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Exotic superconducting states in altermagnets

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Altermagnets host symmetry-allowed superconducting states with spin-up and spin-down condensates that pair with different anisotropies; one has a Josephson current with 4π periodicity in the spin-up channel and 2π in the spin-down channel.

desk verdict The symmetry tabulation is a real contribution; the physical-realization claims are conditional on GL couplings that are assumed, not derived. read the letter →

arxiv 2507.10700 v1 pith:QO5KKY3I submitted 2025-07-14 cond-mat.supr-con cond-mat.mes-hallcond-mat.str-el

Kirill Parshukov , Andreas P. Schnyder This is my paper
classification cond-mat.supr-concond-mat.mes-hallcond-mat.str-el
keywords altermagnetismspin-pointgroupsunconventionalsuperconductivitynon-unitarypairingspin-tripletJosephsoneffectMajoranamodesGinzburg-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

The paper classifies all symmetry-allowed superconducting pairing states in altermagnets, the recently discovered compensated collinear magnets with spin-split bands, using irreducible co-representations of spin-point groups in the limit of negligible spin-orbit coupling. It argues that this framework uncovers pairing states that ordinary magnetic point-group classification misses, most notably a non-unitary “altermagnetic superconductor” whose spin-up and spin-down condensates have different spatial anisotropies. The paper also identifies a half-and-half metal-superconductor, in which only one spin species forms Cooper pairs, and spin chiral superconductors with spin-polarized edge modes. A sympathetic reader would care because the central predicted consequence is a measurable, spin-resolved fractional ac Josephson effect: in a junction of two altermagnetic superconductors, the spin-up current has 4π periodicity while the spin-down current has 2π periodicity.

What carries the argument

The central object is the irreducible co-representation (corep) of the spin-point group, the symmetry group of a collinear compensated magnet with negligible spin-orbit coupling, written as $G_S = X + X(T2^\sigma\|E)$ with unitary halving subgroup $X = X_{1/2} + (2^\sigma\|g)X_{1/2}$ and $X_{1/2} = SO(2)\times H$. The pairing functions are obtained by two complementary routes, unitary-group projectors combined with Dimmock indicators and direct corep projectors, and the two methods agree. The property that carries the argument is that the unique coreps come in spin-polarized triplets $D_\pm = (1, \pm i, 0)^T\gamma_\pm(k)$ related by the altermagnetic symmetry $(2^\sigma\|g)$, so a superposition can have different spatial form factors in the two spin channels; four- and six-dimensional coreps allow relative phases that break the antiunitary $(T2^\sigma\|E)$ symmetry and create chiral states. The Ginzburg-Landau free energy is built by decomposing products of coreps into trivial components, and minimization with signs such as $\beta_2<0$ selects the altermagnetic superconductor as a stable state.

What would settle it

Compute the quartic Ginzburg-Landau coefficients from a microscopic model for a candidate altermagnet such as κ-(BEDT-TTF)2Cu[N(CN)2]Br or CrSb and ask whether $\beta_2<0$ and the side conditions in the Supplemental Material hold; alternatively, voltage-bias a junction of two altermagnetic superconductors and look for a spin-resolved ac current whose spin-up component has $4\pi$ periodicity in the phase while the spin-down component has $2\pi$ periodicity.

Watch

Extended reading notes

Core claim

The central claim is that when spin-orbit coupling is negligible, the correct symmetry framework for a compensated collinear magnet is the spin-point group rather than the magnetic point group, because spin and lattice operations act independently. Using irreducible co-representations of these groups, the paper constructs and tabulates all symmetry-allowed pairing functions for altermagnets. Three states emerge that magnetic point-group classification would miss: (i) an “altermagnetic superconductor” with pairing $d(k) = D_+(r_x k_x + r_y k_y) + e^{i\xi}D_-(-r_x k_x + r_y k_y)$, in which the $S=+1$ and $S=-1$ condensates have different spatial anisotropies, making the order parameter non-unitary with $i\,d(k)\times d(k)^* = 8 r_x r_y k_x k_y$; (ii) a half-and-half metal-superconductor where only one spin direction pairs; and (iii) a spin chiral superconductor whose relative phase breaks the antiunitary $(T2^\sigma\|E)$ symmetry, giving a Chern number in each spin subspace. For the altermagnetic superconductor the paper computes a fractional ac Josephson current, with spin-up current $I_\uparrow = -(\pi\Delta_0/4)(2e/\hbar)\sin(\varphi/2)$ having 4π periodicity while the spin-down current has 2π periodicity, tied to spin-polarized Majorana boundary modes in one spin channel only. The paper shows these states are stable minima of Ginzburg-Landau free energies and treats small spin-orbit coupling as a perturbation that lifts the remaining phase degeneracies.

Load-bearing premise

The paper's exotic states are stable only under hand-chosen signs of the quartic coefficients in the Ginzburg-Landau functional (notably $\beta_2<0$ for the altermagnetic superconductor and $\beta_5<0$ with side conditions for the spin chiral state), and those coefficients are never derived from a microscopic Hamiltonian; if the true signs are opposite, the headline states are not realized and the Josephson prediction does not apply.

Editorial extensions

If this is right

  • If the altermagnetic superconductor forms, its non-unitary order parameter should produce spin-polarized zero-energy Majorana boundary modes wherever the two spin channels' nodes are separated on a sample edge.
  • A voltage-biased junction of two such superconductors should carry an ac Josephson current with a $4\pi$-periodic spin-up part, $I_\uparrow=-(\pi\Delta_0/4)(2e/\hbar)\sin(\varphi/2)$ in the transparent limit, and a $2\pi$-periodic spin-down part.
  • The half-and-half metal-superconductor would open a gap on only one spin Fermi surface while the opposite spin remains metallic, effectively a superconductor and a half-metal coexisting in the same crystal without applying a magnetic field.
  • Spin chiral states from four- or six-dimensional coreps would give integer Chern numbers in each spin subspace and chiral edge modes in two dimensions, while the altermagnetic-symmetry-preserving state belongs to class AIII with spin-polarized nodes protected by winding numbers in one and three dimensions.

Reading between the lines

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

  • Because the $4\pi$/$2\pi$ splitting is tied to spin rather than to an external magnetic field, the spin-resolved Josephson current offers a zero-field fingerprint that could distinguish the altermagnetic superconductor from ordinary triplet or chiral superconductors.
  • A natural next step the paper does not take is to compute the quartic Ginzburg-Landau coefficients microscopically; that calculation would not only test whether the altermagnetic superconductor is realized but would also map out which sign regime favors the half-and-half state over it, with the tabulated basis functions remaining the correct language in either case.
  • The half-and-half state is conceptually a bulk analogue of a superconductor/half-metal interface, so it may exhibit unusual Andreev reflection or spin-transport signatures at contacts, a direction the paper mentions only through topology and the Josephson response.
  • The complete tables of corep basis functions across the spin-point groups of compensated collinear magnets should let experimentalists screen candidate altermagnets for which pairing symmetries are symmetry-allowed before any microscopic calculation, and the construction could be extended to treat finite spin-orbit coupling beyond perturbation theory.
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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. This paper studies the symmetry-allowed superconducting pairing states of altermagnets—collinear compensated magnets with negligible spin-orbit coupling—using irreducible co-representations (coreps) of spin-point groups. The authors tabulate pairing basis functions for all 58 spin-point groups describing collinear compensated magnetic order, using two independent projector constructions that are reported to give identical results. They focus on three states: a non-unitary 'altermagnetic superconductor' with different spatial anisotropies for the S=+1 and S=−1 triplet condensates (d(k)=D+(rxkx+ryky)+eiξD−(−rxkx+ryky)); a 'half-and-half metal-superconductor' in which only one spin species pairs; and a 'spin chiral superconductor' arising from four-dimensional coreps that breaks the antiunitary (T2∥E) symmetry. Each state is analyzed with a phenomenological Ginzburg-Landau free energy, a topological classification (classes A and AIII, Table I), and physical consequences including spin-polarized Majorana boundary modes and a spin-resolved ac Josephson effect with a 4π-periodic spin-up current and a 2π-periodic spin-down current (Eq. (10)). A tight-binding model of the organic altermagnet κ-(BEDT-TTF)2Cu[N(CN)2]Br is presented as a candidate realization.

Significance. The complete corep-based classification is a potentially valuable resource for the field of altermagnet superconductivity, and the agreement between the two independent projector schemes (Approaches 1 and 2, Fig. 1) is a genuine internal consistency check on the tabulation. If the analysis is correct, the paper establishes that the spin-point-group framework, rather than conventional magnetic point groups, is the appropriate symmetry setting for pairing in altermagnets, and it produces concrete falsifiable predictions: spin-polarized zero-energy boundary modes, the spin-asymmetric 4π/2π ac Josephson response, and candidate materials. The GL theory is honestly labeled as phenomenological, and the topological classification is standard. The principal reservations are not about the classification itself but about the completeness of the GL stability conditions and the ungrounded parameter assumptions behind the material-specific predictions, which are detailed below.

major comments (2)
  1. [SM §SIV, §SVI (Eqs. (7), (S50)-(S54))] The stability conditions for the headline states are stated incompletely. (a) For the two-component functional Eq. (7), the solution |φ1|=|φ2|≠0 requires not only β2<0 but also 4β1+β2>0 for the free energy to be bounded and the amplitude to be real, and it is the global minimum only for β2<−β1; for −β1<β2<0 the boundary minimum (1,0)/(0,1) has lower free energy. The claim in SM §SIV that 'If β2<0 there is a minimum at |φ1|=|φ2|' omits these conditions. (b) For the four-component functional, the inequalities (S50)-(S53) do not imply positivity of the denominator 8β1+β2+β3+β4+β5 in Eq. (S54), so the amplitude can be imaginary within the stated parameter range; for example, (β1,β2,β3,β4,β5)=(1,−100,−100,−100,−1) satisfies all stated inequalities while giving a negative denominator. The Hessian and global-boundedness checks for the claimed minimum are also not shown. Since 'these correspond to stable minima of the free energies' is a central assertion of the paper, the complete stability conditions, a comparison of all candidate minima, and a proof that the quartic functional is bounded below in the stated range should be provided.
  2. [Material example and Eqs. (7), (10); SM §SVIII] The physical realization and the headline Josephson prediction are conditional on GL parameters that are never derived or estimated. The (1,1) altermagnetic superconducting state wins over the half-and-half state only for β2<0 (and, per the analysis above, β2<−β1); the spin chiral state requires β5<0 together with (S50)-(S53); and the phase-locked states require αSOC<0 and αSOC1=αSOC2<0. None of these coefficients is computed from a microscopic Hamiltonian or estimated for the proposed candidates, even though the tight-binding model for κ-(BEDT-TTF)2Cu[N(CN)2]Br is already parameterized in the paper (SM §SVIII). Eq. (10)'s 4π/2π ac Josephson prediction applies only to the (1,1) state, so if the true quartic couplings favor β2>0, the ground state would be the half-and-half metal-superconductor and the predicted fractional Josephson effect would not occur. The stress-test concern on this point lands. I recommend either a weak-coupling computation of the GL coefficients within the presented model, or an explicit statement that the material example and Eq. (10) apply only in the assumed parameter regime and not yet to the specific compounds.
minor comments (5)
  1. [Abstract, p. 1, and Fig. 3 caption] There are several typos that should be corrected: 'superconductig' in the abstract, 'non-unitray' in the introduction, 'alternamagnetic' in the Josephson section, 'Quazi-two-dimensional' in the Fig. 3 caption, and 'Heisenbergs trasse' in the affiliation line.
  2. [Eq. (10) and SM §SIX] The transparent-limit expression for the spin-down current I↓ contains a factor cos(φ/2)/|cos(φ/2)| and is discontinuous at φ≡π (mod 2π); the limiting procedure from the full expression Eq. (S75) and the treatment of the branch point should be stated explicitly so that the 2π periodicity claim is unambiguous.
  3. [SM §SIX] The Josephson calculation uses a parabolic, spin-degenerate normal-state dispersion and thereby neglects the altermagnetic spin splitting of the Fermi surfaces that motivates the whole paper; a brief justification that the 4π versus 2π distinction survives the inclusion of spin-split Fermi surfaces would strengthen the prediction.
  4. [References, Ref. [44]] Ref. [44] (Feng and Zhang, PRB 111, 054520) already develops superconducting order parameters in spin space groups; the authors should state explicitly which of their tabulated results, if any, overlap with that work and what is genuinely new here.
  5. [SM §SIX, near Eq. (S73)] The assumption that the junction does not relax into the thermodynamic limit is central to observing 4π periodicity; a brief comment on quasiparticle poisoning and the conditions under which the fractional ac Josephson effect is actually observable would help avoid overinterpretation.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: coreps basis functions, GL minima, and Josephson response are self-contained derivations; only a minor, non-load-bearing self-citation appears for the material example.

full rationale

The derivation chain is self-contained rather than circular. The pairing basis states are constructed within the paper from irreducible co-representations of the spin-point groups using the unitary halving-group projectors (Eq. 4) and the corep projectors (Eq. 5), with the Dimmock indicator classification and standard group-theory references as independent support; the two independent construction routes are stated to give identical tables. The GL free energies (Eqs. 7 and 8) are the symmetry-allowed invariants for the constructed coreps, and the minima analyses are ordinary Landau calculations. The stability of the headline states does depend on assumed signs of the quartic coefficients (β2<0, conditions (S50)-(S53), αSOC1=αSOC2<0), and the paper labels the theory as phenomenological and leaves microscopic parameter computation to future work (SM SVI); this is a correctness/realization risk about which states are realized, not a circularity. The topological classification (SM SVII) and the fractional ac Josephson currents (SM SIX) are computed from the BdG Hamiltonian built from the chosen pairing state, with the 4π-periodic spin-up current arising from the solved bound-state energy E↑ ∝ cos(φ/2), not from any assumed output. The only self-citation is Ref. [23] for the 2D spin-space group assignment of κ-(BEDT-TTF)2Cu[N(CN)2]Br in the material example; this is material-specific and not load-bearing for the general symmetry classification or the model computations, so the overall circularity score is low.

Assumptions & free parameters 3 free parameters · 6 assumptions · 3 invented entities

The classification rests on the domain assumption that the altermagnet normal state has exact spin-point-group symmetry with negligible SOC, on the one-band local pairing ansatz of Eq. (6), and on standard co-representation theory (Dimmock indicators, corep orthogonality). The stability results additionally require hand-chosen signs of GL couplings (beta2, beta5, alpha_SOC) which are listed as free parameters. No invented forces or particles are introduced; the three named superconducting states are the paper's output and carry falsifiable handles (spin-resolved Josephson periodicity, spin-polarized edge modes, proposed material).

free parameters (3)
  • beta2 (GL quartic coupling) = assumed < 0 for altermagnetic SC; > 0 for half-and-half state
    Sign of the |phi1|^2|phi2|^2 term in Eq. (7) selects which of the two condensates is stable; never computed from microphysics.
  • beta5 (GL quartic coupling, 4D corep) = assumed < 0 with conditions (S50)-(S53)
    Stabilizes the equal-amplitude solution with theta1=theta2=theta3=theta4 for the spin chiral state; chosen by hand in SM Sec. SVI.
  • alpha_SOC1, alpha_SOC2 = assumed equal and < 0
    SOC-induced quadratic terms introduced in SM Secs. SV-SVI to lift the continuous relative-phase degeneracy and select the chiral minimum.
assumptions (6)
  • domain assumption Spin-orbit coupling is negligible, so spin and lattice degrees of freedom transform independently under the spin-point group elements
    Main text, 'Spin-point group basis functions' (before Eq. (1)); defines the regime in which SPGs and their coreps apply.
  • domain assumption The magnetic order is collinear, fully compensated, has an SO(2) spin-rotation symmetry and an antiunitary spin-flip times time-reversal symmetry (T 2s || E)
    Main text Eqs. (1)-(3) and the group list in SM Sec. SX; excludes non-collinear and uncompensated magnets.
  • domain assumption The Cooper pair wave function has the one-band form i[psi + d·tau] tau_y and transforms as a corep of the normal-state spin-point group
    Main text Eq. (6); excludes inter-band, multi-orbital, and finite-momentum pairing, although SM Sec. SVIII notes multiband cases need extension.
  • standard math Dimmock's construction of co-representations and the corep orthogonality relations (Karavaev coefficients) are valid and applicable
    SM Secs. SI-SIV; standard group theory from refs [56-58].
  • domain assumption kappa-(BEDT-TTF)2Cu[N(CN)2]Br is altermagnetic with 2D spin-space group 121g1g
    Material example section; follows from the authors' own prior analysis (ref [23]) and is not directly measured.
  • domain assumption CrSb is described by spin-point group 16/1m1m1m with magnetic point group P6'/m'm'm once SOC pins moments along z
    SM Sec. SVI, from ARPES references [26,27].
invented entities (3)
  • Altermagnetic superconductor (non-unitary spin-triplet state with S=±1 components related by (2s||g)) independent evidence
    purpose: Central new state with spin-dependent spatial anisotropy; source of spin-polarized topological response
    Falsifiable handles: spin-polarized gapless point nodes with Majorana boundary modes (Fig. 4a) and 4pi-periodic spin-up ac Josephson current (Eq. (10)); candidate material proposed.
  • Half-and-half metal-superconductor independent evidence
    purpose: State where one spin species is superconducting and the other remains metallic
    Falsifiable via spin-resolved tunneling, Andreev reflection, or specific heat; analogous to 3He A1 phase but claimed in a zero-field compensated magnet.
  • Spin chiral superconductor independent evidence
    purpose: Multicomponent order parameter (from 4D/6D coreps) breaking the antiunitary symmetry per spin subspace, giving per-spin Chern numbers
    Falsifiable via chiral edge modes, intrinsic anomalous Hall-type responses, or Kerr effect; predicted for CrSb-type symmetry (SM Sec. SVI).

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Cite this review

Pith. "Pith review of Exotic superconducting states in altermagnets." pith.science (2026). https://pith.science/paper/QO5KKY3I

@misc{pith2026250710700,
  author       = {Pith},
  title        = {Pith review of: Exotic superconducting states in altermagnets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QO5KKY3I}},
  note         = {Machine review of arXiv:2507.10700}
}
read the original abstract

The interplay between magnetism and superconductivity is one of the central topics of condensed matter physics, which has recently been put into new light by the discovery of altermagnets. Here, we study this interplay from a fundamental symmetry perspective using irreducible co-representations of the altermagnetic spin-point groups. We construct and tabulate all symmetry-allowed pairing functions for altermagnets, which uncovers numerous exotic pairing states. We focus on three of them, namely: (i) a non-unitary superconductor with different spatial anisotropies for the spin-up and spin-down condensates, (ii) a half-and-half metal-superconductor where only electrons with one of the two spin components form Cooper pairs, and (iii) a spin chiral superconductor with spin-polarized edge states. Interestingly, the first of these three superconductors exhibits an unusual fractional ac Josephson current for only one of the two spin polarizations. We present phenomenological Ginzburg-Landau theories for these unconventional superconductors and show that they correspond to stable minima of the free energies. We examine their topological properties, study the effects of small spin-orbit coupling, consider possible material examples, and investigate their topological responses.

Figures

Figures reproduced from arXiv: 2507.10700 by the authors.

Figure 1
Figure 1. Diagram describing the two approaches employed to obtain the b [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Three types of exotic superconducting states [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. (a) Quazi-two-dimensional κ-(BEDT-TTF)2X compound lattice with the magnetic order. (b) The band structure of the compound tight-binding model with (ta, tb, tp, tq, h) = (−0.207, −0.067, −0.102, 0.043, 0.2) eV, where h is the strength of the interaction with the magnetic moments. The BZ and Fermi surface with two spin-polarized pockets are shown in the insets. at low temperatures, and goes through the superconduct￾in… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: (a) BdG excitation spectrum for the stable [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

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