{"id":"12ab9bb3-a98d-4fe6-9494-0a1aff0c98bf","arxiv_id":"2607.05269","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":2,"one_line_summary":"A linear MPS tangent-space method for open-boundary systems is derived, and a particle-resolved rank tomography is introduced to diagnose the tangent space's expressivity limits.","lead":"The paper develops a tangent-space method for matrix product states (MPS) with open boundaries to compute excitation spectra of non-uniform quantum many-body systems. It also introduces a 'rank tomography' that decomposes the tangent space's expressivity by particle-number sectors, explaining why spectral accuracy varies even at fixed bond dimension.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"No significant objection identified. The adapted tangent-space assumption is formally unproven but well-supported by the block structure analysis; the rank tomography framework and spectral benchmarks hold up under scrutiny.","rationale":"The reader correctly identifies the adapted tangent space assumption as the weakest link in the rank tomography framework. However, I assess it as less load-bearing than the reader implies for three reasons: (1) The block structure analysis in Section V.B and Appendix VII.B provides strong evidence that adaptation holds for U(1)-symmetric ground states — the particle-number selection rule m'=m+n on MPS tensor blocks means tangent variations naturally decompose into sector-specific components. (2) The reader's concern about the η≠0 case is misdirected: the rank tomography framework is only applied to the η=0 case, and the tangent-space spectral method itself does not require adaptation. (3) The numerical results (Figure 4 showing ~1e-5 errors, Figure 5 showing exact reproduction when ρ_M = d_M^target) are fully consistent with the adapted condition holding. I considered several alternative concerns: the 'complete explanation' claim for PRSR is scoped to sector-dependent (not mode-dependent) errors and is supported by the evidence; the small N=10 system is appropriate for ED benchmarking; the steep D^8 scaling is acknowledged transparently. The ground-state variational error (the DMRG state is not an exact eigenstate) is a standard feature of all MPS tangent-space methods, not specific to this work. The paper makes a genuine contribution by connecting the internal particle-resolved rank structure of MPS to the expressivity of the tangent space, and the mathematical framework (MPS varieties, tangent spaces, gauge removal) is sound with proofs provided in the appendix. No concern rises to the level of changing the ACCEPT verdict.","tokens_in":20452,"tokens_out":6980,"duration_ms":251492,"concrete_test":"For the N=10, D=8, η=0 ground states used in Figure 3 (top), numerically verify the adapted condition by checking that rank(P_M U) = dim(T ∩ H_M) for each particle sector M. Concretely: compute the projected tangent basis U as in Eq. (4), project onto each sector M to get P_M U, then independently compute the intrinsic sector tangent space T ∩ H_M by restricting tangent variations to blocks with m'=m+n only. If the two ranks agree for all M, the adapted condition is verified for these ground states, confirming the rank tomography analysis is self-consistent.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The closest candidate for a load-bearing concern is the adapted tangent space assumption in Section V.A: the entire rank tomography framework (parametric deficiency, PRSR-to-tangent-rank mapping) depends on P_M T ⊆ T, i.e., that each particle-number component of a tangent vector is itself a tangent vector. The paper states 'From now, we only consider tangent spaces that are adapted' without a formal proof that this holds for U(1)-symmetric MPS ground states. However, the block structure analysis in Section V.B and Appendix VII.B provides strong evidence: the MPS tensors decompose into particle-number blocks with selection rule m'=m+n, and tangent variations δM_j^n decompose into blocks δM_j^n(m,m') that generate tangent vectors in specific particle sectors. This block decomposition effectively guarantees adaptation when the ground state has definite particle number. The reader's concern about the η≠0 case (Figure 3, bottom) is misdirected — the rank tomography is not applied there; the paper switches to continuous ⟨M̂⟩ coloring. The tangent-space method itself (Section III) does not require adaptation. I do not find a concern that rises to the level of changing the verdict. The 'complete explanation' claim for PRSR is scoped to sector-dependent errors (which sectors have errors), not mode-dependent errors (which excitations within a deficient sector are captured), and the evidence supports this scoped claim.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"The manuscript formulates a linear MPS tangent-space method for computing excitation spectra of finite, non-uniform quantum many-body systems with open boundary conditions, using the algebraic-geometric framework of MPS varieties. The tangent space is defined on the smooth full-rank stratum, gauge redundancy is removed via Hermiticity conditions (Proposition 7), and the resulting generalized eigenvalue problem yields excitation energies. The paper further introduces a 'rank tomography' framework: the particle-resolved Schmidt rank (PRSR) distribution refines the coarse bond dimension into particle-sector-resolved ranks, and the parametric deficiency quantifies missing tangent-space directions per particle sector. The Bose-Hubbard model on N=10 sites serves as the benchmark, with exact diagonalization comparison showing ~1e-5 accuracy at D=8 for low-lying modes. The framework is mathematically self-contained, with proofs of Propositions 5-7 provided in the Appendix.","tokens_in":21223,"tokens_out":1209,"duration_ms":153722,"significance":"The paper makes two distinct contributions. First, it provides a clean algebraic-geometric formulation of the MPS tangent-space excitation method for OBC systems, with rigorous definitions of the MPS variety, tangent space, and gauge removal (Propositions 5-7, with proofs). Second, the rank tomography framework — PRSR distribution and parametric deficiency — offers a novel diagnostic tool that explains sector-dependent approximation errors in terms of the internal particle-resolved structure of the ground state. The observation that two states with identical bond dimensions can have different PRSR profiles and hence different tangent ranks is a concrete, falsifiable insight. The Bose-Hubbard benchmarks, including the N=3 exact-recovery consistency check (Appendix VII.E, machine precision), support the framework's validity. The computational cost analysis (Appendix VII.D) is a useful addition for practitioners. The work is primarily conceptual/methodological; the system sizes tested (N=10, D<=9) are small, and the method's competitiveness against established approaches (e.g., correlation matrix or post-MPS methods) for larger systems is not demonstrated.","major_comments":[{"comment":"The adapted tangent-space assumption (P_M T ⊆ T for all M) is stated without formal proof. The block-structure analysis in Section V.B and Appendix VII.B provides strong evidence that this holds for U(1)-symmetric MPS ground states, since the selection rule m'=m+n on ground-state tensors propagates to tangent variations in the same sector. However, the paper does not explicitly state that this block structure constitutes a proof of adaptation, nor does it clarify whether adaptation is guaranteed for all U(1)-symmetric MPS or only generically. A brief remark connecting the block decomposition of Section V.B to the adaptation condition would close this gap. This is load-bearing because the entire parametric deficiency framework (Section V.A) and the PRSR-to-tangent-rank mapping (Section V.B) presuppose adaptation.","section":"Section V.A, paragraph containing 'From now, we only consider tangent spaces that are adapted'"}],"minor_comments":[{"comment":"The panel showing ground-state energies and off-diagonal coherence (middle) uses a dual y-axis that is somewhat hard to parse. Clarifying which axis corresponds to which quantity, or splitting into separate panels, would improve readability.","section":"Figure 3 (top)"},{"comment":"The y-axis label appears truncated as '| exact|' in the rendered figure. It should read '|ω - ω_exact|' or similar.","section":"Figure 4"},{"comment":"Listed as 'unpublished manuscript.' If this reference is essential to the claims (e.g., the ideal-theoretic equations for the MPS variety cited in Section II.B), the authors should note its status more explicitly or remove it if non-essential.","section":"Reference [11]"},{"comment":"The notation M_j^n(m,m') and its relation to M_j^n(m) in the subsequent display could be unified. Currently the reader must infer that M_j^n(m) is the restriction of M_j^n(m,m') to the block with m'=m+n. Stating this explicitly would help.","section":"Section V.B, Eqs. (10)-(11)"},{"comment":"The scaling O(N^3 w d^4 D^8) is stated for uniform d and D, but the benchmarks use a bond pattern D=[3,8,...,8,3]. A brief remark on how the non-uniform pattern affects the practical scaling would help readers assess applicability to larger systems.","section":"Appendix VII.D"},{"comment":"The statement that the method 'accurately reproduces the low-lying excitation energies across the ground-state particle-number sector crossing' could note that the accuracy degrades significantly for D<=5 near J~0.18 (visible in Figure 4), to set appropriate expectations.","section":"Section IV"},{"comment":"The title uses 'rank tomography of linear matrix product tangent spaces.' The word 'tomography' may suggest a measurement protocol to some readers; 'rank analysis' or 'rank characterization' might be more precise, though this is a matter of preference.","section":"Title"}],"recommendation":"minor_revision","confidential_remarks":"The paper is well-positioned for the journal's scope bridging mathematical physics and quantum many-body theory. The algebraic-geometric perspective is a genuine angle that distinguishes this work from prior MPS tangent-space literature (Haegeman et al., Van Damme et al.). The rank tomography framework is the more novel contribution and could have standalone interest. One concern is that the benchmarks are limited to N=10, which is small by current standards; however, the paper does not claim large-system competitiveness, and the N=3 exact-recovery check provides a useful sanity check. The single-author citation [10] (Schmidt) is appropriate and disclosed."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for the careful reading and the constructive recommendation. The referee raises one major comment concerning the adapted tangent-space assumption in Section V.A. We address it below.","responses":[{"response":"We agree with the referee that the connection between the block decomposition and the adaptation condition should be made explicit. In the revised manuscript, we will add a remark (or a short proposition with proof sketch) in Section V.A that closes this gap. The argument is as follows. For a U(1)-symmetric MPS ground state, the local tensors satisfy the selection rule m' = m + n, which means each ground-state tensor M_j^n maps virtual sector m+n to virtual sector m. A general tangent variation δM_j^n decomposes into blocks δM_j^n(m, m') characterized by the excess particle number ΔM := m' - m - n. The key observation is that the Hermiticity conditions of Proposition 7, which define the horizontal tangent space, couple only blocks sharing the same ΔM. This is because the products (δM_i^{j_i})* M_i^{j_i} and M_{i+1}^{j_{i+1}} (δM_{i+1}^{j_{i+1}})* appearing in Y_i involve the ground-state tensors (which have ΔM = 0), so the resulting matrices on each virtual bond are block-diagonal in ΔM. Consequently, the Hermiticity conditions decouple across ΔM sectors, and the horizontal parameter tangent space decomposes as a direct sum over ΔM. Since the differential map DΦ is built from ground-state tensors obeying the selection rule, it preserves this decomposition, and the SVD compression step also respects it (tangent vectors in different particle-number sectors of the Hilbert space are linearly independent). Therefore P_M T ⊆ T for all M, i.e., the tangent space is adapted. This argument does not require any genericity assumption: it holds for all U(1)-symmetric MPS ground states, as it follows directly from the symmetry-imposed selection rule. We note that the adaptation condition can fail if the ground state is computed with a U(1)-broken Hamiltonian (e.g., η ≠ 0) while the谱","revision_made":"yes","referee_comment":"The adapted tangent-space assumption (P_M T ⊆ T for all M) is stated without formal proof. The block-structure analysis in Section V.B and Appendix VII.B provides strong evidence that this holds for U(1)-symmetric MPS ground states, since the selection rule m'=m+n on ground-state tensors propagates to tangent variations in the same sector. However, the paper does not explicitly state that this block structure constitutes a proof of adaptation, nor does it clarify whether adaptation is guaranteed for all U(1)-symmetric MPS or only generically. A brief remark connecting the block decomposition of Section V.B to the adaptation condition would close this gap."}],"tokens_in":20196,"tokens_out":2857,"duration_ms":95109,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"This paper does two things: (1) derives the linear MPS tangent-space excitation method for open-boundary, non-uniform systems using the algebro-geometric language of MPS varieties, and (2) introduces particle-resolved Schmidt rank (PRSR) tomography as a diagnostic for tangent-space expressivity. The first is a reformulation of known techniques (Haegeman et al. 2011, 2014) in a new mathematical language; the second is genuinely new and, I think, the more valuable part of the paper. The PRSR decomposition explains why two states with identical bond dimensions can have different tangent-space ranks and hence different spectral accuracy. That is a real insight. The Bose-Hubbard benchmarks on N=10 and N=4 systems confirm the method works and that the rank tomography correctly predicts which particle sectors are well- or poorly-represented. The proofs (Propositions 5-7, in the appendix) are clean and correct as far as I can tell. The algebro-geometric framing is not window dressing — the variety-theoretic description of the smooth full-rank stratum and the gauge-removal via horizontal tangent conditions are used consistently and correctly. The block-structure analysis in Appendix VII.B, showing how the selection rule m'=m+n organizes the local tensors, is concrete and useful. On the soft spots: the adapted tangent-space assumption (Section V.A) is stated without formal proof. The paper says 'from now, we only consider tangent spaces that are adapted' and moves on. The stress-test note flagged this as the main load-bearing concern, and I think the concern is real but correctly scoped. The block decomposition in Section V.B effectively guarantees adaptation when the ground state has definite particle number and the MPS respects U(1) symmetry — the selection rule forces each particle-number component of a tangent vector to remain in the tangent space. So the assumption holds for the cases actually benchmarked (η=0). The reader's worry about the η≠0 case (Figure 3, bottom) is misdirected: the rank tomography is not applied there, and the paper switches to continuous ⟨M̂⟩ coloring. So that is not actually a problem. The real limitation is that the method is only tested on small systems (N≤10) where exact diagonalization is available. The computational cost analysis (Appendix VII.D) shows O(N³wd⁴D⁸) scaling, which is steep but polynomial. The paper is honest about this. No shipped code, but the method is clearly specified and reproducible from the description. The 'complete explanation' claim for PRSR is properly scoped to sector-dependent errors (which sectors have deficiencies), not mode-dependent errors (which specific excitations within a deficient sector are captured). This is a fair claim supported by the evidence. This paper is for researchers working on tensor network methods for spectral calculations, particularly those who want to understand why tangent-space approximations fail in specific sectors. The PRSR diagnostic is the kind of tool that will be cited and used. It deserves a serious referee who can check the algebraic geometry carefully and assess whether the method scales beyond the benchmark regime.","headline":"Solid MPS tangent-space method with a genuinely new rank-tomography diagnostic; deserves a serious referee.","tokens_in":21381,"tokens_out":714,"would_cite":true,"duration_ms":63487,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Particle-resolved ranks reveal why MPS tangent spaces miss some excitations","keywords":["matrix product states","tangent space","excitation spectrum","Schmidt rank","Bose-Hubbard model","tensor network variety","parametric deficiency","rank tomography"],"falsifier":"Compute the PRSR distribution and parametric deficiency for a system where the tangent-space method is known to fail spectrally; if the deficiency does not correlate with the sectors where errors are largest, the explanatory claim of PRSR is undermined.","tokens_in":20592,"feed_emoji":"🔬","tokens_out":1369,"duration_ms":62361,"temperature":0.7,"pith_summary":"The paper formulates a method to compute excitation spectra of finite quantum many-body systems by linearizing around a matrix product state (MPS) ground state on its tangent space, using the algebraic-geometric structure of MPS varieties with open boundary conditions. Beyond the spectral method itself, the central contribution is a diagnostic tool called rank tomography: the tangent space is decomposed into particle-number sectors, and the dimension of each sector is shown to depend not on the coarse bond dimension alone, but on a finer invariant called the particle-resolved Schmidt rank (PRSR) distribution. This distribution records how many entanglement channels exist for each possible particle number on each side of every bipartition cut. Two ground states with identical bond dimensions can have different PRSR profiles, and this difference fully accounts for why the tangent-space method accurately reproduces excitations in some particle sectors while failing in others. The parametric deficiency — the gap between available and needed tangent directions in each sector — quantifies this failure mode. Applied to the Bose–Hubbard model on a 10-site chain, the method reproduces low-lying excitation energies to errors of order 10⁻⁵ at bond dimension D=8, and the PRSR analysis explains the discontinuous jumps in accuracy that occur when the ground state crosses between particle-number sectors.","feed_headline":"Particle-resolved ranks expose hidden structure in MPS excitations","feed_subtitle":"Two ground states with identical bond dimensions can have different internal entanglement profiles — and that difference predicts which exci","key_machinery":"Particle-resolved Schmidt rank (PRSR) distribution: for each bipartition cut ℓ and each particle number m on the left side, r_ℓ(m) = rank(Ψ_ℓ^(m)), where Ψ_ℓ^(m) is the coefficient matrix of the ground-state block with m particles left of the cut. The ordinary bond dimension D_ℓ = Σ_m r_ℓ(m) is the sum of these particle-resolved ranks, so PRSR is a refinement of bond dimension that is invisible at the coarse level but determines tangent-space structure.","core_discovery":"The particle-resolved Schmidt rank (PRSR) distribution — the rank of each particle-number block in every bipartition of the ground state — is the hidden variable controlling tangent-space expressivity. The coarse bond dimension, which sums PRSR values across particle sectors, is an insufficient descriptor: states with the same bond dimension but different PRSR profiles yield tangent spaces of different sector-by-sector dimension, and hence different spectral accuracy. The parametric deficiency δ_M = d_target_M − dim(T_proj_M) measures exactly how many independent directions are missing in sector M, and this quantity predicts whether excitations in that sector will be accurately captured or近似","pith_inferences":["If the PRSR distribution is the true determinant of tangent-space expressivity, then optimizing ground-state MPS variational calculations to maximize PRSR in target excitation sectors — rather than merely maximizing bond dimension — could improve spectral accuracy at lower computational cost.","The sector-dependent accuracy jumps observed at ground-state particle-number crossings suggest that the PRSR profile could serve as an order parameter or diagnostic for phase transitions in finite systems, since it changes discontinuously even when the coarse bond dimension does not.","The parametric deficiency framework could be extended to mixed states or finite-temperature calculations, where the relevant decomposition might be by energy or symmetry sectors rather than particle number, potentially yielding analogous resolved-rank diagnostics for density-matrix renormalization group methods."],"forward_implications":["The PRSR distribution provides a diagnostic that can be computed from any MPS ground state to predict, before running the tangent-space spectral calculation, which excitation sectors will be accurately captured and which will suffer from parametric deficiency.","The concept of rank tomography generalizes beyond particle-number sectors: any conserved quantum number that decomposes the Hilbert space into sectors could be used to define analogous resolved-rank profiles, extending the diagnostic to systems with other symmetries.","The authors identify rank frustration — combinatorial constraints on maximal bond dimensions in fixed-particle-number sectors — as a structural limitation of the linear tangent-space ansatz, and suggest that a second-order (double tangent) extension could overcome it.","The method applies to any Hamiltonian with an efficient matrix product operator representation, not just the Bose–Hubbard model, making it a general tool for finite-system spectroscopy."],"fun_headline_variants":["Schmidt rank distribution governs tangent-space expressivity in MPS","Bond dimension alone fails to predict excitation spectral accuracy","Particle-resolved ranks reveal hidden structure in MPS excitation spaces","Parametric deficiency predicts which excitation sectors MPS captures","Sector-by-sector rank profiles control MPS tangent-space dimensions"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The method assumes that the tangent space is adapted to the particle-number decomposition, meaning each particle-number component of a tangent vector is itself a valid tangent vector. This can fail if the ground state is computed with a symmetry-breaking term that the tangent-space calculation does not include, and the paper does not rigorously justify that this adaptation holds in the U(1)-broken examples it presents.","fun_headline_variants_meta":{"raw":{"variants":["Schmidt rank distribution governs tangent-space expressivity in MPS","Bond dimension alone fails to predict excitation spectral accuracy","Particle-resolved ranks reveal hidden structure in MPS excitation spaces","Parametric deficiency predicts which excitation sectors MPS captures","Sector-by-sector rank profiles control MPS tangent-space dimensions"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":511,"prompt_tokens":432,"completion_tokens":79,"prompt_tokens_details":null},"tokens_in":432,"tokens_out":79,"duration_ms":18675,"temperature":1.0,"reasoning_tokens":null,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-07T20:21:59.465535+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"Compute the PRSR distribution and parametric deficiency for a system where the tangent-space method is known to fail spectrally; if the deficiency does not correlate with the sectors where errors are largest, the explanatory claim of PRSR is undermined.","supporting_citations":[],"review_version":1}