{"id":"c4118df8-716e-4088-b6bb-c32289e07e2a","arxiv_id":"2507.10700","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Using co-representations of spin-point groups, the authors classify all symmetry-allowed superconducting pairing states of altermagnets and predict three exotic states, including a non-unitary altermagnetic superconductor with a spin-resolved 4π-periodic ac Josephson effect.","lead":"A symmetry analysis of altermagnets, a class of magnetic materials with no net magnetization but spin-split bands, predicts three new kinds of superconducting states, including one where spin-up and spin-down electrons pair with different spatial patterns and one where only a single spin direction becomes superconducting. If realized, these states could produce spin-selective currents and topological edge modes without external magnetic fields or spin-orbit coupling.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The stability of the headline superconducting states rests on hand-chosen signs of GL couplings, which are never derived from a microscopic model; without those estimates the physical realization claims are unverified.","rationale":"The reader's weakest_assumption correctly identifies the load-bearing concern: the headline states are stable only for hand-chosen GL coefficient signs, and no microscopic calculation or material-specific estimate is provided. This is exactly where the central physical claim is least secure. The symmetry classification core is credible and internally consistent (two projector approaches agree), and the paper explicitly presents the GL analysis as phenomenological, so this is not an error but a condition on realization. My read does not change the CONDITIONAL verdict; it reinforces it. A concrete microscopic computation of β2 and β5 for the proposed organic material would settle whether the exotic states are more than symmetry curiosities. The Josephson derivation in SM §SIX contains a minor typo (Eq. S72 writes √(D cos(φ/2)); the final current is consistent with E ∝ cos(φ/2), so the qualitative 4π prediction is unaffected), which I do not treat as a load-bearing issue.","tokens_in":215635,"tokens_out":15803,"duration_ms":181414,"concrete_test":"Compute the GL quartic coefficients β2 and β5 for the κ-(BEDT-TTF)2Cu[N(CN)2]Br model of SM §SVIII within weak-coupling mean-field theory, using the standard derivation of the GL free energy from the Cooper susceptibility with a generic on-site Hubbard U or nearest-neighbor V. If β2>0, or if the conditions (S50)-(S53) fail, then the altermagnetic superconductor and the spin chiral state are not local minima, and the Josephson 4π prediction of Eq. (10) would not be realized in this material; a positive result would validate the physical relevance of the states.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The altermagnetic superconductor (1,1) and the spin chiral state are shown to be stable minima of the GL functional only for β2<0 (Eq. 7, SM §SIV) and β5<0 with conditions (S50)-(S53) (SM §SVI), respectively, while phase degeneracy is lifted only for αSOC1=αSOC2<0 (SM §SV-SVI). These coefficients are not computed from any microscopic Hamiltonian or estimated for the proposed materials, so if the true quartic couplings have opposite signs, the (1,1) state is a saddle point rather than the global minimum, the spin chiral state is not realized, and the predicted spin-polarized 4π Josephson effect (Eq. 10) does not apply. This is not an internal inconsistency - the paper honestly labels its GL analysis as phenomenological - but it makes the central physical claim conditional on parameters that are currently assumed, not derived.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":215769,"tokens_out":25864,"duration_ms":265887,"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":[{"comment":"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.","section":"SM §SIV, §SVI (Eqs. (7), (S50)-(S54))"},{"comment":"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.","section":"Material example and Eqs. (7), (10); SM §SVIII"}],"minor_comments":[{"comment":"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.","section":"Abstract, p. 1, and Fig. 3 caption"},{"comment":"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.","section":"Eq. (10) and SM §SIX"},{"comment":"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.","section":"SM §SIX"},{"comment":"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.","section":"References, Ref. [44]"},{"comment":"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.","section":"SM §SIX, near Eq. (S73)"}],"recommendation":"major_revision","confidential_remarks":"The paper fits a condensed-matter theory journal well. The overlap with Ref. [44] on spin-space-group superconducting order parameters should be addressed explicitly in the revision, but it is not a novelty-blocking issue given the different focus on altermagnets and on physical responses. The stability-condition gap in the SM (§SVI) is the decisive fixable flaw; if the authors supply the missing inequalities and at least one microscopic estimate of the GL coefficients, the paper would be publishable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear X,\n\nThe thing to know about this one: the symmetry-classification core is real, and the exotic states are clearly identified, but their stability and the Josephson prediction are conditional on GL couplings whose signs are assumed, not derived. If β2 comes out positive in a real material, the altermagnetic superconductor state is a saddle point and the 4π-periodic spin-up Josephson effect does not occur. The paper acknowledges its GL analysis is phenomenological, but this is the load-bearing part of the physical narrative.\n\nWhat is genuinely new: the complete tabulation of coreps and pairing basis functions for all 58 spin-point groups of collinear compensated magnets, cross-checked by two independent projector methods. That is a useful reference. The three headline states—half-and-half metal-superconductor, altermagnetic superconductor with different spin-up/down anisotropies, and spin chiral superconductor—are distinct and not present in conventional magnetic point group classifications. The topological classification follows the standard A and AIII classes and is clean.\n\nSoft spots, in rough order of softness. The GL stability of each headline state is shown for carefully chosen quartic couplings (β2<0, β5<0, αSOC1=αSOC2<0), with no microscopic calculation or material-specific estimate. If the real couplings have opposite signs, the (1,1) state is not the global minimum and the spin-resolved Josephson current (Eq. 10) does not apply. This is not an internal inconsistency, but it means the physical realization claims are conditional on parameters that are currently free. The candidate material suggestion rests on the authors' own symmetry assignment in an earlier paper, not independently verified. The novelty relative to refs [40] and [44] is not benchmarked; the reader has to trust the claimed difference. I could not check the SM tables entry-by-entry because the version I saw was truncated.\n\nNet: as a phenomenological roadmap, this is solid and worth publishing after revision that either computes or more carefully delimits the parameter regime for each headline state. The classification tables alone justify a serious referee. I'd send it to review.","headline":"The symmetry tabulation is a real contribution; the physical-realization claims are conditional on GL couplings that are assumed, not derived.","tokens_in":216383,"tokens_out":3183,"would_cite":true,"duration_ms":35558,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["altermagnetism","spin-point groups","unconventional superconductivity","non-unitary pairing","spin-triplet pairing","Josephson effect","Majorana modes","Ginzburg-Landau theory"],"falsifier":"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.","tokens_in":215345,"feed_emoji":"⚛️","tokens_out":10556,"duration_ms":113306,"temperature":0.7,"pith_summary":"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.","feed_headline":"Altermagnets may give spin-up Cooper pairs a 4π period","feed_subtitle":"Spin-up Cooper pairs would oscillate with 4π period while spin-down pairs keep 2π, a testable spin-resolved response.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the spin-point-group construction whose co-representations this paper extends by deriving the representation matrices and basis functions not tabulated there.","marker":"[50]"},{"why":"Provides the Dimmock rules used to convert unitary representations of the halving group into non-unitary co-representations.","marker":"[56]"},{"why":"Gives the Ginzburg-Landau framework for unconventional superconducting pairing that the paper uses to establish stable minima.","marker":"[62]"},{"why":"Identifies the A1 phase of 3He as the analogue for the half-and-half metal-superconductor state.","marker":"[59]"},{"why":"Provides the topological classification scheme that assigns winding numbers and Chern numbers to the spin-subspace Bogoliubov-de Gennes Hamiltonians.","marker":"[64]"},{"why":"Supplies the tight-binding model parameters used for the κ-(BEDT-TTF)2X material example.","marker":"[68]"},{"why":"Documents the altermagnetic spin splitting in CrSb, the candidate material for the spin chiral superconductor.","marker":"[27]"},{"why":"Reports the proximity of superconductivity and compensated collinear order in κ-(BEDT-TTF)2Cu[N(CN)2]Br, motivating the material example.","marker":"[65]"}],"fun_headline_variants":["Altermagnets host exotic superconductors with spin-split pairs","Spin-sensitive superconductivity in altermagnets revealed","Altermagnets allow spin-up only superconducting state","Fractional Josephson effect for one spin in altermagnets","Non-unitary pairing and spin-chiral states in altermagnets"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Altermagnets host exotic superconductors with spin-split pairs","Spin-sensitive superconductivity in altermagnets revealed","Altermagnets allow spin-up only superconducting state","Fractional Josephson effect for one spin in altermagnets","Non-unitary pairing and spin-chiral states in altermagnets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00026,"raw_usage":{"total_tokens":1679,"prompt_tokens":1122,"completion_tokens":557,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":738,"completion_tokens_details":{"reasoning_tokens":471}},"tokens_in":738,"tokens_out":557,"duration_ms":5681,"temperature":1.0,"reasoning_tokens":471,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:29:38.874504+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Ambegaokar \\ and\\ author N","cited_arxiv_id":null,"evidence_quote":"Identifies the A1 phase of 3He as the analogue for the half-and-half metal-superconductor state."}],"review_version":1}