{"id":"ab78f008-59d0-47d8-aba1-af3358d49c60","arxiv_id":"2501.12291","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"The reply shows that keeping the double-discarded projection in CBE is important and that CBE+alpha converges more reliably than pure CBE or pure 3S on the tested fermionic models.","lead":"This reply defends the controlled bond expansion (CBE) method for DMRG against a published comment, arguing that the 2-site tangent space projection is essential and that combining CBE with 3S mixing (CBE+alpha) is robust. It adds toy-model examples showing that dropping this projection can create avoidable errors in time-dependent DMRG.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the DD-projection counterexample is valid as an existence proof for a finite expansion budget, and the CBE+alpha recommendation is appropriately conditional.","rationale":"The reader's weakest assumption identifies a real scope limitation of the H2 toy model, but the paper's claim is deliberately 'can cause avoidable errors,' and the toy model proves that statement. The numerical CBE+alpha benchmarks are the weaker part of the reply: free-fermion Hamiltonians, hand-picked alpha, no code release, and no correlated models. This justifies the conditional verdict but not a rejection. The proposed H2 K=2 check would directly test whether the no-D failure is merely a budget artifact, and would help future readers decide how seriously to take the DD projection in practice. No ad hominem or internal inconsistency found; the reply engages constructively with MO and even accepts randomized SVD as an interesting suggestion.","tokens_in":9398,"tokens_out":13970,"duration_ms":147234,"concrete_test":"Re-run the Sec. S-2B H2 TDVP example under the no-D expansion with K=2 expansion vectors (and with randomized SVD oversampling) and compare the order-delta overlap with exact e^{-i delta H2}|00>. If the |22> contribution is recovered at K=2, the demonstrated failure is specific to K=1, clarifying that full DD projection is an efficiency safeguard under tight budgets rather than a necessary condition for correctness; this would strengthen the case for the CONDITIONAL verdict.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reply's core claim stands under scrutiny. The Sec. S-2B H2 example is an explicit existence proof: with expansion budget K=1, omitting P_DD selects the V^{1s} direction |1> (coefficient 1) over the P_DD direction |2> (coefficient omega_0<1), so the first-order TDVP term |22> is lost. Since the authors state that K=1 is a toy model for realistic finite expansion budgets, the 'can cause avoidable errors' claim is justified; it does not assert that no-D always fails. The projector identities (S1)-(S3) are cited to Ref. [4] rather than re-derived, but no internal inconsistency is apparent. The main limitation is empirical: the CBE+alpha superiority in Sec. S-1 is shown on free-fermion models with hand-chosen alpha and no released code, which supports the CONDITIONAL verdict but does not undermine the logical reply to MO.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript is a formal reply to McCulloch and Osborne's (MO) Comment on the authors' controlled bond expansion (CBE) paper (Phys. Rev. Lett. 130, 246402). It makes three main points. First, it concedes MO's terminological point: CBE-DMRG and 2s-DMRG are not variational in the strict sense, because the truncation step (iv) can slightly increase the energy. Second, it defends the P^DD projection against MO's proposal to omit it, arguing from projector identities (S1)-(S3) and two explicit two-site toy models (Sec. S-2B) that omitting the projection targets directions already contained in the one-site tangent space: this is inefficient for DMRG and can in TDVP cause avoidable first-order projection errors when the expansion budget is finite. Third, it reiterates the recommendation to combine CBE with 3S mixing (CBE+α), presenting free-fermion ground-state calculations on a ring and a cylinder (Sec. S-1, Fig. S-1) in which CBE+α with α=1 converges faster and more robustly than pure CBE or pure 3S with α=10^-4. The reply also welcomes MO's randomized-SVD suggestion and states that the authors intend to test it.","tokens_in":9571,"tokens_out":23192,"duration_ms":215667,"significance":"The reply's central rebuttal is effective. The H2 example in Sec. S-2B is a complete, checkable existence proof: with an expansion budget of one vector, the no-DD rule applied to |10⟩+ω0|22⟩ picks the unit-coefficient V^{1s} direction |1⟩ over the P^DD direction |2⟩ (coefficient ω0<1), losing the |22⟩ contribution at order δ; the DD-only rule captures it. The projector identities (S1)-(S3), cited from Ref. [4], are standard two-site tangent-space decompositions and are internally consistent on inspection. The reply is careful: the 'can cause avoidable errors' claim is conditional, the randomized-SVD variant is explicitly not implemented, and all numerical parameters are documented. If the claims hold, the reply refutes the comment's main critique and clarifies the complementary roles of CBE and 3S. The principal weakness is evidential: the practical reach of the K=1 toy model and the advertised robustness of CBE+α are asserted more strongly than the data establish.","major_comments":[{"comment":"The introductory paragraph of Sec. S-2 states that the DD projection is 'necessary for a successful CBE update,' and the discussion following the H2 example generalizes to 'realistic situations' with finite expansion budgets. The H2 example itself is valid as an existence proof: with expansion budget K=1, the no-DD selection rule applied to (P_DD+P_DK)|δΨ⟩ = |10⟩+ω0|22⟩ picks the coefficient-1 direction |1⟩ over ω0|22⟩, so the |22⟩ contribution is lost at order δ, whereas CBE's DD-only selection captures it; I verified the coefficient ordering and the stated sequences. However, the step from this toy model to the practical conclusion that omitting the DD projection can cause avoidable errors in actual CBE-TDVP rests on the unquantified assertion that K=1 is representative of typical expansion budgets. The reply states this premise but does not support it with data from the CBE-TDVP algorithm of Ref. [10], nor does it show that randomized SVD with the budgets actually used in practice would reproduce the K=1 prioritization. Since the abstract's central claim is hedged as 'can lead to avoidable errors,' the existence proof suffices logically; my request is that the 'necessary' phrasing and the surrounding text carry the same finite-budget qualifier, or that the authors add a demonstration at a realistic expansion budget.","section":"S-2 B (intro and H2 example)"},{"comment":"The numerical support for the CBE+α recommendation compares CBE+α with α=1 against pure 3S with α=10^-4, and several conclusions are worded more strongly than the evidence. The claim that small α is 'required in pure 3S calculations' is asserted rather than demonstrated: no α-scan for 3S is shown, so the reader cannot tell whether 3S with a moderately larger α would escape the local minimum in the inset of Fig. S-1(a) or match CBE+α in Fig. S-1(d). Similarly, the conclusion from Fig. S-1(d) that 3S's convergence issues 'cannot be simply cured by somewhat increasing D*' rests on a single free-fermion model and a single initialization sequence, without error bars or repeated runs. The abstract's closing claim of 'superior efficiency and robustness' therefore goes beyond what the presented data establish. I suggest either adding an α-scan for pure 3S and, ideally, a second model or initialization for the D*-extrapolation comparison, or explicitly scoping the robustness claim to the models and parameter choices shown.","section":"S-1 (Fig. S-1)"}],"minor_comments":[{"comment":"In the paragraph following Eq. (S2), the sentence 'which differs from (S2) by having 1 instead of a D projection at site ℓ+1' should refer to Eq. (S1), not (S2); as printed, the comparison is self-referential. In addition, the subscript on P^DK in Eq. (S2) reads 'ℓ,ℓ+2', which does not match the bond (ℓ,ℓ+1) discussed in the surrounding text; please reconcile the index notation with Ref. [4].","section":"S-2 B (Eqs. S1-S3)"},{"comment":"The projector identities (S1)-(S3) are asserted to follow from Eqs. (33), (37), (53), and (54) of Ref. [4] but are not re-derived; given that these identities carry much of the weight of the toy-model argument, a brief restatement of the underlying decomposition of the two-site tangent space would make the reply self-contained.","section":"S-2 B (Eqs. S1-S3)"},{"comment":"The caption of Fig. S-1 interleaves the descriptions of panels (a)-(d) awkwardly: '(a) half-filling, (b) filling N=0.9L' is followed immediately by '(c) Spinful free fermions…'; reformat the caption so that each panel is described by a self-contained sentence.","section":"Fig. S-1 caption"},{"comment":"The sentence 'further, it allows the use of a large α = O(1), which is in our view required to get a calculation stuck in a local minimum going again by escaping a metastable solution' is grammatically tangled and should be rewritten; the informal 'very large(!)' in the main text can also be regularized.","section":"S-1 (first paragraph)"},{"comment":"The quantitative claims in the discussion of Fig. S-1(d) ('almost 4 times lower,' 'an order of magnitude more accurate') are not checkable from the plotted data alone; reporting the extrapolated energies and error measures in the text or a table would make the comparison verifiable.","section":"S-1 (Fig. S-1(d) discussion)"},{"comment":"The response to MO's suggestion about keeping D(1+δ) states in 3S ends with 'We are not entirely sure what MO mean by that'; since this leaves the point unresolved, it would help to state explicitly which reading of the suggestion the reply's comparison addresses.","section":"S-1 (end of section)"}],"recommendation":"minor_revision","confidential_remarks":"For the editor: the logical rebuttal of MO's critique is sound, and the H2 toy example is a genuine, hand-checkable counterexample; that part of the reply should stand. The main editorial judgment call is the strength of the Sec. S-1 empirical claims: the asymmetry in α between CBE+α and 3S is intentional and defensible, but the 'required in pure 3S' assertion and the 'cannot be cured by increasing D*' conclusion would benefit from an α-scan or scoped wording. The reply's reliance on its own Refs. [2,4,10] is appropriate in a reply-to-comment context and does not conceal any missing external basis for the central identities. I see no scope or novelty concerns for the journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis reply to McCulloch and Osborne is worth a look if you have any stake in DMRG bond expansion. The main new thing is the H1/H2 toy-model pair in Sec. S-2B: a clean existence proof that dropping the DD projection can cause avoidable projection errors in TDVP when the expansion budget is finite. With K=1, omitting DD selects the |1> direction and loses the |22> term, while CBE with DD keeps it. That is a real, checkable argument, and it holds up.\n\nThe paper also does a few things well. It concedes the variational phrasing issue openly, welcomes randomized SVD as a possible simplification, and gives new benchmarks (Fig. S-1) comparing CBE, 3S, and CBE+alpha on free-fermion rings and a cylinder. The CBE+alpha recommendation is not new—it restates the authors' own PRL advice—but the new benchmarks make the practical case more concrete. The projector identities are cited to their PRA/B paper rather than re-derived, but those are linear algebra identities and the dependence is transparent.\n\nThe soft spots are real but not fatal. The empirical comparison is limited to non-interacting fermion models, with hand-chosen alpha values (1 for CBE+alpha, 1e-4 for 3S), and no code or data are released. That makes the 'superior robustness' claim plausible rather than proven. The H2 example is explicitly a toy model with K=1; in practice a randomized SVD with larger K could pick up both directions, blunting the force of the example. But the authors acknowledge that this is a toy model, and the point stands as an existence proof: omitting DD targets directions already in the 1-site tangent space, which is inefficient and can be worse.\n\nThe citation pattern is fine. Self-citation of the projector formalism is appropriate since those are checkable identities, and the comparison includes the external 3S algorithm and MO's proposal.\n\nWho gets value: anyone implementing CBE or comparing 3S and CBE for DMRG or TDVP. The reply is short, explicit, and settles the logical core of the argument.\n\nRecommendation: this deserves a serious referee. It's a legitimate scientific reply with a valid core. I'd accept it for review, and I'd ask for a short note addressing the free-fermion-only limitation and ideally a pointer to code, but I would not block it on those. For a reply, it is thorough and honest.","headline":"A focused, technically solid reply that makes the DD-projection case with a clean toy-model proof; the CBE+alpha benchmarks are new but limited to free fermions.","tokens_in":10096,"tokens_out":2624,"would_cite":true,"duration_ms":25253,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The Reply defends the DD projection in CBE and shows that omitting it can cause avoidable errors in TDVP, recommending CBE+alpha as the robust combination.","keywords":["controlled bond expansion","DMRG","matrix product states","tangent space projector","TDVP","3S mixing","subspace expansion","projection error"],"falsifier":"Run a single TDVP time step for $H_2$ starting from $|00\\rangle$ under the no-DD expansion rule but retain two expansion vectors; if the resulting order-$\\delta$ state contains both $|10\\rangle$ and $|22\\rangle$, the avoidable-error argument for that model would be falsified.","tokens_in":9210,"feed_emoji":"⚛️","tokens_out":7626,"duration_ms":66219,"temperature":0.7,"pith_summary":"This Reply defends the controlled bond expansion (CBE) algorithm against a comment that suggested omitting the projection to the 2-site tangent space during bond expansion. The authors argue that the DD projection is not optional: it selects expansion directions that are genuinely absent from the 1-site tangent space, and skipping it can waste expansion vectors on directions that a single-site sweep will handle anyway. For time evolution via TDVP, they show with explicit examples that omitting the projection can produce avoidable projection errors, including missing an order-$\\delta$ contribution in a simple model. They also argue that CBE and 3S mixing solve different problems and recommend combining them as CBE+$\\alpha$, which they demonstrate converges faster and escapes local minima more reliably than either method alone. A sympathetic reader would take away that CBE is the right way to expand bonds and that the complementarity of expansion and mixing is the practical message.","feed_headline":"Skipping a projection in CBE costs an order-delta term","feed_subtitle":"Omitting the two-site tangent-space projection can drop state components; CBE plus 3S mixing avoids it.","key_machinery":"The load-bearing object is the decomposition of the 2-site tangent-space projector into local pieces, summarized by the identities $P^{\\mathrm{DD}}_{\\ell,\\ell+1}=P^{2s}_\\ell(1-P^{1s})$, $P^{\\mathrm{DK}}_{\\ell,\\ell+1}=P^{2s}_\\ell(1-P^{1s}_{\\ell+1})$, and $P^{\\mathrm{DD}}_{\\ell,\\ell+1}+P^{\\mathrm{DK}}_{\\ell,\\ell+1}$ for the projector proposed in the comment. The first identity says the DD subspace lies outside the 1-site tangent space, so expansion vectors chosen there are new directions; the second and third show the proposed alternative also targets $P^{\\mathrm{DK}}_{\\ell,\\ell+1}$, which is inside $V^{1s}$ and will be visited by the subsequent single-site update. The Reply uses these identities, together with the two-level toy Hamiltonian $H_2=H_1+\\omega_0(|22\\rangle\\langle00|+|00\\rangle\\langle22|)$ with one allowed expansion vector, to demonstrate the difference between CBE and the no-DD rule.","core_discovery":"In the authors' view, both ground-state DMRG and TDVP update a matrix product state by rotating within the 1-site tangent space $V^{1s}$, and the projection error they suffer is the component of $H|\\Psi\\rangle$ lying outside it. The CBE update reduces this error by adding a few directions from $V^{2\\perp}=V^{2s}\\setminus V^{1s}$, selected through the projector $P^{\\mathrm{DD}}_{\\ell,\\ell+1}=P^{2s}_\\ell(1-P^{1s})$. The Reply's central discovery is that a proposed alternative, which replaces the $D$ projection at site $\\ell+1$ by the identity and therefore also targets directions in $P^{\\mathrm{DK}}_{\\ell,\\ell+1}\\subset V^{1s}$, is not just inefficient but in TDVP can cause avoidable errors: the $H_2$ example with a single expansion vector shows that the no-DD rule picks the $|1\\rangle$ direction and loses the $|22\\rangle$ contribution at order $\\delta$, whereas CBE's DD projection keeps it. The Reply therefore claims that the full DD projection should be retained, and that CBE should be combined with a strong 3S mixing step ($\\alpha=O(1)$), since mixing helps escape metastable minima while CBE ensures fast descent within the current minimum.","pith_inferences":["Beyond the paper's examples, if the commenters' randomized-SVD idea is adopted, the same DD projection could likely be applied before the SVD; the $H_2$ example suggests that without it, a small SVD budget can select the wrong directions.","The single-vector truncation in the toy model is the regime where the DD projection matters most; with a large expansion budget, the no-DD rule may approach the same result, so the practical difference may shrink as computational costs allow more vectors.","The same tangent-space argument should apply to other MPS time-evolution methods that rely on local tangent-space updates, not just TDVP."],"forward_implications":["CBE-TDVP should keep the full DD projection; omitting it risks avoidable projection errors whenever the expansion budget is small.","CBE and 3S mixing are complementary: CBE descends quickly to the nearest local minimum, while large-$\\alpha$ mixing escapes metastable minima, so CBE+$\\alpha$ with $\\alpha=O(1)$ during bond growth is recommended.","Pure CBE can get stuck in local minima, and pure 3S with small $\\alpha$ can also fail to escape, whereas the combination converges in fewer sweeps to lower energy.","When converging a series of ground states at increasing bond dimensions, CBE+$\\alpha$ reaches a given accuracy at roughly half the bond dimension needed by 3S in the examples shown."],"supporting_citations":[{"why":"The comment whose proposal to omit the DD projection is the target of the Reply.","marker":"[1]"},{"why":"Defines the CBE algorithm and its steps, which the Reply defends and extends.","marker":"[2]"},{"why":"The strictly single-site 3S mixing algorithm used for comparison and combined into CBE+alpha.","marker":"[3]"},{"why":"Supplies the projector identities for kept and discarded spaces that carry the tangent-space argument.","marker":"[4]"},{"why":"Provides the projector-splitting integrator underlying the TDVP sweeps where the avoidable errors are demonstrated.","marker":"[8]"},{"why":"Introduces CBE-TDVP, the setting in which the Reply's explicit error examples occur.","marker":"[10]"}],"fun_headline_variants":["Omitting tangent-space projection costs order-delta term","CBE+3S mixing beats skipping two-site projection in DMRG","Why dropping a projection in DMRG costs an order-delta error","Retain the DD projection: CBE plus alpha for robust ground states","No-DD rule fails TDVP; CBE+α preserves crucial components"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The demonstration that skipping the DD projection causes an avoidable error assumes only one expansion vector is kept in the $H_2$ toy model; if realistic runs keep many vectors or a randomized SVD recovers both directions, the claimed necessity of the full projection is not established.","fun_headline_variants_meta":{"raw":{"variants":["Omitting tangent-space projection costs order-delta term","CBE+3S mixing beats skipping two-site projection in DMRG","Why dropping a projection in DMRG costs an order-delta error","Retain the DD projection: CBE plus alpha for robust ground states","No-DD rule fails TDVP; CBE+α preserves crucial components"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000178,"raw_usage":{"total_tokens":1335,"prompt_tokens":1021,"completion_tokens":314,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":637,"completion_tokens_details":{"reasoning_tokens":220}},"tokens_in":637,"tokens_out":314,"duration_ms":3366,"temperature":1.0,"reasoning_tokens":220,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T17:18:20.830199+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a single TDVP time step for $H_2$ starting from $|00\\rangle$ under the no-DD expansion rule but retain two expansion vectors; if the resulting order-$\\delta$ state contains both $|10\\rangle$ and $|22\\rangle$, the avoidable-error argument for that model would be falsified.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The comment whose proposal to omit the DD projection is the target of the Reply."},{"cited_title":"Hubig, I","cited_arxiv_id":null,"evidence_quote":"The strictly single-site 3S mixing algorithm used for comparison and combined into CBE+alpha."},{"cited_title":"Gleis, J.-W","cited_arxiv_id":null,"evidence_quote":"Supplies the projector identities for kept and discarded spaces that carry the tangent-space argument."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces CBE-TDVP, the setting in which the Reply's explicit error examples occur."}],"review_version":1}