{"id":"75e658e6-c838-491a-9768-ac0ed39d20ff","arxiv_id":"2411.13795","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"PT-symmetric antiferromagnetic bilayers can host hidden altermagnetism, a local altermagnetic spin splitting that is globally zero and is uncovered by an electric field, as predicted for bilayer Cr2SO.","lead":"This paper introduces hidden altermagnetism, a state in PT-symmetric antiferromagnets where each inversion-partner sector is altermagnetic, so spins split locally but cancel globally. The authors predict that PT-symmetric bilayer Cr2SO realizes this effect and that an electric field can reveal the hidden spin splitting.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The PT-symmetric AFM1 ground state is upheld by only 1–3 meV per cell, so the hidden altermagnetism prediction rests on a numerically fragile magnetic order.","rationale":"The reader's weakest-assumption analysis correctly identifies the fragility of the AFM1 magnetic ground state as the most load-bearing concern. The energy separation of 1–3 meV per cell between AFM1 and AFM2 is uncomfortably small relative to known DFT errors, and the paper provides no convergence tests against U, exchange-correlation functional, or vdW correction. Because the entire hidden altermagnetism scenario—PT symmetry, global spin degeneracy, layer-locked local spin splitting, and the eEd field response—is predicated on AFM1 being the ground state, this concern directly threatens the central claim. My independent reading of the symmetry argument (P = C2z·Mh, local [C2∥O] altermagnetism, PT-enforced degeneracy) finds no internal inconsistency; the concept is plausible and the band-structure calculations are coherent. The lack of a direct zero-field layer-resolved spin polarization plot is a presentation gap rather than a fatal flaw, since the symmetry analysis plus the monolayer altermagnetism strongly imply nonzero local spin polarization. Thus I do not find a stronger concern that would change the verdict; CONDITIONAL remains the appropriate judgment until the magnetic ground state is shown to be robust.","tokens_in":9216,"tokens_out":23297,"duration_ms":213812,"concrete_test":"Recompute ΔE = E(AFM2) − E(AFM1) for both S- and O-terminated bilayers using PBE+U with U = 2.0, 3.0, 3.55, and 4.0 eV, with and without DFT-D3, and with the HSE06 hybrid functional, keeping all other computational settings fixed. If the sign of ΔE flips or its magnitude drops below ~1 meV (numerical noise) for any reasonable parameter set, the PT-symmetric AFM1 ground state is not robust and the hidden altermagnetism prediction loses its material basis. As a complementary check, extract the interlayer exchange parameter J from ΔE and estimate the ordering temperature; if this temperature is below ~1 K, the predicted phase is unlikely to be experimentally accessible.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires the bilayer Cr2SO ground state to be AFM1 (intralayer AFM + interlayer AFM), which possesses global PT symmetry. The paper reports that AFM1 lies lower than AFM2 by only 1.1 meV per cell (S-terminated) and 3.0 meV per cell (O-terminated). These values are within the typical error bars of DFT for the exchange-correlation functional, the Hubbard U parameter, the vdW correction, and k-point or cutoff convergence. No tests of U dependence, functional choice, or vdW-scheme dependence are reported. If a different but equally reasonable parameterization raises AFM2 above AFM1, the PT symmetry is lost and the hidden altermagnetism disappears entirely. The weak interlayer exchange implied by this small energy difference also suggests a very low magnetic ordering temperature, making the effect potentially unobservable in a bilayer sample. The remaining numerical results—global spin degeneracy at zero field and the eEd scaling of the field-induced splitting—are internally consistent, but they all presuppose the AFM1 ground state. Therefore, the material realization of hidden altermagnetism is not established beyond reasonable DFT uncertainty.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the concept of \"hidden altermagnetism\" for PT-symmetric antiferromagnets: the global PT symmetry forbids net spin splitting, but each of the two inversion-partner sectors is individually altermagnetic, producing nonzero local spin polarization. The authors propose a bilayer stacking construction, identify bilayer Cr2SO as a candidate, and use DFT+U to show that an out-of-plane electric field breaks global P, revealing momentum-dependent spin splitting with layer character. They report a field-induced splitting of about 203 meV at 0.03 V/Å, close to the eEd estimate of 207 meV, and discuss SOC-induced valley splitting.","tokens_in":9372,"tokens_out":2907,"duration_ms":31753,"significance":"If the material prediction holds, the concept of hidden altermagnetism is a valuable extension of hidden spin polarization to antiferromagnetic systems, and the bilayer stacking recipe provides a clear design principle. The paper explicitly separates the general symmetry construction from the material example, and the eEd consistency check for the field-induced splitting is a useful quantitative anchor. However, the central material prediction rests on a magnetic ground state whose energy margin is only 1–3 meV per cell, and the zero-field hidden local spin polarization is not explicitly demonstrated with layer-resolved data. The conceptual contribution is sound; the numerical material claim needs strengthening.","major_comments":[{"comment":"The AFM1 ground state, which carries the PT symmetry required for hidden altermagnetism, is reported to be lower than AFM2 by only 1.1 meV per cell (S-terminal) and 3.0 meV per cell (O-terminal). These margins are within typical GGA+U errors arising from the Hubbard U value, the exchange-correlation functional, and the vdW correction scheme. No U-dependence, functional-dependence, or vdW-scheme tests are reported, and no convergence data for k-points/cutoff are given for this energy difference. Since the entire hidden-altermagnetism prediction vanishes if AFM2 becomes the ground state, the material realization claim is not yet robust. Please compute ΔE(AFM2−AFM1) as a function of U (e.g., 2–5 eV), with alternative functionals (e.g., SCAN or HSE) and alternative vdW corrections, and report the magnetic moments and energy convergence.","section":"Material realization"},{"comment":"Hidden altermagnetism requires that, at zero electric field, each inversion-partner sector individually shows altermagnetic spin splitting while the global PT symmetry keeps the total bands spin-degenerate. The manuscript argues this from symmetry and shows only global band structures in Figure 2(e,f); no layer-resolved or sublattice-resolved spin projection is provided at E=0. To support the central claim for bilayer Cr2SO, please show layer- or sector-resolved spin-polarized bands (or real-space spin density integrated per sector) at E=0, demonstrating that the two sectors have opposite local spin polarization while the global spectrum remains degenerate.","section":"Figure 2 and Section 'Material realization'"},{"comment":"The interlayer exchange energy implied by the 1–3 meV/cell AFM1–AFM2 difference suggests a low magnetic ordering temperature for the bilayer. Since the paper states the hidden altermagnetism 'can be confirmed in experiment' and an electric field can be used to 'observe' it, the experimental feasibility depends on the magnetic order surviving at accessible temperatures. Please provide at least an estimate of the magnetic ordering temperature (e.g., via a simple mean-field estimate or Monte Carlo treatment of the interlayer exchange) or explicitly temper the experimental-observability claim until the magnetic ground state is better established.","section":"Discussion and Conclusion"}],"minor_comments":[{"comment":"There are several typographical errors: 'takeing' should be 'taking', 'demonstrat' should be 'demonstrate', and 'Halll' should be 'Hall' in 'Anomalous Halll/Nernst effect'.","section":"Introduction"},{"comment":"The axis label in Figure 4 appears garbled with font-encoding artifacts ('E/uni00000003/uni0000000b/...'); please replace with a properly typeset label such as 'E (V/Å)'.","section":"Figure 4"},{"comment":"Reference [41] to the Supplemental Material is given as 'at []' with an empty placeholder; please provide the correct link or DOI.","section":"References"},{"comment":"In the Figure 3 caption, 'E=+0.00' should simply be 'E=0.00' or 'E=0' for clarity.","section":"Figure 3"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the scope of the journal and the concept is a reasonable extension of hidden spin polarization. The main risk is not the symmetry argument but the numerical fragility of the AFM1 ground state, which is the load-bearing assumption for the material prediction. I would encourage the editor to ask for the robustness tests and the zero-field layer-resolved demonstration described in the major comments."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core idea here is genuinely new and deserves a referee. Hidden altermagnetism is a natural extension of hidden spin polarization to PT-symmetric antiferromagnets, and the paper states the symmetry clearly: global PT keeps bands doubly degenerate, while each inversion-partner sector shows local altermagnetic spin splitting. The stacking recipe is simple and generic, and the bilayer Cr2SO example is a reasonable first-principles test. The eEd estimate matching the computed spin splitting within 2% is a nice internal consistency check.\n\nThe main soft spot is exactly what the stress test flagged. The AFM1 ground state sits only 1.1 meV (S-terminated) and 3.0 meV (O-terminated) per cell below AFM2. That is inside typical DFT error bars for the functional, Hubbard U, vdW correction, and k-point convergence. No tests of those settings are reported. If a reasonable parameter choice flips the order, PT symmetry is lost and the hidden altermagnetism vanishes. This does not kill the concept, but it makes the specific material prediction conditional rather than established. The paper also infers local spin polarization from the band structure and layer projection rather than showing an explicit real-space spin density map, and it gives no input structures or data files for reproduction. These are real shortcomings, but they are fixable and they do not undermine the symmetry argument.\n\nThe self-citations to Refs. 24, 27, and 29 are relevant to electric-field control and valley polarization, so I would not call them a red flag. The paper is honest about what it computes, and the narrative is straightforward. It reads as a concept paper with a supporting example, not as a definitive material discovery. Who gets value from this? People working on altermagnetism, hidden spin polarization, or 2D antiferromagnetic spintronics will want to know the idea, even if they treat the Cr2SO prediction with caution.\n\nMy recommendation: send it to peer review. The concept is important enough to warrant referee time, and the magnetic ground-state fragility is exactly what reviewers should probe. I would ask for convergence tests with respect to U and vdW schemes, an explicit real-space spin polarization plot, and ideally the relaxed structures. If those come back clean, the paper is solid.","headline":"A clean symmetry-based idea with a plausible but numerically fragile material realization; the hidden altermagnetism concept is worth taking seriously.","tokens_in":150,"tokens_out":1386,"would_cite":true,"duration_ms":23316,"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":"PT-symmetric bilayer Cr2SO hides altermagnetism in each layer and reveals it with an electric field.","keywords":["hidden altermagnetism","hidden spin polarization","PT symmetry","bilayer Cr2SO","altermagnetism","spin splitting","electric field control","antiferromagnetic spintronics"],"falsifier":"A measurement that finds the ground state of bilayer Cr2SO to be AFM2 rather than AFM1, or a spin-resolved photoemission experiment under 0.03 V/A that fails to see a layer-projected spin splitting of about 203 meV at X/Y, would disprove the central claim.","tokens_in":8945,"feed_emoji":"🧲","tokens_out":4545,"duration_ms":41298,"temperature":0.7,"pith_summary":"The paper proposes that a PT-symmetric antiferromagnet—a magnet with both space-inversion and time-reversal symmetry—can hide altermagnetism: its two inversion-partner layers each carry a d-wave spin-split band structure, yet the two split patterns cancel globally, leaving zero net spin polarization. The claim matters because it extends the idea of hidden spin polarization to an antiferromagnetic setting, creating a route to materials with no stray field but with layer-resolved spin texture that can be switched by an electric field. Using density-functional calculations, the paper identifies bilayer Cr2SO as a concrete candidate and shows that an out-of-plane electric field breaks the global inversion symmetry to expose a momentum-dependent spin splitting of about 203 meV at 0.03 V/A, close to the simple eEd estimate of 207 meV.","feed_headline":"Electric field reveals hidden altermagnetism in Cr2SO","feed_subtitle":"PT-symmetric bilayer hides layer-local spin splitting; 0.03 V/A exposes ~203 meV.","key_machinery":"The mechanism is the combination of global PT symmetry, which enforces zero net spin polarization, and local $[C_2\\|O]$ symmetry, which enforces altermagnetic spin splitting within each sector. The paper also uses a construction rule: take an altermagnetic monolayer as sector B, mirror-reflect it to form the second layer, then rotate by $C_{2z}$ to make the bilayer inversion-symmetric with $P=C_{2z}M_h$. For bilayer Cr2SO, the altermagnetic splitting at the band edges is quantified, and its field dependence is captured by the linear estimate $eEd$, with $e$ the electron charge, $E$ the applied field, and $d=6.91$ Å the interlayer Cr-Cr distance.","core_discovery":"The central claim is that a magnetic crystal with global PT symmetry can nevertheless be composed of two inversion-partner sectors that are each altermagnetic, a state the author calls hidden altermagnetism. In such a crystal, energy bands satisfy $E^{\\uparrow}(\\mathbf{k})=E^{\\downarrow}(\\mathbf{k})$ globally because $PT$ enforces degeneracy, while within each sector the combined spin-and-lattice symmetry denoted $[C_2\\|O]$ (a two-fold spin rotation perpendicular to the spin axis paired with a lattice rotation or mirror) produces momentum-dependent spin splitting. For PT-symmetric bilayer Cr2SO in its AFM1 magnetic ground state, the paper demonstrates this hidden phase and shows that an out-of-plane electric field separates the two layers energetically; the resulting spin splitting between the first and second conduction bands at X/Y reaches 203 meV at 0.03 V/A, matching the eEd estimate of 207 meV within a few percent. With spin-orbit coupling, the in-plane magnetization gives a small valley splitting of about 4 meV in the conduction band and 2 meV in the valence band between the X and Y valleys.","pith_inferences":["Editorially, the near-degeneracy of AFM1 and AFM2 (1.1 and 3.0 meV per cell) suggests the hidden phase may be switchable by strain or small magnetic fields, which would make the material more than a proof-of-principle.","Editorially, spin-resolved photoemission under an applied gate field is the most direct test; if the predicted 203 meV splitting at 0.03 V/A is observed with opposite spin character on the two layers, the concept is confirmed.","Editorially, the same reasoning could extend to twisted bilayer altermagnets, where the twist angle tunes interlayer coupling and therefore the field strength needed to reveal the hidden splitting."],"forward_implications":["If bilayer Cr2SO realizes hidden altermagnetism, then an out-of-plane electric field in either direction can expose a layer-resolved spin splitting of order 200 meV at achievable field strengths.","The hidden state has globally spin-degenerate bands and zero net magnetization, so it behaves like a conventional antiferromagnet until the field is applied.","Because the building block can be any two-dimensional altermagnet, the same stacking procedure could produce hidden altermagnetism in other Cr2O2-type, V2Se2O-type, V2SeTeO-type, and Fe2Se2O-type monolayers.","The approximately linear relation between spin splitting and field, captured by $eEd$, gives a simple design rule for estimating the required field in other bilayers.","Under spin-orbit coupling, the field-tuned bilayer shows valley polarization between X and Y valleys, with its sign controlled by the magnetization direction."],"supporting_citations":[{"why":"Defines hidden spin polarization in centrosymmetric crystals, the phenomenon this paper extends to antiferromagnets.","marker":"[3]"},{"why":"Introduces altermagnetism as a phase with nonrelativistic spin splitting from compensated antiparallel order, the local property being hidden here.","marker":"[15]"},{"why":"Provides the gate-field control of spin in altermagnets with spin-layer coupling and the eEd estimate used to describe the field-induced splitting.","marker":"[25]"},{"why":"Predicts that Janus monolayer Cr2SO is an altermagnet with spin-valley locking, serving as the building block for the bilayer.","marker":"[29]"},{"why":"Explains valley polarization in two-dimensional tetragonal altermagnets, used here to describe the SOC valley splitting in the bilayer.","marker":"[27]"}],"fun_headline_variants":["Hidden altermagnetism exposed by electric field","Electric field unveils hidden altermagnetism in Cr2SO","Zero net spin, hidden layer altermagnetism","Altermagnetism goes incognito in PT-symmetric crystals","Hidden spin splitting exposed by an electric field"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the bilayer choosing the intralayer/interlayer antiferromagnetic AFM1 order as its ground state, but that order wins by only 1.1 meV (S-terminal) and 3.0 meV (O-terminal) per unit cell, an energy difference smaller than typical density-functional error bars.","fun_headline_variants_meta":{"raw":{"variants":["Hidden altermagnetism exposed by electric field","Electric field unveils hidden altermagnetism in Cr2SO","Zero net spin, hidden layer altermagnetism","Altermagnetism goes incognito in PT-symmetric crystals","Hidden spin splitting exposed by an electric field"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000344,"raw_usage":{"total_tokens":1891,"prompt_tokens":952,"completion_tokens":939,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":568,"completion_tokens_details":{"reasoning_tokens":864}},"tokens_in":568,"tokens_out":939,"duration_ms":6911,"temperature":1.0,"reasoning_tokens":864,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:51:48.629384+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A measurement that finds the ground state of bilayer Cr2SO to be AFM2 rather than AFM1, or a spin-resolved photoemission experiment under 0.03 V/A that fails to see a layer-projected spin splitting of about 203 meV at X/Y, would disprove the central claim.","supporting_citations":[],"review_version":1}