{"id":"a4cdb1a1-622e-41ed-9cf0-247b5d02b0c6","arxiv_id":"2607.10757","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"Odd-parity p-wave order decouples Ising RKKY from macroscopic beating, generates a massive nonrelativistic out-of-plane DM, and drives an intermediate 1/R nodal decay of in-plane DM under Rashba SOC.","lead":"Theory shows that odd-parity p-wave magnets make out-of-plane Ising RKKY ignore the magnetic beating while in-plane terms beat strongly, and produce a nonrelativistic out-of-plane DM plus an anomalous intermediate 1/R decay of in-plane DM along nodes. This gives a concrete route to directionally tunable non-collinear spin textures without relying only on weak spin-orbit coupling.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The manuscript is a clean, fully analytic theory paper whose strongest claim is a direct consequence of the odd-parity structure of M_k^p. Because G_z(R) is odd, the interference sign in the RKKY trace flips relative to even-parity altermagnets, automatically insulating J_zz from the cos(2k_M R) envelope while loading it onto the in-plane Heisenberg channels (Eqs. 24–28). The same odd G_z produces a nonrelativistic DM_z even at λ=0. Along the nodes the retained Rashba gap converts the sin(2k_R R) factor into a linear rise that cancels one power of R, yielding a well-defined intermediate 1/R window that is cut off at R*∼1/(2k_R); this is not a divergence but a controlled crossover already plotted in Fig. 6. All approximations are stated, sub-leading O(k_R/k_F) terms are isolated in Appendix C, and the continuum model is the natural setting for the claimed long-distance physics. The reader’s assessment of low correctness risk and ACCEPT is therefore unchanged; the suggested numerical check would only further confirm an already transparent analytic result.","tokens_in":24535,"tokens_out":612,"duration_ms":6811,"concrete_test":"Independently recompute the nodal in-plane DM envelope (Eq. 35) by numerical quadrature of the exact 2D Fourier transform of G_0 and G_x (Eqs. 4–6) at fixed energy E_F for a dense set of R along k_M=0, without stationary-phase or Hankel asymptotics; confirm that |J_DM,x| tracks ∼1/R for 1/k_F ≪ R ≲ 1/(2k_R) and reverts to ∼1/R^{2} thereafter.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central parity-driven claims (Ising decoupling from macroscopic p-wave beating, nonrelativistic DM_z, and the intermediate 1/R nodal window for in-plane DM) follow directly and consistently from the odd real-space parity of G_z and the hybridized poles retained in the stationary-phase Green’s functions. The reader’s weakest assumption (validity of k_F R ≫ 1 stationary phase plus continuum two-band model inside the window 1/Q ≪ R ≲ 1/(2k_R)) is real but standard; the paper already regularizes the nodal lines with the Rashba gap, supplies the full unapproximated expressions in Appendix C, and shows the graceful recovery of 1/R^{2} beyond R*. No internal inconsistency or hidden assumption that would overturn the qualitative role-reversal or the existence of the intermediate window is present.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript presents an analytical theory of the RKKY interaction between magnetic impurities in a two-dimensional p-wave magnet with Rashba spin-orbit coupling. Using a continuum two-band Hamiltonian and a real-space Green’s-function approach based on stationary-phase and residue calculus, the authors derive closed-form asymptotic expressions for the full exchange tensor. They show that the odd parity of the p-wave exchange field produces a structural decoupling: the out-of-plane Ising component J_zz is insulated from macroscopic p-wave beating and oscillates at the shifted Fermi wavevector, while the in-plane Heisenberg components exhibit directionally tunable beating. Hybridization further generates a three-component Dzyaloshinskii–Moriya interaction (with a nonrelativistic out-of-plane piece) together with symmetric off-diagonal anisotropies; along nodal lines the in-plane DM components display an intermediate-distance 1/R decay before recovering the conventional 2D 1/R^{2} asymptote.","tokens_in":24767,"tokens_out":1051,"duration_ms":38823,"significance":"If the analytic results hold, the work supplies a clear, parity-based distinction between odd-parity p-wave magnets and both conventional ferromagnets and even-parity d-wave altermagnets in the RKKY channel. The closed-form envelopes (Eqs. 24–26, 29–34), the explicit nonrelativistic origin of J_DM,z, and the regularized nodal 1/R window constitute concrete, falsifiable spatial predictions that can guide impurity-based probes and the design of anisotropic non-collinear textures. The thorough appendices (A–C) that retain the hybridized poles and supply the unapproximated O(k_R/k_F) corrections are a methodological strength and make the leading-order claims transparent and reproducible.","major_comments":[{"comment":"Sec. III.D, Eqs. (35)–(37) and Fig. 6: The intermediate 1/R window is a central claim. The linearization sin(2k_R R)≈2k_R R is formally correct for R≪1/k_R, yet the paper should quantify the relative error of this approximation (and of the leading Hankel asymptotics) across the stated window 1/Q≪R≲1/(2k_R). A short comparison of the leading envelope against the full Appendix-C expressions along a nodal cut would confirm that sub-leading phase and amplitude corrections do not erase the dimension-reducing crossover.","section":"III.D, Eqs. (35)–(37), Fig. 6"},{"comment":"Appendix A and Sec. II.C: The stationary-phase evaluation retains only the forward/backward points and the standard 2D prefactor. Near the nodal lines the RSOC-hybridized contours change curvature; a brief remark on whether this modifies the stationary-phase amplitude (or the range of validity of k_F R≫1) inside the intermediate window would strengthen the quantitative reliability of the 1/R claim.","section":"Appendix A, Sec. II.C"}],"minor_comments":[{"comment":"The adjective “massive” used for the nonrelativistic p-wave shift (abstract, Sec. III.C) is potentially confusing; “macroscopic” or “O(1)” would be clearer.","section":"Abstract, Sec. III.C"},{"comment":"Fig. 3 caption and main text: the geometric quenching of J_xy, J_xz and J_DM,x at φ=0 is stated clearly, but a one-sentence reminder that this is a coordinate choice (not a physical vanishing of all chiral response) would help non-specialist readers.","section":"Fig. 3"},{"comment":"In the comparison with d-wave altermagnets (Sec. III.A), a more precise citation of the corresponding envelopes from Refs. 70 and 71 would make the claimed “role reversal” sharper.","section":"Sec. III.A"},{"comment":"Typographical consistency: the arXiv identifier appears as 2607.10757 while the date line reads July 14, 2026; ensure journal submission metadata match.","section":"Title page"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript cleanly fills the gap left by the existing d-wave RKKY literature and is written at a level appropriate for a specialized condensed-matter theory journal. The two major points are constructive and do not threaten the central parity argument; I expect a straightforward revision."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing worth knowing is that the odd parity of the p-wave exchange field flips the usual RKKY interference relative to d-wave altermagnets: G_z becomes odd in real space, so J_zz is stripped of the macroscopic cos(2k_M R) beating while the in-plane Heisenberg channels absorb it. At the same time the hybridized poles give a three-component DM vector whose out-of-plane piece is nonrelativistic (present even at λ=0) and whose in-plane pieces show a clean intermediate 1/R decay along the nodes before recovering 1/R^{2}. That package is new; prior RKKY work stopped at even-parity altermagnets.\n\nThey do the calculation properly. Stationary-phase + residue Green’s functions (Appendix A), standard second-order trace (B), and Hankel asymptotics with the full unapproximated O(k_R/k_F) terms kept in C. The parity argument is transparent from the sign of G_z(R)G_z(-R). Figures 2, 5 and 6 make the role reversal and the nodal crossover visible, and the regularization by the Rashba gap prevents any fake divergence. No free parameters are fitted to data; the dimensionless ratios are just illustrative.\n\nSoft spots exist but are ordinary. Everything is continuum two-band and k_F R ≫ 1, so the claimed intermediate window 1/Q ≪ R ≲ 1/(2k_R) sits inside the asymptotic regime by construction. They already supply the unexpanded expressions and show the graceful return to 1/R^{2}, so the qualitative claim survives. Material realism (how large Q/k_F and how clean the nodes really are) is left for later; that is fine for a theory paper of this type.\n\nThis is for people working on unconventional magnets, RKKY, or non-collinear textures who want closed-form envelopes they can actually use. The math and citation pattern look solid. I would send it to referees without hesitation and would cite the parity decoupling and the nodal 1/R window myself.","headline":"Solid analytic RKKY paper that cleanly shows odd-parity p-wave order reverses the Ising/Heisenberg beating roles and produces a nonrelativistic DM_z plus a real intermediate 1/R nodal window; soft spots are standard continuum/stationary-phase limits, not load-bearing flaws.","tokens_in":25384,"tokens_out":574,"would_cite":true,"duration_ms":8988,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"Odd-parity p-wave magnets decouple Ising exchange from magnetic beating and produce an intermediate 1/R Dzyaloshinskii-Moriya decay along nodal lines.","keywords":["p-wave magnets","RKKY interaction","Dzyaloshinskii-Moriya interaction","Rashba spin-orbit coupling","parity-driven decoupling","nodal 1/R decay","non-collinear spintronics"],"falsifier":"Measure the spatial decay of the in-plane DM interaction between two magnetic impurities placed exactly along a p-wave nodal line: if the envelope fails to follow 1/R over the window 1/Q ≪ R ≲ 1/(2k_R) and then reverts to 1/R^{2}, the central nodal claim is false.","tokens_in":25423,"feed_emoji":"🧲","tokens_out":807,"duration_ms":8451,"temperature":0.7,"pith_summary":"This paper shows that the odd-parity spin splitting of a two-dimensional p-wave magnet fundamentally rearranges the indirect RKKY coupling between magnetic impurities once Rashba spin-orbit coupling is present. Because the exchange field is odd under momentum reversal, the out-of-plane Ising channel is insulated from the macroscopic magnetic modulation and oscillates only at the ordinary Fermi wavevector, while the in-plane Heisenberg channels inherit a strong directional beating envelope. The same hybridized bands also generate a three-component Dzyaloshinskii-Moriya interaction: its out-of-plane piece is nonrelativistic and set by the p-wave momentum shift, whereas the in-plane pieces are purely Rashba-driven. Along the nodal directions the competition between vanishing p-wave shift and the residual Rashba gap produces an anomalous intermediate-distance window in which those in-plane chiral components decay as 1/R before recovering the conventional 2D 1/R^{2} tail. The result supplies a concrete microscopic route to directionally tunable non-collinear spin textures that cannot appear in ordinary ferromagnets or even-parity altermagnets.","feed_headline":"Odd-parity magnets reverse RKKY beating and give 1/R DM decay","feed_subtitle":"Nodal lines turn the in-plane chiral coupling into a temporary one-dimensional range before 2D asymptotics return","key_machinery":"Analytical long-distance real-space Green’s functions obtained by a spin-dependent stationary-phase expansion of the hybridized continuum Hamiltonian; the odd real-space parity of G_z forces a sign reversal in the RKKY trace that routes the magnetic beating exclusively into the in-plane channels and regularizes the nodal DM response with the residual Rashba gap.","core_discovery":"In a 2D p-wave magnet with Rashba spin-orbit coupling the odd parity of the exchange field reverses the usual assignment of spatial beating: the out-of-plane Ising interaction remains free of the macroscopic magnetic wavevector while the in-plane Heisenberg terms carry a strong cos(2k_M R) envelope; simultaneously the hybridized bands produce a three-component DM interaction whose out-of-plane piece is nonrelativistic and whose in-plane pieces exhibit a dimension-reducing 1/R decay along the nodal lines over an extended intermediate window before recovering 1/R^{2}.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Odd-parity p-wave magnets reverse RKKY beating and give 1/R DM decay","Parity decouples Ising RKKY from beating; in-plane DM decays as 1/R","p-wave nodal lines yield temporary 1/R DM decay before 1/R^{2} return","Odd parity flips RKKY assignment and drives nonrelativistic out-of-plane DM","Hybridized p-wave bands produce three-component DM with 1/R nodal decay"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The long-distance stationary-phase evaluation of the continuum Green’s functions, together with neglect of sub-leading Rashba phase shifts, remains accurate throughout the intermediate nodal window where the claimed 1/R crossover is supposed to live.","fun_headline_variants_meta":{"raw":{"variants":["Odd-parity p-wave magnets reverse RKKY beating and give 1/R DM decay","Parity decouples Ising RKKY from beating; in-plane DM decays as 1/R","p-wave nodal lines yield temporary 1/R DM decay before 1/R^{2} return","Odd parity flips RKKY assignment and drives nonrelativistic out-of-plane DM","Hybridized p-wave bands produce three-component DM with 1/R nodal decay"]},"model":"grok-4.5","effort":"low","cost_usd":0.004476,"raw_usage":{"total_tokens":1432,"prompt_tokens":933,"num_sources_used":0,"completion_tokens":127,"cost_in_usd_ticks":44760000,"prompt_tokens_details":{"text_tokens":933,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":372,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":933,"tokens_out":127,"duration_ms":6697,"temperature":1.0,"reasoning_tokens":372,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T09:29:04.187959+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Measure the spatial decay of the in-plane DM interaction between two magnetic impurities placed exactly along a p-wave nodal line: if the envelope fails to follow 1/R over the window 1/Q ≪ R ≲ 1/(2k_R) and then reverts to 1/R^{2}, the central nodal claim is false.","supporting_citations":[],"review_version":1}