{"id":"cc68596b-deb0-4706-8971-520b898b586b","arxiv_id":"2607.13923","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"d-wave axial phonons with an angular-momentum texture arise in orbital altermagnets via molecular Berry curvature, without spin-orbit coupling, enabling angular-momentum Seebeck and splitter effects.","lead":"This theory paper predicts that orbital altermagnets—crystals with a d-wave loop-current magnetic order—can generate phonons (lattice vibrations) carrying angular momentum in a d-wave pattern, without any spin-orbit coupling. If confirmed, this gives a new, SOC-free platform for angular-momentum transport effects such as a heat-driven 'angular-momentum splitter'.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The phonon Hamiltonian drops the reactive electron-phonon self-energy from the same t' coupling; its scale is comparable to the chosen force constants, so the quantitative d-wave texture and transport magnitudes are model-dependent.","rationale":"The most load-bearing condition for the central claim is not the symmetry analysis, which is solid, nor the existence of a d-wave MBC, which follows from the 4'/mm'm magnetic point group; it is whether the phonon Hamiltonian used to compute l_kσ and χxx is the correct Born-Oppenheimer Hamiltonian. The paper's minimal-coupling construction in Eq. (31) is standard, but it is only one part of the electron-phonon feedback: the same coupling that produces the Berry-curvature gauge field also renormalizes the phonon potential. This reactive self-energy is of the same order as the model's spring constants for the quoted parameters, so it is not a small perturbation. The reader's weakest_assumption mentioned 'other electron-phonon self-energy contributions beyond the MBC', which is close; my concern sharpens this into a concrete, checkable omission within the same adiabatic framework. I do not think this overturns the qualitative d-wave prediction: the symmetry-enforced structure of G_k and the vanishing at Γ are independent of D_k, and the d-wave transport tensor is fixed by symmetry. However, the quantitative predictions — branch-resolved l_kσ, acoustic vs optical magnitudes, and the only estimated observable χxx — are not robust until δD is included. That is a limitation, not a refutation, so the reader's ACCEPT verdict remains appropriate.","tokens_in":40019,"tokens_out":22111,"duration_ms":247352,"concrete_test":"Numerically compute the one-loop zero-frequency phonon self-energy from the same nearest-neighbor t' coupling, δD_{κα,κ'β}(k) = -Σ_{occ,unocc} <occ|∂H/∂u|unocc><unocc|∂H/∂u|occ>/(E_unocc-E_occ), add it to D_k in Eq. (38), and re-diagonalize H^eff in Eq. (42). If the d-wave angular momentum texture and the sign structure of χxx survive (with only moderate quantitative shifts), the central claim holds as stated; if the acoustic angular momentum disappears or the optical branches are substantially reshuffled, the MBC-only result is an artifact of the omitted reactive term.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim rests on Eq. (32), where the electron-phonon coupling enters only through the molecular-Berry-curvature gauge term p - hbar A, while D_k in Eq. (38) is a bare force-constant matrix. The same t' coupling necessarily also generates a zero-frequency reactive (Born-Oppenheimer potential) correction to the phonon dynamical matrix, of order δD ~ t'^2/E_gap. With t' = 1 eV/Å and the smallest direct gap ~0.6 eV at X/Y, δD ~ 27 N/m, i.e., ~25% of the dominant nearest-neighbor force constant n11 = -100 N/m and much larger than the tiny G^2 term that is retained. If δD is comparable to or larger than the bare spring constants, the phonon eigenvectors and frequencies entering Eq. (45) acquire substantial corrections, changing the magnitudes of l_kσ and the acoustic/optical branch ratios, and therefore also the quantitative χxx in Eq. (49). The d-wave symmetry itself is protected by the magnetic point group, so the qualitative texture and the sign structure of χ likely survive; but the specific values and the relative acoustic vs optical amplitudes are not established unless δD is included. The manuscript does not provide code or data that would let a reader verify whether the omitted self-energy reshuffles the modes.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs a minimal two-band tight-binding model of a loop-current orbital altermagnet on the checkerboard lattice, computes the electronic orbital magnetic moment texture and shows that it has d-wave symmetry under the magnetic point group 4'/mm'm. It then computes the molecular Berry curvature (MBC) from a nearest-neighbor electron-phonon coupling and uses the minimal-coupling prescription to modify the phonon Hamiltonian. Solving the coupled phonon eigenproblem yields finite phonon angular momentum l_kσ for all four branches with a d-wave texture. The paper further derives longitudinal (Seebeck) and transverse (splitter) angular-momentum transport responses to a temperature gradient, estimating χ_xx ~ 10^-4 k_B for τ = 1 ns. The central claim is that orbital altermagnets provide a spin-orbit-coupling-free route to d-wave axial phonons and to associated heat-driven angular-momentum currents.","tokens_in":40395,"tokens_out":6786,"duration_ms":66582,"significance":"If correct, this is a meaningful conceptual extension: it connects altermagnetic order, beyond the spin sector, to phonon angular momentum and predicts a directional transport signature. The symmetry analysis is careful: Appendix A gives explicit derivations of the MBC and of the symmetry constraints; the d-wave texture is symmetry-enforced rather than fitted; and the paper makes a concrete, falsifiable prediction — the sign of the angular-momentum Seebeck response reverses under a 90° rotation of the temperature gradient, and a purely transverse symmetric response appears along the nodal lines. These qualitative predictions are robust. However, the quantitative magnitudes of l_kσ and of χ_xx are model-dependent and, as argued below, are not yet established because of an omitted reactive electron-phonon self-energy and the heuristic current definition.","major_comments":[{"comment":"The phonon Hamiltonian in Eq. (32) retains the MBC gauge term (p - ħA)^2 but keeps D_k in Eq. (38) as a bare force-constant matrix. The same electron-phonon coupling t' that generates the MBC also generates a reactive (Born-Oppenheimer) correction to the phonon potential, δD ~ t'^2/E_gap. With t' = 1 eV/Å and the smallest direct gap ~0.6 eV near X/Y, δD ~ 27 N/m, which is ~25% of the dominant nearest-neighbor constant n11 = -100 N/m and much larger than the retained G^2 term. Since l_kσ in Eq. (45) depends on phonon eigenvectors and frequencies, the quantitative amplitudes, acoustic/optical ratios, and hence χ_xx in Eq. (49) are not established unless δD is included in D_k or shown to be negligible. The d-wave sign structure is protected by the magnetic point group, so the qualitative claim survives, but the numerical values in Figs. 4 and 5 are model-dependent.","section":"Sec. IV, Eq. (32) and Eq. (38)"},{"comment":"The angular-momentum current is evaluated as j_z = V^{-1} Σ l_kσ v_kσ f_kσ, and χ_xx is computed with a constant relaxation time τ chosen between 1 ps and 1 ns. The authors acknowledge in the Discussion that this definition ignores the non-commutativity of the angular-momentum and velocity operators and the non-conservation of phonon angular momentum. The symmetry argument for the splitter effect (Eqs. (50)-(51)) is robust, but the specific value χ_xx ~ 10^-4 k_B and the temperature dependence shown in Fig. 5(c) are not a controlled prediction. The authors should either derive the current from the microscopic phonon Hamiltonian (including the MBC correction to the phonon velocity) or explicitly present the result as an order-of-magnitude symmetry-based estimate.","section":"Sec. V, Eqs. (47)-(49)"},{"comment":"The electron-phonon coupling is restricted to the nearest-neighbor hopping t; the second-neighbor hoppings t_A and t_B and the staggered potential V are treated as frozen. This is acknowledged in the text, but it is load-bearing for the quantitative MBC: Eqs. (30c,d) derive from this approximation and are not symmetry-enforced. In a realistic orbital altermagnet, bond-length modulation of t_A and t_B will contribute to the MBC and will change the detailed k-dependence of G_k and hence of l_kσ, even though the d-wave symmetry remains. The paper should either extend the electron-phonon coupling to the second-neighbor bonds or provide a quantitative estimate of the resulting corrections before presenting specific values for l_kσ and χ_xx.","section":"Sec. III, Eqs. (19)-(20) and App. A 3a"}],"minor_comments":[{"comment":"The equation 'G_Ay_By(k) = G_Ay_By(k) = 0' is tautological; presumably one of the entries should be G_By_Ay(k).","section":"Eq. (30c)"},{"comment":"The heading repeats 'G_Ax_Bx(k=0) and G_Ax_Bx(k=0)' twice; the second should be G_Ay_By(k=0).","section":"App. A 3b, item (c)"},{"comment":"The statement that the product v_x v_y l is 'an odd function in k' is not generally true: under inversion v_x and v_y are both odd, so the product is even. Please specify the symmetry operation (e.g., C4z,+T) that makes the integral vanish.","section":"Sec. V, after Eq. (50)"},{"comment":"The color scale and axis labels are hard to read, especially on the log-scale color bar. Please label the axes and color bar explicitly.","section":"Fig. 5(c)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the journal's scope and presents a conceptually interesting proposal. The main issue is the omitted reactive electron-phonon self-energy, which undermines the quantitative predictions but not the symmetry-protected d-wave texture. I believe this is fixable in revision by recomputing or reinterpreting the FCM and by clearly separating qualitative symmetry results from quantitative model-dependent estimates."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper gives a SOC-free route to d-wave axial phonons, and the core symmetry argument is solid. The magnitudes, however, are on shakier ground because the model drops a reactive electron-phonon correction that is comparable to its own spring constants.\n\nWhat is actually new: they take an established mechanism, molecular Berry curvature transferring electronic angular momentum texture to phonons, and apply it to a loop-current orbital altermagnet. Refs. [42,43] needed SOC or magnon-phonon coupling to get d-wave phonon textures; this paper gets it purely from orbital currents. The transport proposal, a phonon angular-momentum Seebeck effect with a sign change under 90-degree rotation, and the corresponding splitter geometry, is a clean and testable signature. That is a real contribution.\n\nWhat is done well: the formal sections are careful. The MBC derivation using Slater-Condon rules is complete, the symmetry analysis of the MBC matrix is explicit, and the appendix proving no optical splitting at the Gamma point from the magnetic point group is a nice touch. The phonon dynamics and second quantization are standard but transparent. The d-wave texture of the phonon angular momentum follows from the symmetries, so the qualitative result does not rest on the specific model parameters.\n\nSoft spots: first, the neglect of the reactive electron-phonon self-energy. The same t' hopping modulation that generates the MBC also generates a real, zero-frequency correction to the dynamical matrix, of order t'^2/gap. With t' = 1 eV/Å and a gap around 0.6 eV, that is roughly 27 N/m, about 25% of the chosen n11. If D_k is the bare lattice force constant, this omission can renormalize phonon frequencies and eigenvectors enough to change the acoustic-to-optical ratio of the angular momentum, as well as the magnitude of chi_xx. If D_k already represents the renormalized force constant, the paper should state that; as written, it reads as a bare spring matrix. The d-wave texture is still protected by symmetry, but the computed numbers are not robust to this choice. Second, the transport calculation uses a single constant relaxation time and a Boltzmann picture with velocity defined as hbar^{-1} d epsilon/dk, which is known to be incomplete for phonon angular momentum currents; the authors acknowledge some of this. So treat chi_xx as an estimate of scale, not a quantitative prediction. Also, no code or data is provided, which makes the numerical plots hard to check.\n\nThese issues are real but not disqualifying. The central claim, that orbital loop currents can generate d-wave phonon angular momentum without SOC, is well-supported by the symmetry analysis, and the proposed splitter effect is a valuable direction. I would send this to a good referee with a request to make the self-energy issue explicit.","headline":"A clean symmetry-based demonstration of SOC-free d-wave phonon angular momentum in orbital altermagnets, with a testable transport signature; the qualitative result is solid, but the numerical magnitudes need a second look at the reactive electron-phonon self-energy.","tokens_in":40878,"tokens_out":5237,"would_cite":true,"duration_ms":56477,"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":"Phonons can carry d-wave angular-momentum textures in orbital altermagnets without spin-orbit coupling, enabling heat-driven angular-momentum currents.","keywords":["axial phonons","orbital altermagnetism","molecular Berry curvature","d-wave texture","phonon angular momentum","angular-momentum Seebeck effect","angular-momentum splitter effect","loop-current order"],"falsifier":"Measure the momentum-resolved phonon angular momentum or circular-phonon dichroism in a loop-current orbital altermagnet near the X and Y points: if no sign-changing four-lobed texture with zero at the zone center and nodal lines along the diagonals is seen, or if the optical phonon modes split at the zone center, the central claim is false. A thermal-transport check: rotate the temperature gradient by 90 degrees and look for the predicted sign reversal of the angular-momentum current; its absence would also falsify the d-wave mechanism.","tokens_in":1458,"feed_emoji":"🧲","tokens_out":1552,"duration_ms":64554,"temperature":0.7,"pith_summary":"This paper tries to establish that axial phonons, lattice vibrations carrying finite angular momentum, can acquire a d-wave angular-momentum texture in orbital altermagnets without spin-orbit coupling. On a checkerboard lattice with loop-current order, the molecular Berry curvature generated by electron-phonon coupling transfers the electrons' d-wave orbital-magnetic-moment pattern to both optical and acoustic phonons. The resulting d-wave texture implies two thermal-transport responses: a longitudinal angular-momentum Seebeck effect whose sign flips when the temperature gradient is rotated by 90 degrees, and a transverse angular-momentum splitter effect when the gradient lies along the nodal diagonals. A sympathetic reader would care because this removes the need for relativistic interactions and identifies a concrete, symmetry-protected route to phonon angular momentum and its manipulation by heat flow.","feed_headline":"Phonons gain a d-wave angular-momentum texture without spin-orbit","feed_subtitle":"Heat flow can drive angular-momentum currents that flip sign when the temperature gradient is rotated by 90 degrees.","key_machinery":"The molecular Berry curvature (MBC), a gauge-invariant real-space Berry curvature of the Born-Oppenheimer electronic ground state defined from derivatives of the ground state with respect to ionic displacements, is the central object. It enters the phonon Hamiltonian through minimal coupling, with the Berry connection expressed through the MBC, so the curvature acts like a magnetic field for phonons. What carries the argument is the d-wave loop-current tight-binding model on the checkerboard lattice with magnetic point group 4'/mm'm, whose two-band structure lets the MBC be computed analytically and gives the phonon angular momentum its d-wave form.","core_discovery":"The central claim is a transfer mechanism: in the Born-Oppenheimer approximation, electron-phonon coupling generates a molecular Berry curvature that acts as a real-space gauge field on lattice vibrations. For the minimal two-band model of a d-wave loop-current orbital altermagnet, this curvature inherits the symmetry of the electronic orbital magnetic moments, producing a d-wave phonon angular momentum with four alternating-sign lobes and nodal lines along the diagonals. Both optical and acoustic branches show the texture, with the magnetic point group 4'/mm'm forbidding any optical-mode splitting at the Brillouin-zone center and hence no time-reversal-odd axial moment there. The paper conn","pith_inferences":["Editorial inference: the same d-wave transport tensor could be observed as boundary accumulation of phonon angular momentum, potentially converted into an electrical signal through the inverse spin Hall effect at a metallic contact; the paper mentions this detection route, but the device-level consequence is an extension.","Editorial inference: because the molecular Berry curvature scales with electron-phonon coupling and inverse squared electronic gaps, materials with small indirect gaps or strong coupling could push the response from the toy-model estimate into the range of magnonic angular-momentum conductivities; a systematic material search is a testable extension.","Editorial inference: odd-parity p-wave or higher-order g- and i-wave altermagnets would produce different angular-momentum textures and transport anisotropies, suggesting a family of phonon angular-momentum switches controlled by crystal symmetry.","Editorial inference: measuring the sign flip under a 90-degree rotation of the temperature gradient would simultaneously test both the d-wave symmetry and the molecular-Berry-curvature mechanism; absence of the flip would point to missing non-adiabatic corrections."],"forward_implications":["Orbital altermagnets, including loop-current systems such as certain kagome metals and cuprate-like models, become candidate platforms for intrinsic axial phonons without spin-orbit coupling.","A temperature gradient along a high-symmetry axis drives a longitudinal angular-momentum current; rotating the gradient by 90 degrees reverses its sign, a hallmark of d-wave symmetry.","A gradient along a nodal diagonal drives a transverse angular-momentum current with no accompanying heat current, the angular-momentum splitter effect.","The phonon angular-momentum magnitude grows as the inverse square of the relevant indirect electronic gap, so smaller-gap orbital altermagnets should show strongly enhanced responses; the toy-model estimate gives chi_xx around 10^-4 k_B at tau = 1 ns.","The symmetry argument is generic: any d-wave altermagnet whose magnetic point group forbids ferromagnetism should exhibit a vanishing zone-center axial moment and a symmetry-protected d-wave phonon texture."],"fun_headline_variants":["Heat flow splits phonon angular momentum in orbital altermagnets","Phonon d-wave texture emerges without spin-orbit coupling","Orbital altermagnets give phonons a d-wave angular momentum","Temperature gradient drives transverse angular momentum currents","No spin-orbit needed for d-wave phonon angular momentum"],"cache_read_input_tokens":42112,"weakest_assumption_plain":"The whole picture rests on the Born-Oppenheimer molecular-Berry-curvature treatment: adiabatic electrons, with the minimal-coupling term as the only electron-phonon effect on phonon dynamics; if non-adiabatic corrections or additional electron-phonon self-energy terms are not negligible, the d-wave phonon angular momentum and its transport signatures would not survive.","fun_headline_variants_meta":{"raw":{"variants":["Heat flow splits phonon angular momentum in orbital altermagnets","Phonon d-wave texture emerges without spin-orbit coupling","Orbital altermagnets give phonons a d-wave angular momentum","Temperature gradient drives transverse angular momentum currents","No spin-orbit needed for d-wave phonon angular momentum"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000585,"raw_usage":{"total_tokens":2554,"prompt_tokens":677,"completion_tokens":1877,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":421,"completion_tokens_details":{"reasoning_tokens":1792}},"tokens_in":421,"tokens_out":1877,"duration_ms":11423,"temperature":1.0,"reasoning_tokens":1792,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T03:18:25.743606+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the momentum-resolved phonon angular momentum or circular-phonon dichroism in a loop-current orbital altermagnet near the X and Y points: if no sign-changing four-lobed texture with zero at the zone center and nodal lines along the diagonals is seen, or if the optical phonon modes split at the zone center, the central claim is false. A thermal-transport check: rotate the temperature gradient by 90 degrees and look for the predicted sign reversal of the angular-momentum current; its absence would also falsify the d-wave mechanism.","supporting_citations":[],"review_version":1}