{"id":"9773d9e8-edd7-4342-94d5-0f36a1638883","arxiv_id":"2607.21430","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Shiba-dual attractive Hubbard models on bipartite lattices host η-altermagnetism, where Bogoliubov bands split by η-pseudospin with p-, d-, and f-wave structures.","lead":"This paper proposes 'η-altermagnetism': applying the Shiba transformation to altermagnets moves momentum-dependent band splitting from spin into the η-pseudospin channel, so superconducting and charge-ordered states can show p-, d-, or f-wave band splitting. The idea gives a fresh target for cold-atom and materials searches for unconventional pairing and charge order.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The η-ALM ground-state claim rests on a single HFB solution without demonstrating it is the global energy minimum or stable beyond mean-field.","rationale":"The reader's weakest assumption identifies the HFB mean-field approximation as load-bearing. I agree, but I refine the concern: the paper does not even establish that the η-ALM solution is the global minimum of the HFB energy, a necessary condition for the central claim. The Shiba duality maps exact eigenstates, so the ground-state correspondence is rigorous only if the repulsive-side ALM order is itself the true ground state, which the paper does not demonstrate beyond citing mean-field ALM literature. The HFB computations show a self-consistent solution but no energy comparison with competing orders, no finite-size scaling, and no exact benchmarks. The concrete test I propose—a systematic HFB energy comparison with multiple initializations—directly addresses whether the η-ALM state is the global energy minimum within the paper's own framework. If it is not, the central claim is invalid regardless of quantum fluctuations. If it is, the further exact-diagonalization/DMRG check would test whether the order survives beyond mean-field. This does not change the reader's CONDITIONAL verdict; it strengthens the reasons for it.","tokens_in":16037,"tokens_out":16451,"duration_ms":158119,"concrete_test":"Run a systematic HFB energy comparison on the checkerboard lattice with the paper's parameters (t1=1, t2=0.1, U=-4, µ=-2) using multiple initial ansätze: (i) uniform s-wave BCS, (ii) staggered CDW, (iii) the published η-ALM state, (iv) a random Gaussian state. If any non-η-ALM initialization converges to a lower HFB energy, the claim that η-ALM is the ground state fails. For a stronger check, run exact diagonalization or DMRG on a small cluster (e.g., 4×4 checkerboard) and compute the staggered η-pseudospin structure factor S_η(π,π); if S_η does not scale with system size, the order is not long-range.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the ground state of the attractive Hubbard models with anisotropic second-neighbor hopping exhibits η-ALM: a staggered η-pseudospin order with BdG band splitting. The paper's support is (i) an exact Shiba duality argument and (ii) HFB calculations on finite checkerboard and honeycomb clusters (Figs. 1–2). The exact duality guarantees that the attractive model's ground state is the Shiba image of the repulsive model's ground state, but it does not establish that the repulsive model's ground state actually has ALM order; that is assumed from prior ALM literature (Refs. [8,9]), which itself is mean-field/symmetry-based. On the attractive side, the HFB calculation finds a self-consistent solution but does not show it is the global minimum of the HFB energy: no comparison with competing orders (uniform s-wave BCS, CDW, other η-vector orientations), no multiple-initialization search, and no finite-size scaling or exact-diagonalization benchmark. If the η-ALM solution is a metastable local minimum, or if quantum fluctuations destroy the staggered η order, the predicted BdG η-pseudospin splitting would not appear in the true ground state.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the concept of η-altermagnetism (η-ALM) by applying the exact Shiba duality to altermagnetic repulsive Hubbard models. In the half-filled bipartite attractive Hubbard model, the Shiba-dual P̃T̃ symmetry is shown to protect a η-pseudospin Kramers degeneracy in the BdG bands; anisotropic second-neighbor hopping lifts this degeneracy and produces η-pseudospin band splitting, either of odd parity (η-pseudospin splitting) or even parity (spin-η-locked splitting). Hartree-Fock-Bogoliubov calculations on checkerboard and honeycomb lattices are presented as concrete examples, showing p-, d-, and f-wave splitting structures accompanied by staggered charge-density and uniform singlet pairing orders.","tokens_in":16322,"tokens_out":13871,"duration_ms":151550,"significance":"If the mean-field solutions faithfully represent the true ground states, the paper offers a useful conceptual bridge between altermagnetism and superconducting/charge-ordered phases: the equal-spectrum relations (Eqs. 13–16) follow rigorously from Shiba duality, and the even-parity spin-η-locked splitting is a nontrivial prediction that goes beyond a direct index relabeling. The paper also provides an explicit HFB formalism in the Supplemental Material and makes falsifiable band-structure predictions for cold-atom and condensed-matter settings. However, the central ground-state claim currently rests on self-consistent HFB solutions without an unbiased assessment of competing orders or fluctuations, which limits the significance until that gap is addressed.","major_comments":[{"comment":"The self-consistent equation (S12) characterizes stationary points of the HFB energy, not global minima. The manuscript does not report a comparison with competing self-consistent states—such as uniform s-wave BCS, pure staggered CDM, other orientations of the η vector, or the normal state—nor does it report multiple random initializations, finite-size scaling, or an exact-diagonalization/QMC benchmark on small clusters. Since Figs. 1 and 2 are the principal evidence that η-ALM is a ground-state phenomenon, and the exact Shiba duality only maps the repulsive-side state (whose ALM order is itself assumed from mean-field/symmetry analysis), the central claim that these are ground states is under-supported. I would ask for at least a minimal stability analysis: energies of competing HFB solutions and a finite-size study, or an explicit statement that the claims are at the self-consistent HF","section":"§2D-lattice examples; Supplemental Eq. S12"},{"comment":"The abstract states that anisotropic second-neighbor hopping generates η-ALM, but Eq. (14) shows that for even parity the Shiba-dual Hamiltonian contains an explicit spin-dependent second-neighbor hopping, h_even τ3 σ3, rather than the spin-independent hopping introduced in Eq. (11). Thus the even-parity η-ALM examples in Figs. 1 and 2 are not the same family of attractive Hubbard models with spin-independent anisotropic hopping; they require an additional spin-dependent term whose physical origin should be stated in the abstract. Without this qualification, the generality of the proposal is overstated, and readers may incorrectly infer that spin-independent real second-neighbor hopping in the attractive model produces even-parity η splitting.","section":"Abstract and Eq. (14)"},{"comment":"Equation (15) shows that odd-parity η-ALM has a dispersion identical to that of the corresponding ALM under the index change σ_m→η_m. This is an exact consequence of Shiba duality and therefore does not by itself constitute an independent prediction; the genuinely new band-structure content of the paper is the even-parity spin-η-locked splitting of Eq. (16). The manuscript would be improved by stating this clearly and by framing the odd-parity examples as a dictionary translation rather than as novel numerical evidence. This does not invalidate the concept, but it affects how the claims in the introduction and abstract are weighted.","section":"Eq. (15) and discussion of odd-parity η-ALM"}],"minor_comments":[{"comment":"The lattice sizes appear as '162 ×2' and '182 ×2'; these should read '16^2 ×2' and '18^2 ×2'. The superscripts have been lost in the text.","section":"Figs. 1 and 2 captions"},{"comment":"The formula 'E^η_k = Σ_{n=1}^4 E_{nkη_m} n_{nk} in the occupied bands' is not fully defined. Please specify the occupation factor n_{nk}, the normalization, and the relation to the band-splitting magnitude |E_{n,+}-E_{n,-}|. As written, the plotted 'splitting energy' mixes occupied-state weighting and may not be the standard diagnostic for the p-/d-/f-wave structure.","section":"BZ splitting energy definitions, Figs. 1(d) and 2(d)"},{"comment":"The text says 'the mass term h_even ρ3 τ3 - m·τ3 η' gives the quoted correction, while Eq. (14) writes the even-parity term as h_even τ3 σ3. The notation is inconsistent; please clarify which operator is the spin-independent second-neighbor hopping in the BdG representation and which is the spin-dependent Shiba-dual counterpart.","section":"Eq. (14) and following text"},{"comment":"The sentence 'The components correspond to onsite uniform singlet pairing... which are degenerate in the ground state' could be clarified: the full SU_η(2) multiplet is degenerate in the exact ground state, but an HFB solution selects one member. This is related to the symmetry-breaking issue raised in the major comments.","section":"Discussion of degeneracy"}],"recommendation":"major_revision","confidential_remarks":"The paper is a reasonable candidate for publication after revision. The main barrier is the ground-state justification: the HFB solutions are not shown to be global minima and no unbiased check is provided. The scope issue about spin-dependent hopping in the even-parity case should also be fixed in the abstract. I do not see grounds for rejection, but the present version overstates the ground-state certainty."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nHere’s my take on Lin’s η-ALM paper. The genuinely new thing is the proposal that Shiba duality transplants the altermagnetic band-splitting idea from spin to η-pseudospin in attractive Hubbard models. The even-parity spin-η-locked splitting and the p/d/f-wave lattice examples are not in the prior literature, and the symmetry analysis is careful and internally consistent. The equal-spectrum relations (Eqs. 13–16) do follow rigorously from the exact Shiba duality, and the paper makes a real conceptual point: higher-angular-momentum splitting can come from the lattice, not from a bond order.\n\nThe soft spot is exactly the one the stress-test identifies. The concrete claim that these are ground states rests on Hartree-Fock-Bogoliubov solutions on finite clusters, with no comparison to competing orders, no multiple initializations, and no exact or QMC benchmark. The exact duality is not enough, because it maps the attractive model to the repulsive model with the same anisotropic hopping, and the ALM order in that repulsive model is itself assumed from mean-field/symmetry-based literature. So the paper’s central assertion is conditional: if the repulsive ALM order is the ground state, then the attractive η-ALM order is its exact image; the HFB numerics suggest it but don't prove it.\n\nI’d also note that the odd-parity η-ALM is, by the paper’s own Eq. (15), just the ALM dispersion with spin indices relabeled to η indices. That’s not a flaw—it follows from the duality—but it means the genuinely new content is the even-parity spin-η-locked case. The generalization to all bipartite lattices is asserted, not demonstrated, and the paper would be stronger if it at least sketched a finite-size scaling or an exact diagonalization check on a small cluster.\n\nWho is this for? Readers working on altermagnetism, unconventional superconductivity, or cold-atom Hubbard models will find the concept thought-provoking. It’s a plausible, symmetry-grounded extension of ALM, and the algebra is worth refereeing. The HFB evidence needs to be strengthened and the ground-state claim softened before I’d take the physics as established.\n\nRecommendation: send to peer review. A good referee can check the symmetry derivations and push for better numerics. The concept is new enough and the algebra sound enough that it should not be desk-rejected.","headline":"A genuinely new symmetry-based concept—η-ALM—with clean algebraic derivations, but the ground-state evidence is thin and the odd-parity case is mostly a relabeling of known ALM.","tokens_in":16832,"tokens_out":3235,"would_cite":true,"duration_ms":33260,"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":"Shiba duality maps altermagnetic band splitting from spin to η-pseudospin in attractive Hubbard models, defining η-altermagnetism as a Bogoliubov-de Gennes counterpart of altermagnetism in pairing and charge orders.","keywords":["Shiba duality","η-pseudospin","altermagnetism","attractive Hubbard model","Bogoliubov-de Gennes","charge density order","singlet pairing","bipartite lattice"],"falsifier":"An unbiased exact-diagonalization or quantum Monte Carlo study of the checkerboard or honeycomb attractive Hubbard model with t1=1, t2=0.1, U=-4 at half filling could measure the momentum-resolved spectral function and the η-pseudospin splitting. If the splitting vanishes in the thermodynamic limit or the ground state shows no long-range pairing/charge order, the mean-field η-ALM band picture would be refuted.","tokens_in":15869,"feed_emoji":"🧲","tokens_out":8294,"duration_ms":68319,"temperature":0.7,"pith_summary":"The paper argues that the altermagnetic principle—momentum-dependent band splitting in a compensated collinear order—can be transplanted from spin to η-pseudospin via Shiba duality. In half-filled bipartite attractive Hubbard models, the η-pseudospin (whose components are on-site singlet pairing and staggered charge density) plays the role of spin, and anisotropic second-neighbor hopping generates splitting of the Bogoliubov-de Gennes bands. The paper defines this as η-altermagnetism (η-ALM) and shows that odd-parity η-ALM gives pure η-pseudospin splitting while even-parity η-ALM gives spin-η-locked splitting. Hartree-Fock-Bogoliubov computations on checkerboard and honeycomb lattices produce p-, d-, and f-wave splitting structures. If correct, the result extends the altermagnetic band-splitting phenomenology into superconducting and charge-ordered ground states, without requiring higher-angular-momentum order parameters themselves.","feed_headline":"Altermagnetic splitting appears in pairing and charge orders","feed_subtitle":"Shiba duality maps spin splitting to η-pseudospin splitting in attractive Hubbard models.","key_machinery":"The central object is the η-pseudospin, an SU(2) pseudospin defined through Nambu spinors whose components are on-site uniform singlet pairing and staggered charge-density modulation. Shiba duality—a partial particle-hole transformation acting on one spin species—exactly maps repulsive Hubbard models to attractive ones at half filling. The load-bearing symmetry is the Shiba-dual P̃T̃, which protects Kramers degeneracy of η-pseudospins in the BdG bands of η-AFM. Anisotropic second-neighbor hopping (sublattice currents for odd parity, spin-dependent sublattice bonds for even parity) breaks this symmetry and generates the η-pseudospin or spin-η-locked splitting. Hartree-Fock-Bogoliubov theory p","core_discovery":"At half filling on bipartite lattices, Shiba duality exactly maps the repulsive Hubbard model to the attractive one, so the spin order of the repulsive side corresponds to η-pseudospin order on the attractive side. The paper shows that the η-pseudospin band structure mirrors the spin band structure: a Shiba-dual parity-time-reversal symmetry P̃T̃ enforces a Kramers degeneracy of η-pseudospins in the BdG bands of η-AFM. Introducing anisotropic second-neighbor hopping breaks this degeneracy and produces η-ALM. For sublattice currents (odd parity), the BdG bands develop η-pseudospin splitting with the same dispersion as spin altermagnetism; for sublattice spin bonds (even parity), the splitting","pith_inferences":["The mean-field result suggests a broader principle: any order parameter with an SU(2) symmetry that can be rotated by Shiba duality may exhibit altermagnetic-type splitting; one could look for orbital or valley analogues.","If the η-pseudospin splitting survives beyond mean field, it would connect altermagnetism to pair-density waves and charge order, potentially unifying seemingly unrelated higher-angular-momentum states.","A testable extension: tune a cold-atom attractive Hubbard system with anisotropic next-nearest-neighbor hopping and measure the spectral function; the predicted spin-η-locked splitting in the even-parity case could be distinguished by spin-resolved probes.","The exact correspondence also implies that known even-parity ALM materials might have attractive-Hubbard analogues realizable in optical lattices, enabling quantum simulation of the pairing-sector splitting."],"forward_implications":["Altermagnetic band-splitting phenomenology now has a pairing/charge-order counterpart: odd- and even-parity η-ALMs with p-, d-, and f-wave splitting structures on bipartite lattices.","The Shiba-dual correspondence implies that any ALM model on a bipartite lattice maps to an η-ALM model, so the known ALM taxonomy can be translated to the attractive side.","Doping the attractive model (equivalent to a Zeeman field on the repulsive side) produces canted η-ALM with additional alternate splitting associated with pairing orders beyond the s-wave η3 splitting of the pure model.","In ultracold-atom implementations, the predicted BdG band splitting could be observed with momentum-resolved Raman or radio-frequency spectroscopy, providing a direct experimental signature.","If η-ALM is realized in materials, junctions may show orientation-dependent Andreev reflection and Josephson effects, similar to altermagnet-superconductor junctions, with possible η-pseudospin torque."],"fun_headline_variants":["Shiba duality births η-altermagnetism in pairing orders","Pairing orders acquire η-altermagnetic splitting","η-altermagnetism emerges via Shiba duality in Hubbard pairs","From spin to pseudospin: altermagnetism in pairing orders","Shiba dual altermagnetism in pairing and charge orders"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the Hartree-Fock-Bogoliubov mean-field ground state—obtained by fully occupying the negative-energy BdG band—faithfully represents the true ground state of the interacting attractive model; in two dimensions, quantum fluctuations could destroy the sharp η-pseudospin splitting.","fun_headline_variants_meta":{"raw":{"variants":["Shiba duality births η-altermagnetism in pairing orders","Pairing orders acquire η-altermagnetic splitting","η-altermagnetism emerges via Shiba duality in Hubbard pairs","From spin to pseudospin: altermagnetism in pairing orders","Shiba dual altermagnetism in pairing and charge orders"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001388,"raw_usage":{"total_tokens":5474,"prompt_tokens":780,"completion_tokens":4694,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":524,"completion_tokens_details":{"reasoning_tokens":4610}},"tokens_in":524,"tokens_out":4694,"duration_ms":31389,"temperature":1.0,"reasoning_tokens":4610,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T07:27:33.823547+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"An unbiased exact-diagonalization or quantum Monte Carlo study of the checkerboard or honeycomb attractive Hubbard model with t1=1, t2=0.1, U=-4 at half filling could measure the momentum-resolved spectral function and the η-pseudospin splitting. If the splitting vanishes in the thermodynamic limit or the ground state shows no long-range pairing/charge order, the mean-field η-ALM band picture would be refuted.","supporting_citations":[],"review_version":1}