{"id":"37496946-2c82-47f6-93d0-6cac76b63ee4","arxiv_id":"2608.02473","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"ET-CTMRG incorporates excitation tensors into corner-transfer-matrix projector optimization, yielding stable iPEPS excitation spectra and quantitative agreement with neutron data for K2Co(SeO3)2.","lead":"This paper introduces a modified tensor-network method (ET-CTMRG) that builds renormalization projectors from both ground-state and excitation tensors, removing numerical instabilities in iPEPS spectral calculations. It then computes spin excitation spectra for the supersolid magnet K2Co(SeO3)2 and reports agreement with inelastic neutron scattering data.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"ET-CTMRG projector optimality is proven only for an aggregated cost containing nonphysical cross terms; the physical error is merely bounded above, not minimized.","rationale":"The reader's weakest assumption identifies exactly the same load-bearing concern: the projectors are optimized for a cost function that includes nonphysical cross terms, and the physical error is only bounded, not minimized. This concern is central because the method's core innovation is to tailor the CTMRG projectors to the excited-state subspace; if the optimization can be dominated by unphysical contributions, the method's accuracy for the effective Hamiltonian is not guaranteed in general. The paper provides strong empirical evidence for specific models—the ε/ε_phy near-coincidence in Fig. S1, the systematic convergence with χ, and the stability of the spectrum as N_r grows—which supports the method's practical effectiveness. However, these do not constitute a proof of the projector's preservation of the physical subspace. The concrete test proposed would settle whether the nonphysical terms are harmless in practice, especially in the strongly anisotropic regime where the method is most needed. Since the reader already issued a CONDITIONAL verdict based on this and related concerns, my stress-test does not change the verdict: the concern is valid but addressable, and the paper's evidence does not warrant rejection while the ambiguity remains.","tokens_in":21424,"tokens_out":14986,"duration_ms":145142,"concrete_test":"For the K2Co(SeO3)2 parameters (or a strongly anisotropic XXZ model), compute ε and ε_phy as functions of χ using ET-CTMRG at a representative momentum (e.g., the M point). If ε_phy remains comparable to ε and both converge systematically with χ, the concern is mitigated. Additionally, perform an alternative projector optimization that minimizes ε_phy directly—either by solving the SVD problem with the cost restricted to the physical pair set S of Eq. (S22), or by weighting nonphysical pairs to zero—and compare the resulting excitation spectrum with the reported ET-CTMRG spectrum. If the spectra differ substantially, the nonphysical cross terms bias the projectors and the paper's optimality claim is insufficient to establish the method's reliability.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that the rank-χ projectors from the SVD of K†K̃ preserve the excited-state subspace relevant to Heff. The paper proves optimality for the cost function ε of Eq. (7), which sums over all 16 pairs (i,j) in Eq. (6), including nonphysical combinations such as M1–M1 and M2–M2 (both half-blocks containing the same excitation species, B or B†). The physical error ε_phy of Eq. (S21), restricted to the 9 physical pairs S of Eq. (S22), is only bounded above by ε. Minimizing an upper bound does not guarantee that the physical error is small or even reduced; the nonphysical terms could dominate the SVD and bias the projectors. Fig. S1 shows ε and ε_phy nearly coincident for the square-lattice Heisenberg antiferromagnet at one momentum and D=3, but this is a single empirical coincidence, not a general guarantee. For K2Co(SeO3)2, the model where GS-CTMRG fails dramatically, no ε/ε_phy comparison is shown; the method's success there could be sensitive to the nonphysical terms. If the nonphysical terms dominate in some parameter regime, the projectors would not be tailored to physical excitations, undermining the claim that ET-CTMRG 'faithfully' renormalizes the excited-state manifold. The generality of the central claim therefore rests on an unproven assumption about the behavior of the aggregated cost function.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces ET-CTMRG, a modification of the CTMRG projector construction used in iPEPS-based calculations of dynamical spectral functions. Instead of building the truncation projectors P and Q solely from the ground-state half-block M0 as in GS-CTMRG, the method defines aggregated half-block matrices K and K̃ from all configurations containing zero or one B/B† excitation, and obtains the optimal rank-χ projectors from the SVD of K†K̃. The authors show analytically that this minimizes a combined Frobenius cost ε, demonstrate numerically that for the square-lattice Heisenberg antiferromagnet the truncation error decreases by many orders of magnitude with χ and the spectrum stabilizes with the number of retained states Nr, and apply the method to the triangular-lattice XXZ model of K2Co(SeO3)2, where GS-CTMRG is severely unstable. The computed spectra are compared with inelastic neutron scattering data at three fields in the supersolid Y phase; good overall agreement is reported, with a caveat in the Supplemental Material about the M-point mode energy.","tokens_in":21816,"tokens_out":3830,"duration_ms":38002,"significance":"If the claims hold, ET-CTMRG is a valuable methodological advance: it addresses a known instability of iPEPS spectral-function calculations, is applicable at arbitrary momenta without unit-cell enlargement, and retains the computational structure of GS-CTMRG. The paper is transparent: the core derivation is presented in the main text and SM, benchmarks include several paradigmatic models, and the material comparison uses Hamiltonian parameters taken from previous thermodynamic/INS work rather than fitted to the target spectra. The stable convergence with Nr and χ, and the order-of-magnitude error reduction, are concrete falsifiable strengths. The main limitations are the lack of a formal guarantee that the aggregated cost preserves the physical excited-state subspace (only an upper-bound relation is proven) and a small but explicit discrepancy in the application that moderates the \"excellent quantitative agreement\" headline.","major_comments":[{"comment":"The statement in SM S2B that \"εphy must be smaller than ε, and hence minimizing ε optimizes εphy simultaneously\" is not logically correct. From ε² = εphy² + εnonphys², minimizing ε minimizes an upper bound on εphy, but it does not guarantee that εphy is minimized or even reduced; the nonphysical pairs (1,1) and (2,2) could in principle dominate the SVD and bias the projectors. Fig. S1 shows near-equality of ε and εphy for the SLHAF at one momentum and D=3, which is useful empirical evidence but not a general guarantee. For K2Co(SeO3)2, the model where GS-CTMRG fails most dramatically, no ε/εphy comparison is shown. Since the central claim is that ET-CTMRG \"faithfully\" renormalizes the excited-state manifold, the authors should either prove a stronger relation or explicitly state that the optimality is with respect to the aggregated cost, and provide ε/εphy comparisons for the models and","section":"SM S4, Fig. S8"},{"comment":"The manuscript's own extrapolation in Fig. S8 gives an M-point lower-mode energy of approximately 0.06 meV, which the text notes is \"approximately half of the energy at which the maximum in the INS intensity appears in Fig. 5(d)\"; the QMC value quoted is about 0.08 meV. This is a significant quantitative discrepancy on the lowest, most prominent branch at a symmetry point. The abstract's \"excellent quantitative agreement\" and the main text's \"quantitative accuracy\" are therefore overstated. While the overall spectral shape and field dependence agree well, the M-point branch energy is off by roughly a factor of two from the experimental peak position. The authors should qualify the agreement claim (e.g., \"good qualitative and semi-quantitative agreement\") and discuss whether this discrepancy is within the expected finite-D error or points to the parameter refinement suggested in the SM.","section":"SM S4, Fig. S8"}],"minor_comments":[{"comment":"The phrase \"this leads to a strategy call\" is unclear; presumably a strategic choice or decision is meant. Please rephrase.","section":"Main text, 'Benchmark and application'"},{"comment":"The text refers to \"Eq. (10) of the main text\" for the INS intensity, but in the main text the expression appears as Eq. (9). Please correct the cross-reference.","section":"SM S4, first paragraph"},{"comment":"The sentence \"the norm [Eq. (S10)] and energy [Eq. (S9)]\" is imprecise: Eq. (S9) is the effective Hamiltonian matrix, and the excitation energy is obtained from the generalized eigenvalue problem of Eq. (S11). Please rephrase to avoid suggesting that Eq. (S9) itself is the energy.","section":"SM S2B, around Eq. (S9)"},{"comment":"No data-availability or code-availability statement is provided. For a numerical-methods paper, a statement on whether the ET-CTMRG implementation is publicly available would improve reproducibility.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically sound in its derivation and benchmarks, but the headline claim of \"excellent quantitative agreement\" with INS is weakened by the authors' own SM statement that the M-point mode energy is about half the experimental peak energy. The referee report requests qualification of that claim and a sharpening of the mathematical relation between ε and εphy. Neither issue is fatal; both can be addressed within the manuscript's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core of this paper is real: the authors identify that ground-state-only CTMRG projectors make the effective Hamiltonian, not the norm matrix, the unstable object, and they construct excitation-tailored projectors via a clean SVD of K†K̃. The algebra from Eq. (6) to Eq. (7) checks out, the Eckart–Young step is standard, and the benchmarks are compelling: orders-of-magnitude error reduction, stable spectra as N_r and χ grow, and a norm-matrix replacement test that supports the Heff diagnosis. The K2Co(SeO3)2 spectra are the first stable iPEPS results for this material and they capture the two branches, rotons, and field evolution qualitatively.\n\nThe soft spots are proportionate. The stress-test concern about nonphysical cross terms in the cost function is legitimate: optimality is proven only for the aggregated ε, and εphy is merely bounded above. The authors know this—they define εphy and show it nearly coincides with ε in Fig. S1 for one model at one momentum. That is evidence, not a guarantee, and the paper would be stronger if it showed the same comparison for the anisotropic model where GS-CTMRG fails. It does not undermine the method, but it qualifies the word \"faithful.\"\n\nThe bigger issue is the abstract's \"excellent quantitative agreement.\" The SM S4 statement that the M-point mode energy (~0.06 meV) is about half the INS peak energy is a factor-of-two discrepancy, and the authors themselves suggest the Hamiltonian parameters could be refined. That is honest, but it contradicts the abstract's claim. The paper should either soften the claim or show that the factor of two is within the combined numerical and experimental uncertainty.\n\nAlso minor: no code or data is released, and the closest prior stabilization method [26] is mentioned but not benchmarked against. Both are addressable in revision.\n\nWho is this for? Anyone doing iPEPS spectral calculations on 2D magnets, especially anisotropic or frustrated models where GS-CTMRG is unstable. I would cite it if I worked in that area. It deserves a serious referee: the method is novel, the benchmarks are strong, and the K2Co(SeO3)2 application is a useful step even if the quoted agreement is too strong. I would send this to peer review and ask for a careful revision on the claims and a epsilon/epsilon_phy comparison in the hard case.","headline":"ET-CTMRG is a genuinely new fix for a known instability in iPEPS spectral functions, with strong benchmarks and a mostly convincing material application; the main gap is an overstatement of the K2Co(SeO3)2 agreement.","tokens_in":22334,"tokens_out":1258,"would_cite":true,"duration_ms":14142,"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":"Folding excitation states into the renormalization step makes tensor-network spectra stable and reproduces inelastic neutron-scattering measurements for the triangular-lattice supersolid K2Co(SeO3)2.","keywords":["tensor networks","excitation spectra","dynamical structure factor","corner-transfer matrix renormalization group","iPEPS","spin supersolid","triangular lattice XXZ model","inelastic neutron scattering"],"falsifier":"A decisive check is to compute εphy, the physical Frobenius error using only the physical combinations of Mi and M̃j, for the triangular-lattice XXZ model of K2Co(SeO3)2 at increasing χ. If εphy plateaus at a large value while ε keeps decreasing, the optimal projectors are being driven by nonphysical cross terms and the method's accuracy guarantee for that regime fails.","tokens_in":21302,"feed_emoji":"🧲","tokens_out":7072,"duration_ms":71808,"temperature":0.7,"pith_summary":"The paper argues that numerical instabilities in tensor-network computations of excitation spectra for two-dimensional quantum magnets come from one specific step: the renormalization tensors used to coarse-grain the environment are built only from the ground state, so they systematically discard the excited-state subspace. It introduces ET-CTMRG, in which the truncation projectors are optimized using both ground-state and single-excitation tensors folded into one matrix K. On benchmark Heisenberg antiferromagnets the truncation error drops by orders of magnitude and the spectrum converges as more states are kept; for the strongly anisotropic triangular-lattice material K2Co(SeO3)2 the method produces spectra that match inelastic neutron scattering, where the ground-state-only version fails. The paper pinpoints the effective Hamiltonian matrix, not the norm matrix, as the source of instability.","feed_headline":"Renormalizing with excitations stabilizes quantum spin spectra","feed_subtitle":"Ground-state-only truncation destabilized spin spectra; including excitations restores order and matches neutron data.","key_machinery":"The central object is the matrix K (and its partner K̃), built from the upper and lower halves of the corner-transfer-matrix network: it aggregates the ground-state half-block M0 with half-blocks containing one excitation tensor B, one B†, and both B and B† (M1, M2, M3). Replacing M0 by K in the CTMRG cost function—so that the truncation projectors P and Q come from the truncated SVD of K†K̃—carries excitation information into every renormalization step at essentially the same computational cost as the ground-state-only version.","core_discovery":"The central claim is that the numerical instability in iPEPS spectral functions is caused by CTMRG truncation projectors being constructed from ground-state tensors alone. The paper replaces the ground-state half-block matrix M0 with an aggregated matrix K built from all half-block configurations containing zero, one, or two excitation tensors, so the projectors P and Q solve the minimization problem with cost ||K†(I−PQ)K̃||. The optimal rank-χ projectors are the truncated SVD factors of K†K̃. This excitation-tailored renormalization makes the effective Hamiltonian matrix accurate enough that solving the generalized eigenvalue problem Heff v = E Neff v yields stable, converged spectra. Bench","pith_inferences":["The same principle—building truncation projectors from excited-state configurations rather than only the reference state—could be tested in other tensor-network settings, such as fermionic or bosonic response-function calculations, to probe its generality beyond spin models.","The aggregated cost function includes nonphysical cross terms; a practical check would be to monitor the physical Frobenius error εphy separately for strongly anisotropic or frustrated models, where the numerical coincidence of ε and εphy seen in one benchmark may not persist.","Because excitation-tailored projectors must be recomputed at every CTMRG step, the method forgoes the cached-projector speed-up available in ground-state-only schemes; a low-rank update or recycling of the previous step's projectors could recover some speed while retaining excitation information.","The paper's suggested M-point dip energy near 0.06 meV, roughly half the measured intensity peak, implies that the model parameters (such as Jxy/Jz, determined thermodynamically) could be refined by fitting the full computed spectrum to the neutron data."],"forward_implications":["ET-CTMRG yields stable, convergent excitation spectra for models where ground-state-only CTMRG is unstable, enabling full-momentum studies with large retained-state counts.","For Heisenberg antiferromagnets, the truncation error is reduced by orders of magnitude at the same CTM bond dimension, and the spectrum converges systematically as more states are kept.","The instability in spectra is traced to the effective Hamiltonian matrix, not to ill-conditioning of the norm matrix; replacing only the norm matrix does not cure the instability.","For K2Co(SeO3)2 in the supersolid Y phase, the computed spectral function agrees quantitatively with inelastic neutron-scattering data, including the two low-energy magnon branches and the two-magnon continuum.","The method avoids unit-cell enlargement and second-order derivatives while keeping the computational structure close to that of ground-state CTMRG."],"fun_headline_variants":["Excitation-tailored renormalization stabilizes spin spectra","Including excitations in renormalization fixes spin spectrum instability","ET-CTMRG yields accurate quantum spin spectra","Ground-state truncation destabilized spin spectra; fix: excitation-tailored"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The method's accuracy rests on the premise that the rank-χ projectors obtained from the SVD of K†K̃ faithfully preserve the excited-state subspace that actually controls Heff—even though the optimized cost function contains nonphysical cross terms whose effect is only bounded by the physical truncation error, not guaranteed to match it.","fun_headline_variants_meta":{"raw":{"variants":["Excitation-tailored renormalization stabilizes spin spectra","Including excitations in renormalization fixes spin spectrum instability","ET-CTMRG yields accurate quantum spin spectra","Ground-state truncation destabilized spin spectra; fix: excitation-tailored"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00014,"raw_usage":{"total_tokens":991,"prompt_tokens":728,"completion_tokens":263,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":472,"completion_tokens_details":{"reasoning_tokens":194}},"tokens_in":472,"tokens_out":263,"duration_ms":5129,"temperature":1.0,"reasoning_tokens":194,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T06:43:16.312019+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A decisive check is to compute εphy, the physical Frobenius error using only the physical combinations of Mi and M̃j, for the triangular-lattice XXZ model of K2Co(SeO3)2 at increasing χ. If εphy plateaus at a large value while ε keeps decreasing, the optimal projectors are being driven by nonphysical cross terms and the method's accuracy guarantee for that regime fails.","supporting_citations":[],"review_version":1}