
  • If the leading-instability claim survives in real materials, the search for intrinsic superconducting altermagnets can focus on good metals among known altermagnets regardless of pairing mechanism, since conventional phonon attraction is claimed sufficient.
  • The predicted near-zero magnetization plateau around $\Phi=\Phi_0/2$ offers a detection route that avoids direct spin-current measurement and could be tried first in proximity-engineered altermagnet rings.
  • A possible loophole the paper leaves open: a finite-momentum FFLO state would be an alternative to zero-momentum triplet pairing, and the two platforms would be hard to distinguish by magnetization alone.
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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 / 6 minor

Summary. This paper argues that superconducting altermagnets constitute a platform for persistent spin currents. Starting from two microscopic lattice models (a d-wave and a g-wave altermagnet), the authors solve BdG gap equations and conclude that for sufficiently large altermagnetic splitting η the leading pairing instability is equal-spin triplet p_x±i p_y, giving two decoupled condensates with independent U(1) symmetries in the non-relativistic limit. From this they derive a superfluid-density anisotropy leading to a spin-current dynamo effect, a Ginzburg-Landau theory with a microscopically computed Josephson coupling J between the two condensates, and a ring geometry in which half-integer flux stabilizes pure spin supercurrent. They support the GL predictions with self-consistent BdG simulations in strips and rings, including the effects of SOC and magnetic disorder, and report that spin currents oscillate but do not decay.

Significance. The two-condensate picture is physically appealing and, if correct, would make superconducting altermagnets a zero-magnetization dissipationless spin-transport platform with a concrete experimental fingerprint (vanishing magnetization near half-integer flux). The paper's strengths include self-consistent BdG simulations with self-consistently determined order parameters, a microscopic calculation of the Josephson coupling J in Appendix B, the pendulum analogy that cleanly explains why SOC produces oscillations rather than decay, and explicit disorder tests in both BdG and GL settings. The results are, however, conditional on the claim that the leading superconducting instability is zero-momentum equal-spin triplet pairing, and that claim is only verified in two toy models under a restricted mean-field ansatz.

major comments (2)
  1. [Sec. II.C, Eqs. (8), (14a)-(14b)] The central premise of the paper is that the leading superconducting instability of altermagnets is equal-spin triplet p_x±i p_y with zero center-of-mass momentum. The evidence for this is the solution of the gap equations (14a)-(14b) for two lattice models, but the calculation is restricted by the ansatz (8) to uniform chiral p-wave order and a competing extended s-wave order. Finite-momentum FFLO pairing, which the authors cite as a possible alternative in the Outlook (Ref. [68]) and which has been argued to compete with zero-momentum pairing in altermagnets, is not included in the free-energy comparison. Because all of the persistent spin-current phenomena derived in Sections III and IV assume two decoupled zero-momentum condensates, this omission is load-bearing. I request that the authors either carry out a stability analysis of FFLO in the same models or explicitly reformulate the conclusions as conditional on the absence of a competing finite-momentum or singlet instability.
  2. [Sec. II.C, final paragraph] The sentence 'this leads to a firm conclusion that superconductivity in any altermagnet will almost certainly be in the equal-spin triplet channel' overstates what is demonstrated. The numerical evidence in Fig. 2 is obtained for two specific Hamiltonians with one representative attractive interaction and filling each; the generalization to arbitrary altermagnetic band structures, interaction ranges, and multi-band materials is an extrapolation. The qualitative mechanism (singlet pair breaking by spin splitting, triplet robustness) is credible, but the manuscript should state the precise conditions under which the conclusion holds and identify material-specific effects that could overturn it.
minor comments (6)
  1. [Sec. IV.B] The text refers to the 'BdG Hamiltonian Eq. (10)', but Eq. (10) defines C_k and S_k; the BdG Hamiltonian is Eq. (11).
  2. [Figs. 5 and 6] Figures 5 and 6 contain stray 'Out[]=' artifacts in the axis labels or captions, apparently from a notebook export; these should be removed.
  3. [Sec. III.B] The paper reports that the numerical slope in Fig. 3(c) is about a factor of two smaller than the quadratic-dispersion estimate; a brief comment on the origin of this deviation (e.g., finite gap or lattice corrections) would be useful.
  4. [Sec. IV.B] The boundary condition φ(0)=0 used for the pendulum solutions is not physically motivated; the text should explain what experimental setup fixes the initial relative phase.
  5. [Sec. IV.C] The discussion of branch switching in Fig. 7(b) does not address phase-slip dynamics; the statement that the system 'will switch' should be phrased as an equilibrium statement, since the static London theory does not describe the switching kinetics.
  6. [Eq. (39)] The numerical estimate following Eq. (39) uses k_F=0.5 and ξ/λ_L ≈ 1; given the sensitivity of λ_Rc to these inputs, the value λ_Rc ≈ 0.3Δ0 should be labeled an order-of-magnitude estimate.

Circularity Check

0 steps flagged · score 1.0 of 10

No material circularity: the central instability, spin-current relations, and Josephson coupling are derived from the stated models and verified by self-consistent BdG calculations, not fitted or assumed.

full rationale

The paper's derivation chain is self-contained. The central premise, that the leading superconducting instability of the altermagnetic models is equal-spin triplet px±ipy pairing, is established by solving the gap equations (14a)-(14b) for the d-wave and g-wave Hamiltonians with a specified nearest-neighbor attraction, and the free-energy comparison in Sec. II.C is a model calculation rather than an import. The spin-current relations, including Eqs. (15)-(19) and the dynamo relation js = (rho_eta/rho_0) tau^z je, follow algebraically from the symmetry-constrained superfluid-density tensor and are then checked numerically against self-consistent BdG results; the numerical agreement is a consistency check, not a fit. The Josephson coupling J used to predict oscillation periods and critical currents is computed microscopically in Appendix B from the BdG free energy, so the later GL predictions are not tuned to the outcomes. Several self-citations appear, e.g., Refs. [30,32] for the Rashba-selected ground state and Refs. [69,70] for the superfluid-density expression, but these are not load-bearing: the ground-state choice is stated to be insensitive for most spin-current results, and the superfluid-density formula is a standard microscopic expression. The acknowledged omission of finite-momentum FFLO competition in the numerical comparison is a correctness/scope risk rather than a circularity, because the paper explicitly computes what it claims within the stated ansatz and does not redefine the outcome as its input. Overall, no predicted result reduces by construction to its own assumptions.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The central claim is built from model Hamiltonians with hand-chosen parameters (η, μ, V1, λ, m0), a single-band approximation, and the assumed triplet instability. There are no fitted experimental constants; the quantitative predictions are computed from the models rather than tuned to measured data. The main external assumption is that real altermagnets superconduct intrinsically with the modeled pairing.

free parameters (5)
  • altermagnetic splitting η = η=0.2 for Fig. 1; scanned to η=2.0
    Model parameter set by hand; the central claim requires η > Δ0 so that the triplet channel dominates (Sec. II.C).
  • chemical potential μ = μ=-2.6t (d-wave), -2.0t (g-wave)
    Representative band fillings chosen for gap-equation solutions and numerical calculations.
  • attractive interaction V1 = V1=1.93t (d-wave), 1.95t (g-wave)
    Weak nearest-neighbor attraction chosen to give a superconducting instability; the Tc curves and η threshold depend on it.
  • Rashba/Dresselhaus SOC strength λ = λ_R=0.15t in Fig. 6; scanned in calculations
    Strength of SOC relative to Δ0 controls the Josephson coupling J and the spin-current oscillation period; not fitted to experimental data.
  • magnetic disorder amplitude m0 = m0=0, 0.25t, 0.50t
    Random local exchange field strength chosen to test robustness; m0=0.50 exceeds the SC gap Δ0≈0.42.
assumptions (7)
  • standard math BdG mean-field decoupling of the attractive interaction with singlet and equal-spin triplet order parameters
    Used in Sec. II.C to derive the gap equations (13)-(14).
  • domain assumption Single-band effective model with altermagnetic splitting encoded in spin-dependent hopping, neglecting sublattice and interband effects
    Justified in Sec. II.A by Néel temperatures being orders of magnitude larger than SC energy scales; invoked for both d-wave and g-wave models.
  • domain assumption The leading SC instability of an altermagnet with weak attraction is equal-spin triplet px±ipy pairing when η > Δ0
    Established by model gap equations in Sec. II.C and prior Refs. [29-34]; this premise underlies the two-condensate description.
  • domain assumption Non-relativistic limit has U(1)↑ × U(1)↓ symmetry, so spin-up and spin-down condensates are decoupled
    Assumed throughout Secs. III-IV; SOC breaks this symmetry and introduces the J coupling.
  • domain assumption Helical p↑− ⊗ p↓+ state is the selected ground state in the presence of weak Rashba SOC
    Taken from Refs. [30,32]; the authors note most spin-current results are insensitive to the ground-state choice.
  • standard math London approximation with constant order-parameter amplitude is valid for length scales long compared to the coherence length
    Used in Sec. IV.A to reduce GL theory to the phase-only free energy in Eq. (25).
  • domain assumption Superconducting altermagnets with intrinsic triplet order exist in practice
    No superconducting altermagnet has been observed; the paper argues many altermagnets are good metals and likely to superconduct at low temperatures.

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Cite this review

Pith. "Pith review of Persistent spin currents in superconducting altermagnets." pith.science (2026). https://pith.science/paper/2DDWNCKT

@misc{pith2026250722139,
  author       = {Pith},
  title        = {Pith review of: Persistent spin currents in superconducting altermagnets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2DDWNCKT}},
  note         = {Machine review of arXiv:2507.22139}
}
read the original abstract

Superconductors are famously capable of supporting persistent electrical currents, that is, currents that flow without any measurable decay as long as the material is kept in the superconducting state. We introduce here a class of materials -- superconducting altermagnets -- that can both generate and carry persistent {\em spin} currents. This includes spin-polarized electrical supercurrent as well as pure spin supercurrent that facilitates spin transport in the absence of any charge transport. A key to this remarkable property is the realization that the leading superconducting instability of altermagnetic metals consists of two independent condensates formed of spin-up and spin-down electrons. In the non-relativistic limit the two condensates are decoupled and can thus naturally support persistent currents with any spin polarization, including pure spin supercurrents realized in the charge counterflow regime. We describe a novel ``spin-current dynamo effect'' that can be used to generate pure spin supercurrent in such systems by driving a charge current along certain crystallographic directions. Away from the non-relativistic limit, when spin-orbit interactions and magnetic disorder are present, we find that the spin current generically develops spatial oscillations but, importantly, no dissipation or decay. This is in stark contrast to spin currents in normal diffusive metals which tend to decay on relatively short lengthscales. We illustrate the above properties by performing model calculations relevant to two distinct classes of altermagnets and various device geometries.

Figures

Figures reproduced from arXiv: 2507.22139 by the authors.

Figure 1
Figure 1. FIG. 1. Spin split Fermi surfaces of (a) [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Superconducting critical temperatures in singlet [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Spin-polarized supercurrent generation in a [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Spin-current dynamo effect: Generation of pure spin supercurrent from perpendicular charge current. Panels (a) and [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Evolution of a spin current [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Effect of Rashba SOC on pure spin supercurrent in a [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. (a) A SC ring threaded by magnetic flux Φ. Red [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Spin current in a long wire calculated from the GL [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]

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Forward citations

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