{"id":"c60cac9a-764c-4d11-9d81-0166eae9c963","arxiv_id":"2608.08827","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A bilayer acoustic crystal with a constructed pseudo-time-reversal operator realizes the first time-reversal-invariant altermagnetic acoustic system, observed via pseudospin-split bands and orthogonal propagation.","lead":"Researchers built an acoustic crystal that mimics the band-splitting of altermagnets, a magnetic phase with zero net magnetization, without breaking time-reversal symmetry. The device routes sound waves along perpendicular paths depending on a constructed pseudospin label, which could enable new acoustic signal-routing devices.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The realized acoustic structure's effective Hamiltonian is fitted to COMSOL, and the sign-reversed interlayer hopping (-κ) required by pseudo-TRS is never independently verified.","rationale":"The paper is a well-executed theory-plus-experiment study: the measured band structures, iso-frequency contours, and field maps agree with COMSOL simulations, and the tight-binding model is internally consistent with d-wave anisotropy terms that vanish in the symmetric limit. The reader's conditional verdict is appropriate. The most load-bearing concern is the unverified acoustic-to-tight-binding mapping and, more specifically, the sign of the B-sublattice interlayer hopping, which is structural to the pseudo-TRS claim. The tight-binding parameters are fitted to the same COMSOL simulation of the full geometry, so the fit does not independently establish that the realized acoustic network implements H↓ = T_p H↑ T_p^-1. This concern does not invalidate the experimental observations, but it limits how strongly they can be attributed to the altermagnetic model until the effective couplings are independently calibrated. Since this is the kind of missing independent support the reader already flagged, the verdict remains CONDITIONAL (no change).","tokens_in":11272,"tokens_out":17520,"duration_ms":200900,"concrete_test":"Measure the transmission and reflection spectra of a single interlayer cavity pair: two cavities of the A type (no hole / hole) connected by the design tube, and the corresponding B pair, at the operating frequency. Extract the off-diagonal hopping matrix element (magnitude and sign) from a two-level fit to the measured eigenfrequencies, and compare with +κ and -κ used in Fig. 2(b). Independently repeat for one NNN pair with the large tube (37.52 mm²) and one with the small tube (4.69 mm²) to check the assumed 8:1 hopping ratio and sign. If the B-pair hopping does not have the opposite sign to the A-pair, or the NNN ratio deviates substantially, then the claim that the fabricated crystal realizes H↓ = T_p H↑ T_p^-1 is not supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim depends on the realized cavity-tube network being described by Eqs. (1)-(3) with H↓ = T_p H↑ T_p^-1, i.e., the B-sublattice interlayer hopping is -κ while A uses +κ, and the on-site potentials are reversed. The acoustic realization section asserts the standard linear map (hole area sets -2V, tube cross-sections set hoppings) and carries over signs from refs [48-53], but the tight-binding parameters in Fig. 2(b) are obtained by fitting the same COMSOL simulation of the full structure. The fit enforces consistency between the acoustic bands and a four-band TB model, but it does not verify that the local two-block relation H↓ = -H↑ is physically implemented by the tube geometry; in particular, no fabrication-level detail explains how identical interlayer tubes connect A with +κ and B with -κ. If the realized interlayer coupling on B has the opposite sign from the assumed -κ, or if the hole-induced potential shift is strongly nonlinear, the realized crystal can still show anisotropic bands and orthogonal contours, but the pseudo-TRS construction at the heart of the claim would not be the active mechanism; the experiment would demonstrate an ordinary anisotropic bipartite acoustic lattice rather than the proposed altermagnetic model. Thus the weakest point is not the TB algebra but the independent verification of the acoustic-to-Hamiltonian map that connects the experiment to the model.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a tight-binding model of a bilayer centered square lattice with two sublattices (A, B) representing two pseudospin species. Each sublattice carries layer-dependent on-site potentials ±V and interlayer couplings ±κ, and the pseudo-time-reversal operator T_p = σ_y K connects the two sublattice blocks. When the next-nearest-neighbor couplings satisfy r1≠r3 and r2≠r4, the Hamiltonian acquires d-wave anisotropic terms proportional to (cos k_y - cos k_x), producing momentum-dependent pseudospin splitting, orthogonal iso-frequency contours, and sublattice–pseudospin locking. The authors realize this model in a two-layer acoustic cavity–tube crystal and report experimental measurements of split bulk bands, orthogonal iso-frequency contours, sublattice-localized fields, and pseudospin filtering at 3.48 kHz.","tokens_in":11597,"tokens_out":10446,"duration_ms":119138,"significance":"If the acoustic geometry faithfully implements the proposed Hamiltonian, this work would be a valuable classical-wave analogue of altermagnetism, showing that d-wave anisotropic splitting and sublattice-polarized transport can be engineered without breaking physical time-reversal symmetry. The experimental effort is substantial: measured bulk bands for the two pseudospin sectors, iso-frequency contours, real-space field maps, and a filtering demonstration, all compared with COMSOL and tight-binding calculations. The tight-binding derivation itself is internally consistent, and the d-wave terms correctly vanish when r1=r3 and r2=r4. The paper also contains a candid internal note, after Eq. (2), that the layer-selective field confinement is a parameter effect rather than a symmetry-protected feature. However, the central claim that this is a time-reversal-invariant altermagnetic acoustic crystal rests on several load-bearing assumptions that are not yet independently verified, in particular the acoustic realization of the sign of the interlayer hopping, the non-circularity of the fitted parameters, and the conservation of the pseudospin label in the full Hamiltonian.","major_comments":[{"comment":"The tight-binding model requires interlayer hopping +κ on the A sublattice and -κ on the B sublattice, but the text states that the interlayer tubes for κ share the identical cross-sectional area with the NN and NNN tubes. Since the hopping sign in coupled-cavity systems is set by the phase conventions and geometry of the tube connections, identical tubes cannot by themselves produce opposite signs. No fabrication-level detail is given for how the B sublattice interlayer coupling becomes -κ. If this sign is not actually realized, the pseudo-TRS relation H↓ = T_p H↑ T_p^{-1} is not implemented, and the observed anisotropic bands and orthogonal contours could arise from an ordinary anisotropic bipartite acoustic lattice rather than from the altermagnetic construction.","section":"Acoustic realization, Fig. 2(a)-(b)"},{"comment":"The iso-frequency contours in Fig. 2(c) are computed using parameters extracted by fitting the tight-binding Hamiltonian to the COMSOL band structure in Fig. 2(b). The orthogonal contours are therefore not an independent prediction but a consequence of the fitted model. To support the claim that the acoustic crystal realizes the altermagnetic Hamiltonian, the authors should either calibrate the hole-area/tube-area map independently (for example, from single-cavity and dimer simulations) or demonstrate that the fitted parameters are forced by the geometry without free adjustment.","section":"Fig. 2(c) and acoustic realization section"},{"comment":"The pseudospin operator S_z = τ_z ⊗ (Vσ_z + κσ_x)/√(V^2+κ^2) does not commute with the Hamiltonian because the nearest-neighbor term f(k)τ_x⊗σ_0 in Eq. (8) flips the sublattice index τ_z. Consequently, eigenstates at generic k are not pseudospin eigenstates, and the red/blue band labels in Figs. 1(c), 2(b), and 3(b)-(c) require a projection or dominance criterion that is not stated. Without such a criterion, the reported 'pseudospin-dependent band splitting' may be a labeling artifact rather than a symmetry-protected property. The authors should quantify the hybridization due to f(k) at the experimental parameters (|t|/V ≈ 0.025) and specify how the pseudospin channels are separated in the measured data.","section":"Eqs. (2)-(8)"},{"comment":"The 2×2 relation H↓ = T_p H↑ T_p^{-1} is k-independent, but in the full 4×4 Bloch Hamiltonian the operator T_p = (τ_0⊗σ_y)K does not map H(k) to H(-k) because it does not flip the sublattice index. The altermagnetic phase is instead defined through a combined operation involving the translation L(a/2,a/2), which is described only verbally. The authors should present the explicit action of L, T_p, and their combination on H(k), including the condition under which this combined operation is a symmetry and why it is broken when r1≠r3 and r2≠r4.","section":"Eq. (1) vs. Eq. (3)"}],"minor_comments":[{"comment":"The phase convention for the fermionic time-reversal operator is inconsistent: with the standard T_f = -iσ_y K, one has -iT_f = -σ_y K, not σ_y K. Please state the convention used for T_f.","section":"Eq. (1)"},{"comment":"The manuscript explicitly concedes that the layer-selective field confinement is 'a parameter effect rather than a symmetry-protected feature.' This qualification should also appear in the abstract or introduction when the claim of sublattice–pseudospin locking is made.","section":"After Eq. (2)"},{"comment":"The sign convention for the NNN hoppings r_i is not discussed: r1=r2=-0.006 kHz and r3=r4=8r1=-0.048 kHz are negative, while the text only relates their magnitude to the tube cross-sectional area. Please specify how the sign of each hopping is set by the geometry.","section":"Fig. 2(a)-(b)"},{"comment":"The terminology alternates between 'spin-sublattice locking' and 'sublattice–pseudospin locking.' To avoid confusion with physical spin, please use 'sublattice–pseudospin locking' consistently throughout.","section":"Figs. 3 and 4 captions"}],"recommendation":"major_revision","confidential_remarks":"The paper represents a serious experimental and theoretical effort, but the novelty claim of a 'time-reversal-invariant altermagnetic acoustic crystal' will likely be scrutinized by reviewers who view the pseudospin as a sublattice label rather than a genuine spin-like degree of freedom. The authors should be asked to provide an independent calibration of the acoustic-to-tight-binding map, to explain the physical origin of the sign-reversed interlayer hopping, and to define the pseudospin projection used to color the bands. These are fixable within the scope of the manuscript, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here’s my take on arXiv:2608.08827. The paper is a serious, well-executed attempt to bring altermagnetism into acoustics without breaking time-reversal symmetry. The new thing is the construction: two pseudospin degrees of freedom on a bilayer centered square lattice, connected by a pseudo-time-reversal operator that flips pseudospin while preserving physical TRS. That this can be done for scalar sound waves—no helicity, no flow bias—is a genuine step forward. The tight-binding model is internally consistent, the d-wave anisotropy terms vanish in the symmetric limit, and the experiments are impressive: spin-split bands, orthogonal iso-frequency contours, sublattice locking, and directional filtering all show up in the measured data and match the simulated bands.\n\nThe soft spots are real but not fatal. First, the acoustic parameters are extracted by fitting the tight-binding Hamiltonian to the COMSOL band structure, so the iso-frequency contours in Fig. 2(c) are not independent predictions; they are consequences of the fit. The experiments then confirm the simulated contours, which is consistent, but it doesn't independently verify the model. Second, the sign-reversed interlayer hopping (-κ) between B sublattice layers is essential for the pseudo-time-reversal construction, yet nothing in the fabrication or measurement demonstrates that the realized coupling has the opposite sign. Identical tube cross-sections are used for κ on both sublattices; the sign is simply carried over from earlier acoustic tight-binding papers. If the realized sign were wrong, the crystal could still show anisotropic bands and orthogonal contours as an ordinary bipartite lattice, and the 'altermagnetic' attribution would be empty. The pseudospin operator is also defined by sublattice index, so some of the 'spin splitting' is contained in the definition. None of these are disqualifying for a Letter, but they should be stated more carefully. The data availability statement is weak, and the supplementary material wasn't available for this review.\n\nOverall, the central algebra is sound and the experiment is a solid demonstration of anisotropic acoustic wave control. The paper deserves a serious referee. A good referee should ask for (i) public data and the supplementary file, (ii) some independent check of the effective Hamiltonian (for example, a fabricated structure where the interlayer coupling sign is deliberately flipped, or a direct measurement of the two-layer phase relation), and (iii) a more careful statement that this is an engineered analogue, with the caveat that the sign-reversed coupling is assumed. If those requests are met, I'd support publication. The paper is worth discussing in a reading group.","headline":"A serious, well-executed theory-plus-experiment paper proposing the first TRS-preserving acoustic altermagnet analogue, but the experiment-model link is fitted and the sign-reversed interlayer coupling is never independently verified.","tokens_in":12125,"tokens_out":3161,"would_cite":true,"duration_ms":36321,"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":"A bilayer acoustic crystal with two pseudospin channels reproduces altermagnetic band splitting under preserved time-reversal symmetry.","keywords":["time-reversal-invariant altermagnetism","acoustic crystal","pseudospin","pseudo-time-reversal operator","band splitting","sublattice-spin locking","iso-frequency contours","tight-binding model"],"falsifier":"Build a control acoustic sample with $r_1 = r_3$ and $r_2 = r_4$, the paper's antiferromagnetic condition: the model predicts no pseudospin splitting anywhere in the Brillouin zone, so a measured splitting in that sample would refute the altermagnetic interpretation. A second decisive check is to calibrate the cavity-hole and tube-geometry mapping on single-cell test structures and compare the fitted tight-binding parameters against the values used in the band-structure calculations.","tokens_in":11042,"feed_emoji":"🔊","tokens_out":18392,"duration_ms":153317,"temperature":0.7,"pith_summary":"Altermagnets are magnetic phases whose spin-up and spin-down bands split in momentum even though the material carries no net magnetization, a combination traditionally thought to require broken time-reversal symmetry (TRS). This paper claims to remove that requirement in a classical-wave system by giving two pseudospin degrees of freedom (synthetic two-valued labels that play the role of electron spin) to the two sublattices of a bilayer acoustic lattice, connected by a pseudo-time-reversal operator that flips pseudospin while leaving physical TRS intact. The result is the first time-reversal-invariant altermagnetic acoustic crystal, whose measurements show pseudospin-split bands, anisotropic iso-frequency contours oriented orthogonally for the two channels, and real-space sublattice-spin locking. If the claim holds, acoustic crystals become a testbed for altermagnetic physics, and the same two-pseudospin construction transfers to photonic and mechanical wave systems.","feed_headline":"Altermagnet sound splits without breaking time reversal","feed_subtitle":"The two pseudospin channels split and steer sound along perpendicular paths, with no magnetic bias required.","key_machinery":"The carrying object is the pseudo-time-reversal operator $\\mathcal{T}_p = \\sigma_y K$ (with $\\mathcal{T}_p^2 = -1$), built from the layer Pauli matrix $\\sigma_y$ and complex conjugation, acting on a bilayer centered square lattice whose two sublattices carry opposite pseudospin configurations. It reproduces the action of the genuine fermionic time-reversal operator up to a phase, so the system can host pseudo-spin splitting while real time reversal remains unbroken. The control parameters are the four next-nearest-neighbor couplings $r_1, r_2, r_3, r_4$: unequal couplings for identical pseudospins along the $x$ and $y$ directions break the pseudo-time-reversal symmetry and generate the $d$-wave terms $\\Delta_\\tau$ and $\\Delta_\\sigma$, both proportional to $(\\cos k_y a - \\cos k_x a)$, the algebraic signature of the altermagnetic phase. In the experiment the machinery is physical: cavities and connecting tubes realize the tight-binding parameters, with a central hole of area $S_V$ lowering the on-site potential by $2V$ and tube cross-sectional areas setting the hopping amplitudes.","core_discovery":"The central claim is that the defining signatures of altermagnetism can appear in a system that strictly preserves time-reversal symmetry. The construction is a bilayer centered square lattice whose A and B sublattices host opposite pseudospins: pseudospin-up carries on-site potentials $(V,-V)$ across the two layers with interlayer hopping $\\kappa$, while pseudospin-down carries $(-V,V)$ with $-\\kappa$. A pseudo-time-reversal operator $\\mathcal{T}_p = \\sigma_y K$, with $\\mathcal{T}_p^2 = -1$, relates the two sectors and differs from the genuine fermionic time-reversal operator only by a phase factor, yet it leaves physical TRS intact. When the next-nearest-neighbor hoppings satisfy $r_1 \\neq r_3$ and $r_2 \\neq r_4$, the Bloch Hamiltonian develops terms proportional to $(\\cos k_y a - \\cos k_x a)$; this is the $d_{x^2-y^2}$-wave anisotropy of altermagnetism, producing momentum-dependent pseudospin splitting and locking each pseudospin to its own sublattice, while restoring $r_1 = r_3$, $r_2 = r_4$ returns degenerate bands, the antiferromagnetic limit. The authors realize this in an acoustic cavity-tube crystal and report measurements of spin-split bands, orthogonally elongated iso-frequency contours, and spatially separated pseudospin channels that match the model.","pith_inferences":["A decisive follow-up would be to calibrate the cavity-hole and tube-geometry mapping on single cells and compare the fitted tight-binding parameters with the values used in the band-structure fits, since the linear relations are adopted from earlier acoustic work rather than verified in situ.","The operator construction suggests a broader dictionary: any collinear electronic altermagnet could be mimicked by a layered classical lattice with staggered on-site potentials, potentially extending beyond spin-1/2 pseudospins to higher pseudospin textures with richer splitting patterns.","If the linear mapping holds, a reconfigurable version could switch between the altermagnetic and antiferromagnetic phases dynamically, for instance by mechanically or electrically tuning the tube cross-sections, turning the sample into a tunable acoustic router.","Because the split channels carry no net angular momentum or charge, the demonstrated filtering is purely geometric; one concrete extension is a pseudospin-addressed acoustic network in which signals are steered by sublattice site alone."],"forward_implications":["The measured spin-split bands, orthogonally elongated iso-frequency contours, and sublattice-confined pressure fields count as direct experimental evidence of altermagnetic sublattice-pseudospin locking under strictly TRS-preserving conditions.","The crystal works as a pseudospin filter: simultaneous excitation of both channels produces orthogonal propagation, so each output port delivers a single pseudospin species acoustically.","The same two-pseudospin construction with a pseudo-time-reversal operator transfers directly to photonic, mechanical, and other classical-wave lattices, giving a general route to simulate magnetic symmetries without magnetic bias.","The same lattice with $r_1 = r_3$ and $r_2 = r_4$ is the antiferromagnetic phase with degenerate bands, so a single acoustic design hosts both magnetic phases depending only on how the next-nearest-neighbor couplings are arranged."],"supporting_citations":[{"why":"Supplies the acoustic mapping that a central cavity hole of area $S_V$ lowers the on-site potential by $2V$, the relation the experimental design relies on.","marker":"[48]"},{"why":"Supplies the cavity-tube construction by which tube cross-sectional areas set hopping amplitudes in acoustic lattices.","marker":"[49]"},{"why":"Another instance of the cavity-tube hopping mapping, adopted for the interlayer and next-nearest-neighbor couplings of the acoustic crystal.","marker":"[50]"},{"why":"Prior acoustic metamaterial experiment using the same coupling-by-tube-geometry scheme, cited as justification for linear control of the hoppings.","marker":"[52]"},{"why":"Defines altermagnetism and the broken time-reversal symmetry it conventionally requires, the framework the paper claims to circumvent.","marker":"[2]"}],"fun_headline_variants":["Sound waves show altermagnetism without time-reversal break","Acoustic crystal mimics altermagnetism with intact time symmetry","Pseudospin splitting in sound without breaking TRS","Time-reversal-safe altermagnetic crystal for sound","Altermagnetic sound crystal preserves time reversal"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The experimental realization rests on the assumption, adopted from earlier acoustic work (the design of Fig. 2(a) and the surrounding text) rather than calibrated in this paper, that the cavities and tubes implement the tight-binding Hamiltonian with linear relations: a central hole of area $S_V$ lowers the on-site potential by exactly $2V$, and tube cross-sectional areas set hopping amplitudes proportionally; if the realized effective parameters drift substantially from these assumed values, the measured splitting, contours, and filtering would no longer match the altermagnetic model.","fun_headline_variants_meta":{"raw":{"variants":["Sound waves show altermagnetism without time-reversal break","Acoustic crystal mimics altermagnetism with intact time symmetry","Pseudospin splitting in sound without breaking TRS","Time-reversal-safe altermagnetic crystal for sound","Altermagnetic sound crystal preserves time reversal"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000168,"raw_usage":{"total_tokens":1296,"prompt_tokens":1015,"completion_tokens":281,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":631,"completion_tokens_details":{"reasoning_tokens":205}},"tokens_in":631,"tokens_out":281,"duration_ms":3340,"temperature":1.0,"reasoning_tokens":205,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:23:28.877439+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Build a control acoustic sample with $r_1 = r_3$ and $r_2 = r_4$, the paper's antiferromagnetic condition: the model predicts no pseudospin splitting anywhere in the Brillouin zone, so a measured splitting in that sample would refute the altermagnetic interpretation. A second decisive check is to calibrate the cavity-hole and tube-geometry mapping on single-cell test structures and compare the fitted tight-binding parameters against the values used in the band-structure calculations.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the acoustic mapping that a central cavity hole of area $S_V$ lowers the on-site potential by $2V$, the relation the experimental design relies on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the cavity-tube construction by which tube cross-sectional areas set hopping amplitudes in acoustic lattices."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Another instance of the cavity-tube hopping mapping, adopted for the interlayer and next-nearest-neighbor couplings of the acoustic crystal."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Prior acoustic metamaterial experiment using the same coupling-by-tube-geometry scheme, cited as justification for linear control of the hoppings."},{"cited_title":"Šmejkal, J","cited_arxiv_id":null,"evidence_quote":"Defines altermagnetism and the broken time-reversal symmetry it conventionally requires, the framework the paper claims to circumvent."}],"review_version":1}