{"id":"6218ffcf-08f7-4a6e-bb1b-c2452870b3ce","arxiv_id":"2505.04726","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Magnon thermal Hall and spin Nernst conductivities in a Lieb lattice altermagnet scale with the altermagnetic splitting, computable from a new closed-form symplectic quantum geometric tensor.","lead":"This paper derives a compact formula for the quantum geometry of two-band magnon systems and uses it to compute thermal Hall and spin Nernst conductivities in a model altermagnet. The conductivities grow with the altermagnetic splitting, suggesting a possible transport-based probe of altermagnetism.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The transport formulas in Eq. (28) are imported, not derived; the sign structure of the particle-hole doubled response is load-bearing and unverified.","rationale":"The reader's weakest assumption, the unverified transport formula from the particle-hole-symmetric doubled Nambu construction, is the same single point on which the central claim turns. I considered the leading-order Holstein-Primakoff truncation, the indefiniteness of the 'quantum metric' G in Eq. (18), and the sign convention for the Berry curvature; none of these would, if wrong, change the predicted proportionality to J2-J2' as sharply as a wrong sign in Eq. (28). The paper is otherwise carefully structured: the pseudo-unitary algebra in Sec. II, the analytic Berry curvature Eq. (27), and the low/high temperature expansions are internally consistent, and Eq. (33) follows from Eq. (28) once the latter is accepted. The missing derivation is explicitly acknowledged in the text ('Up to an additive constant... doubling... Then Ref. [45] shows...'), so this is a real gap rather than an external preference. A direct Kubo derivation would either confirm Eq. (28) or reveal the sign/weight correction needed; until then, the central result should remain conditional. Hence no change to the reader's CONDITIONAL verdict; agreement is complete.","tokens_in":13306,"tokens_out":28436,"duration_ms":296949,"concrete_test":"Independently derive kappa_xy and alpha_xy for the original 2x2 bosonic Hamiltonian in Eq. (1) using the standard Matsumoto-Shindou-Murakami Kubo linear-response formalism (Refs. [42,43]) applied directly to the two physical magnon modes, without the particle-hole doubling step. Check whether the resulting c1 and c2 combinations match Eq. (28), especially whether the second term in (28b) enters with a plus or a minus, and whether the energy argument is E_-(-k) or E_-(k). A fast numerical cross-check is to compute kappa_xy at B=0 with J2 != J2': Eq. (28b) must integrate to zero because F+ is d-wave odd and delta is d-wave odd, while a sign error in the second term would generically give a nonzero value. If the direct derivation disagrees, recompute Fig. 5 and Eq. (33).","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section III B, just before Eq. (28), the two linear-response formulas are imported rather than derived: the text states that Eq. (24) is not particle-hole symmetric but can be rewritten in that form by doubling the Nambu array, and that Ref. [45] then gives Eq. (28). This is the exact step converting the microscopic Bose operators into the transport coefficients, yet the paper provides no derivation and no check that the doubling preserves the physical heat and spin current operators. The weight of the concern is visible in the sign/band structure: alpha_xy in (28a) is a difference c1(epsilon+)-c1(epsilon-(-k)) while kappa_xy in (28b) is a sum c2(epsilon+)+c2(epsilon-(-k)). If the particle-hole doubled response required the same sign for the second c2 term, the leading small-delta expansion of kappa would be proportional to delta^2 and the claimed B(J2-J2') dependence in Eq. (33b) would not survive; if (28a) carried a plus in c1, alpha would be quadratic in the altermagnetic splitting and the central prediction would collapse. The doubling also has to handle the c-number shift from the bosonic commutation relations and the lower-band Berry curvature with the correct sign. Because the paper's own text flags this as borrowed from Refs. [17] and [45] without re-derivation, the central claim is conditional on an unverified input.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops a symplectic quantum geometric tensor (SQGT) for two-band bosonic Bogoliubov Hamiltonians, exploiting the pseudo-unitary (U(1,1)) structure of the Bogoliubov–Valatin transformation. The authors derive closed-form expressions for the symplectic quantum metric and Berry curvature in terms of the dynamical vector, Eqs. (17)–(21), and show that the band topology is trivial because the dynamical unit vector maps to a contractible hyperbolic sheet. They then apply this framework to a minimal Lieb lattice altermagnet, deriving the magnon bands and an analytic Berry curvature, Eq. (27). Using linear-response formulas from the literature (Eq. (28)), they compute the magnon thermal Hall and spin Nernst conductivities and analyze their temperature and altermagnetic-splitting dependence. The central result, Eq. (33), is that to leading order the spin Nernst conductivity is proportional to (J2'–J2) and the thermal Hall conductivity is proportional to B(J2'–J2), suggesting these transport signals as experimental probes of altermagnetism.","tokens_in":13593,"tokens_out":11430,"duration_ms":106298,"significance":"If the central claim holds, the paper makes two valuable contributions. First, it provides an elegant and fully analytic derivation of the quantum geometric tensor for two-band bosonic Bogoliubov systems, which is a useful tool for future studies of magnon geometry and topology. Second, it predicts a clear, falsifiable dependence of the magnon thermal Hall and spin Nernst conductivities on the altermagnetic splitting, offering a potential experimental route to detect altermagnetism in insulating magnets. The derivation is self-contained up to the transport formulas, no parameters are fitted to the target conductivities, and the leading-order proportionality in Eq. (33) is obtained analytically. The suggestion of La2O3Mn2Se2 as a candidate platform is concrete and reasonable, though quantitative predictions for specific materials are not provided.","major_comments":[{"comment":"The two transport formulas in Eq. (28) are imported from Refs. [42,43,45] without derivation, despite being the load-bearing step that converts the microscopic Hamiltonian into the computed conductivities. The manuscript itself flags this in Sec. III B: it states that Eq. (24) is not particle-hole symmetric and must be rewritten in that form by doubling the Nambu array, after which Ref. [45] gives Eq. (28). This step is not merely a technicality: the sign structure of Eq. (28) (a difference of c1 terms for alpha_xy and a sum of c2 terms for kappa_xy) is essential for the leading-order results in Eq. (33). If the sign of the lower-band contribution in Eq. (28b) were reversed, the leading term in kappa_xy would be quadratic in the altermagnetic splitting rather than linear in B(J2'-J2), and if Eq. (28a) carried a plus sign, alpha_xy would be quadratic and vanish at linear order. The doubling procedure must also handle the c-number shift from bosonic commutation relations and the sign of the lower-band Berry curvature. The authors should provide a derivation of Eq. (28) for the doubled Hamiltonian, or at least a detailed verification that the physical heat and spin current operators are correctly represented and that the sign structure is preserved. Without this, the central claim is conditional on an unverified input.","section":"Sec. III B, Eq. (28)"},{"comment":"The expansion leading to Eq. (33) is described as being 'to second order in delta', but the displayed results are first order in (J2'-J2) for alpha_xy and first order in B(J2'-J2) for kappa_xy. The authors should clarify the small parameters in the expansion and show the terms to the order actually retained. In particular, the role of the magnetic-field part of delta in Eq. (26a) deserves explicit discussion: the alpha_xy result in Eq. (33a) survives at B=0, but at finite B the delta in Eq. (26a) also contains B, and the authors should show that the B-only contribution to alpha_xy integrates to zero due to the odd symmetry of F_xy, as is implicitly assumed. This clarification is important for a reader to understand the precise regime of validity of Eq. (33).","section":"Sec. III B, Eq. (33)"}],"minor_comments":[{"comment":"The description of the numerical integration is too vague: 'An adaptive numerical integration method is used' without specifying the quadrature scheme, grid density, or convergence criteria. For reproducibility of Figs. 5 and 6, the authors should provide details of the method and an estimate of the numerical uncertainty.","section":"Sec. III B, numerical integration"},{"comment":"There is a sign inconsistency in the notation for the altermagnetic parameter: Eq. (33a) and the text around Eq. (26a) use (J2'-J2), while the x-axis of Fig. 6 is labeled (J2 - J2')/J1. The sign is arbitrary, but the manuscript should be internally consistent.","section":"Eq. (33a) vs. Fig. 6"},{"comment":"The sentence 'Both conductivities vanish in the low and high temperature limits when delta|B=0 = 0' should be qualified: at moderate temperatures, Eq. (33) shows that kappa_xy also vanishes when J2'=J2 even if B is finite. This is clear from the figures, but the text could state it explicitly to avoid confusion.","section":"Sec. III B, paragraph after Eq. (28)"},{"comment":"The claim that the symplectic quantum metric 'does not seem to diverge in regions with crossings or overlaps of the two excitation energies' is made for the two-band case; the same statement is later repeated for the Berry curvature. It would be helpful to note explicitly that this is a property of the symplectic (LR) QGT and contrasts with the conventional QGT, as the authors do later.","section":"Sec. II B 1, Eq. (18)"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid analytical contribution to magnon quantum geometry, and the altermagnet application is timely. The main concern is the unverified import of Eq. (28), which is the load-bearing step for the central claim. The authors should be asked to either derive it or provide a clear verification in an appendix. The numerical details should also be improved for reproducibility. If these points are addressed, the paper would be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper has one genuinely new analytical result, a closed-form symplectic quantum geometric tensor for two-band bosonic Bogoliubov Hamiltonians, and one potentially useful physical prediction, that magnon thermal Hall and spin Nernst conductivities in a Lieb altermagnet grow linearly with the altermagnetic splitting. If that prediction holds, it is a viable bulk transport probe for altermagnetism.\n\nWhat is new: Eqs. (17) and (21) express the SQGT and Berry curvature directly in terms of the Lorentzian unit vector of the dynamical matrix, without constructing eigenvectors. That is clean and I have not seen it in the cited literature. The topological triviality argument is a nice touch, and the transport calculation is mostly self-contained, with no fitted parameters. Citation pattern looks fine; the self-citations supply the model Hamiltonian, not the result.\n\nSoft spots, in order of importance. First, Eq. (28) is imported from Refs. [17] and [45], not derived. The step matters because the sign structure in (28) is load-bearing: kappa_xy is a sum of c2 terms, alpha_xy is a difference of c1 terms. The paper says doubling the Nambu array restores particle-hole symmetry, but it does not show how the heat and spin current operators transform under that doubling. If the sign in (28b) were wrong, the leading dependence of kappa_xy on the altermagnetic splitting could become quadratic and the central prediction would not survive. I do not think it is wrong, but the referee material should include an explicit check or derivation. Second, Eq. (20) has a factor-of-two slip: ds^2 equals trace[(dP)^2] only up to a factor, and as written it does not equal the real part of Q in Eq. (17). That is a minor typo-level issue and does not affect transport, but it should be fixed. Third, the numerical integration is described only as 'adaptive'; enough detail for reproducibility would be better. Minor.\n\nThe Holstein-Primakoff leading-order approximation ignores magnon interactions; that is standard for this kind of calculation and not a real flaw.\n\nOverall: the central transport claim is plausible and the analytical framework will be useful to people working on magnon transport or altermagnets. It deserves a serious referee, with the specific request to verify Eq. (28). I would bring it to our reading group and would cite the SQGT result.","headline":"A clean new symplectic QGT for two-band magnons plus a transport prediction that is worth checking; the only load-bearing gap is Eq. (28).","tokens_in":14144,"tokens_out":4315,"would_cite":true,"duration_ms":39151,"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":"The paper claims that in a two-dimensional altermagnet, the magnon thermal Hall and spin Nernst conductivities are proportional to the altermagnetic parameter $(J_2-J_2')$, making them direct probes of altermagnetism.","keywords":["altermagnetism","magnon thermal Hall effect","spin Nernst effect","quantum geometric tensor","bosonic Bogoliubov Hamiltonian","Berry curvature","Lieb lattice","Dzyaloshinskii-Moriya interaction"],"falsifier":"A direct numerical check is to compute the thermal Hall and spin Nernst conductivities from the Kubo formulas on the original two-band Hamiltonian without the doubling step; Eq. (33) predicts exact zero at $J_2=J_2'$, so any nonzero result from that calculation would falsify the paper's central claim.","tokens_in":13118,"feed_emoji":"🧲","tokens_out":5973,"duration_ms":51353,"temperature":0.7,"pith_summary":"This paper establishes that the magnon thermal Hall conductivity and the magnon spin Nernst conductivity in a two-dimensional altermagnet are direct measures of the altermagnetic splitting: both vanish when the altermagnetic parameter $J_2-J_2'$ is zero, and to leading order the spin Nernst signal is proportional to $J_2-J_2'$ while the thermal Hall signal is proportional to $B(J_2-J_2')$. To reach this result the authors derive an analytic expression for the quantum geometric tensor of two-band bosonic Bogoliubov Hamiltonians, based on the pseudo-unitary group structure of the bosonic diagonalization. They apply it to a minimal Lieb-lattice altermagnet with Dzyaloshinskii-Moriya interaction and compute the conductivities as functions of temperature and splitting. Because the altermagnetic splitting can be tuned by strain, the paper proposes these bulk transport signals as an experimental probe of altermagnetism in insulating magnets such as $\\mathrm{La_2O_3Mn_2Se_2}$.","feed_headline":"Altermagnet splitting sets size of magnon Hall currents","feed_subtitle":"Thermal Hall and spin Nernst signals vanish without the altermagnetic parameter, offering a transport probe.","key_machinery":"The key object is the symplectic quantum geometric tensor (SQGT), defined by replacing the usual eigenprojectors with pseudo-orthogonal projectors onto eigenstates of the dynamical matrix $D=\\sigma_z H$. The paper shows that for any two-band bosonic Bogoliubov Hamiltonian, the LR pseudo-orthogonal projectors can be written as $P_\\pm=\\frac{1}{2}(1_2\\pm \\hat{d}\\cdot K)$, where $\\hat{d}$ is a unit vector on a Lorentzian two-sheeted hyperboloid built from the Hamiltonian coefficients. This yields closed-form expressions for the symplectic quantum metric and the symplectic Berry curvature, and it explains why the topology is trivial: the hyperbolic sheet is contractible. The same $P_\\pm$ feed the linear-response integrals for heat and spin currents.","core_discovery":"The central discovery is a direct analytic link between altermagnetic splitting and bosonic Hall transport. For the Lieb-lattice model, the magnon Berry curvature is odd under a 90-degree rotation, so its Brillouin-zone integral vanishes and the bands are topologically trivial; nevertheless the distribution-function-weighted integrals that give the conductivities do not vanish. Expanding those integrals for small splitting gives $\\alpha_{xy}\\propto (J_2-J_2')$ and $\\kappa_{xy}\\propto B(J_2-J_2')$, so a nonzero altermagnetic parameter is necessary and sufficient (within the model) for both transverse responses at any temperature. The same analysis shows the Berry curvature itself is independent of the altermagnetic parameter and depends only on the sum $J_2+J_2'$, the Dzyaloshinskii-Moriya coupling, and the anisotropy.","pith_inferences":["If the doubling of the Nambu array in fact preserves the physical currents, the proportionality $\\alpha_{xy}\\propto(J_2-J_2')$ is probably generic for any collinear two-sublattice antiferromagnet with d-wave anisotropic couplings, not only the Lieb lattice.","Since the Berry curvature is independent of the altermagnetic parameter while the conductivities depend on it through the distribution functions, a combined measurement of the band-resolved Berry curvature and the transport coefficients could isolate the coupling difference.","A natural numerical extension is exact diagonalization of finite clusters with the same parameters; computing the Hall response without assuming particle-hole symmetry would test the load-bearing linear-response step."],"forward_implications":["If the central claim is right, measuring a finite magnon spin Nernst signal at zero magnetic field in an insulating antiferromagnet would indicate altermagnetic splitting even when the magnon bands are not directly resolved.","The thermal Hall conductivity in the same material should require a magnetic field and be proportional to $B(J_2-J_2')$ at leading order, giving a way to separate the altermagnetic parameter from other couplings.","Because the conductivity formulas vanish identically at $J_2=J_2'$, the standard square-lattice antiferromagnet with only $J_1$ and one next-nearest-neighbor coupling is predicted to show no magnon Hall or Nernst effect from this mechanism.","Strain tuning of $J_2-J_2'$ should change the conductivities in a predictable way, which the paper suggests could be tested in layered altermagnetic insulators such as $\\mathrm{La_2O_3Mn_2Se_2}$."],"supporting_citations":[{"why":"Diagonalization of the quadratic boson Hamiltonian; supplies the pseudo-unitary transformation $T^\\dagger\\sigma_z T=\\sigma_z$ and the doubling that connects to particle-hole symmetric response formulas.","marker":"[17]"},{"why":"Introduces the concept of quantum geometry for bosonic Bogoliubov quasiparticles; motivates calling the projector-adapted object the symplectic QGT.","marker":"[18]"},{"why":"Eigenprojector approach to Berry curvature and quantum metric; the paper adapts its trace formula to derive closed-form SQGT expressions without eigenvectors.","marker":"[27]"},{"why":"Defines the magnonic Berry curvature through pseudo-orthogonal projection operators in spin-wave systems; the paper identifies its symplectic Berry curvature with this quantity.","marker":"[28]"},{"why":"Minimal Lieb-lattice model of altermagnetism with two magnetic sublattices and nonmagnetic atoms; provides the spin Hamiltonian and the altermagnetic parameter $J_2-J_2'$.","marker":"[34]"},{"why":"Magnon thermal Hall effect with dipolar interaction; one of the two quantum linear-response derivations whose formulas are used for $\\kappa_{xy}$.","marker":"[42]"},{"why":"Magnon spin Nernst effect in antiferromagnets; derivation of the spin-current response formula used for $\\alpha_{xy}$.","marker":"[43]"},{"why":"Magnon Hall effect in antiferromagnetic lattices; gives the particle-hole-symmetric expressions, including the sign difference between heat and spin currents, used in Eq. (28).","marker":"[45]"},{"why":"A separate altermagnet model with in-plane DMI; the comparison case that yields zero $\\alpha_{xy}$ and $\\kappa_{xy}$, cited to show why out-of-plane DMI matters.","marker":"[47]"}],"fun_headline_variants":["Altermagnetic splitting drives magnon Hall signals","No altermagnet, no magnon Hall response","Altermagnet parameter tunes magnon thermal Hall","Magnon Hall conductivity set by altermagnet term"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that the linear-response formulas derived for particle-hole symmetric Hamiltonians remain valid after the bosonic Hamiltonian is rewritten in a doubled Nambu form; if that doubling changes the physical heat and spin currents, the predicted dependence on the altermagnetic parameter would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Altermagnetic splitting drives magnon Hall signals","No altermagnet, no magnon Hall response","Altermagnet parameter tunes magnon thermal Hall","Magnon Hall conductivity set by altermagnet term"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000798,"raw_usage":{"total_tokens":3433,"prompt_tokens":787,"completion_tokens":2646,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":403,"completion_tokens_details":{"reasoning_tokens":2585}},"tokens_in":403,"tokens_out":2646,"duration_ms":19074,"temperature":1.0,"reasoning_tokens":2585,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:23:47.515039+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct numerical check is to compute the thermal Hall and spin Nernst conductivities from the Kubo formulas on the original two-band Hamiltonian without the doubling step; Eq. (33) predicts exact zero at $J_2=J_2'$, so any nonzero result from that calculation would falsify the paper's central claim.","supporting_citations":[{"cited_title":"Colpa, Diagonalization of the quadratic boson hamilto- nian, Phys","cited_arxiv_id":null,"evidence_quote":"Defines the magnonic Berry curvature through pseudo-orthogonal projection operators in spin-wave systems; the paper identifies its symplectic Berry curvature with this quantity."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Minimal Lieb-lattice model of altermagnetism with two magnetic sublattices and nonmagnetic atoms; provides the spin Hamiltonian and the altermagnetic parameter $J_2-J_2'$."},{"cited_title":"Bures, An extension of kakutani’s theorem on infinite product measures to the tensor product of semifinite w*- algebras, Trans","cited_arxiv_id":null,"evidence_quote":"Magnon thermal Hall effect with dipolar interaction; one of the two quantum linear-response derivations whose formulas are used for $\\kappa_{xy}$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Magnon spin Nernst effect in antiferromagnets; derivation of the spin-current response formula used for $\\alpha_{xy}$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"A separate altermagnet model with in-plane DMI; the comparison case that yields zero $\\alpha_{xy}$ and $\\kappa_{xy}$, cited to show why out-of-plane DMI matters."}],"review_version":1}