Pith. sign in

REVIEW 2 major objections 6 minor 1 cited by

Reply to comment on "Controlled bond expansion for Density Matrix Renormalization Group ground state search at single-site costs"

T0 review · 2 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2501.12291 v1 pith:M4JAJF2B submitted 2025-01-21 cond-mat.str-el

Andreas Gleis , Jheng-Wei Li , Jan von Delft This is my paper · ORCID
classification cond-mat.str-el
keywords controlledbondexpansionDMRGmatrixproductstatestangentspaceprojectorTDVP3Smixingsubspaceprojectionerror
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

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.

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 (2)
  1. [S-2 B (intro and H2 example)] 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.
  2. [S-1 (Fig. S-1)] 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.
minor comments (6)
  1. [S-2 B (Eqs. S1-S3)] 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].
  2. [S-2 B (Eqs. S1-S3)] 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.
  3. [Fig. S-1 caption] 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.
  4. [S-1 (first paragraph)] 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.
  5. [S-1 (Fig. S-1(d) discussion)] 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.
  6. [S-1 (end of section)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the DD-projection argument rests on explicit toy-model algebra, not on fitted inputs or self-referential definitions.

full rationale

The Reply's core claims are direct rebuttals of McCulloch and Osborne's proposal, supported by explicit constructions rather than by fitting or renaming. The H2 example in Sec. S-2B is an existence proof with a stated single-vector expansion budget: it explicitly computes the two candidate expansion directions (|1> with coefficient 1 and |2> with coefficient omega_0<1), shows that omitting the DD projection selects |1> and loses the |22> contribution, and concludes only that avoidable errors 'can' occur under finite expansion budgets. This is not circular: the conclusion follows from the displayed algebra, and the limitation that K=1 is a toy model is acknowledged in the text. The projector identities (S1)-(S3) are cited to the authors' own prior work (Ref. [4]), but they appear explicitly in the supplement as checkable linear-algebra identities with stated assumptions that do not include the target conclusion; they are not equivalent to the Reply's claim that omitting P_DD can be harmful. The CBE+alpha comparisons against pure 3S and MO's proposed alternative are empirical benchmarks on free-fermion models, not predictions forced by construction. No equation or parameter is introduced that reduces to the conclusion by definition, and no load-bearing uniqueness claim is imported solely from the authors' prior papers. Thus the derivation chain is self-contained and no circular step is identifiable.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central claims rest on the projector formalism of Ref. [4], the algorithmic structure of CBE-TDVP from Ref. [10], and a small set of hand-chosen numerical examples. No new physical entities are postulated, and the alpha mixing parameter is the only obvious hand-chosen quantity in the numerical comparisons.

free parameters (2)
  • 3S mixing parameter alpha for pure 3S = alpha = 10^-4
    Chosen as required for pure 3S stability; the asymmetry with alpha=1 for CBE+alpha affects the comparison in Fig. S-1 and is not justified by a systematic scan.
  • CBE+alpha mixing parameter alpha = alpha = 1 for the first n_alpha=22 half-sweeps, then alpha = 0
    Hand-chosen schedule; the reply argues alpha=O(1) is favorable, but no sensitivity analysis is shown.
assumptions (3)
  • domain assumption The 2-site tangent space projection error decomposes locally as P_DD_{ell,ell+1} H|Psi>, with P_DD = P^{2s}_ell (1 - P^{1s}) (Eq. S1-S3).
    Adopted from the authors' Ref. [4]; the reply's explanation of why the DD projection is necessary uses these identities without re-deriving them.
  • standard math Single-site DMRG and 1s-TDVP sweeps cover the full 1s tangent space V^{1s} through flat local regions.
    Invoked in Sec. S-2 A following Refs. [7,8]; needed to argue that directions in V^{1s} are already handled and should be projected out of bond expansion.
  • domain assumption CBE-TDVP's pre-expansion plus projector-splitting backward steps controls Trotter error and lets the toy-model sequences represent the algorithm exactly to O(delta).
    The H1/H2 examples in Sec. S-2 B assume this algorithmic structure from the authors' Ref. [10].

how reviews work

0 comments
Cite this review

Pith. "Pith review of Reply to comment on "Controlled bond expansion for Density Matrix Renormalization Group ground state search at single-site costs"." pith.science (2026). https://pith.science/paper/M4JAJF2B

@misc{pith2026250112291,
  author       = {Pith},
  title        = {Pith review of: Reply to comment on "Controlled bond expansion for Density Matrix Renormalization Group ground state search at single-site costs"},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M4JAJF2B}},
  note         = {Machine review of arXiv:2501.12291}
}
abstract

We reply to McCulloch and Osborne's recent comment on our manuscript (Phys. Rev. Lett. 130, 246402 (2023)) on controlled bond expansion (CBE) for density matrix renormalization group (DMRG) ground state search. We appreciate their suggestion to consider randomized SVD and address their constructive critique on the variational properties of CBE-DMRG. However, we strongly disagree with their proposal to omit the projection to the 2-site tangent space and explain its importance for efficient bond expansion. In particular, in the context of CBE applied to the time-dependent variational principle (TDVP), we show that omitting this projection can lead to avoidable errors. Lastly, we emphasize the complementary roles of 3S mixing and CBE, reiterating our recommendation from Phys. Rev. Lett. 130, 246402 (2023) to combine both methods (CBE+$\alpha$). We provide examples to demonstrate the superior efficiency and robustness of CBE+$\alpha$.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Interplay of spin-orbit coupling, crystal field splitting and correlations: a ghost rotationally invariant slave boson treatment

    cond-mat.str-el 2026-07 conditional novelty 6.0 of 10

    GRISB with up to 21 bath orbitals reproduces DMFT results for spin-orbit-related observables in Sr2RuO4 and shows that U enhances while J suppresses the effective spin-orbit coupling.

Reference graph

Works this paper leans on

11 extracted references · 9 canonical work pages · cited by 1 Pith paper

  1. [4]

    Gleis, J.-W

    A. Gleis, J.-W. Li, and J. von Delft, Projector formalism for kept and discarded spaces of matrix product states, Phys. Rev. B 106, 195138 (2022)

    work page 2022

  2. [10]

    J.-W. Li, A. Gleis, and J. von Delft, Time-dependent varia- tional principle with controlled bond expansion for matrix product states, Phys. Rev. Lett. 133, 026401 (2024)

    work page 2024

  3. [1]

    I. P. McCulloch and J. Osborne, Comment on ”controlled bond expansion for density matrix renormalization group ground state search at single-site costs”, (2024)

    work page 2024

  4. [2]

    Gleis, J.-W

    A. Gleis, J.-W. Li, and J. von Delft, Controlled bond ex- pansion for density matrix renormalization group ground state search at single-site costs, Phys. Rev. Lett. 130, 246402 (2023)

    2023

  5. [3]

    Hubig, I

    C. Hubig, I. P. McCulloch, U. Schollw¨ ock, and F. A. Wolf, Strictly single-site DMRG algorithm with subspace expansion, Phys. Rev. B 91, 155115 (2015)

    work page 2015

  6. [5]

    The Supplemental Material includes Refs

    See Supplemental Material at [url] for more details on CBE+α, an additional comparison of CBE, 3S, and CBE+α, and a discussion on the importance of projecting to the 2-site tangent space in CBE. The Supplemental Material includes Refs. [1–4, 6–10]

    work page

  7. [6]

    Stoudenmire and S

    E. Stoudenmire and S. R. White, Studying two- dimensional systems with the density matrix renormaliza- tion group, Ann. Rev. Cond. Mat. Phys. 3, 111 (2012)

    work page 2012

  8. [7]

    Haegeman, M

    J. Haegeman, M. Mari¨ en, T. J. Osborne, and F. Verstraete, Geometry of matrix product states: Metric, parallel trans- port, and curvature, Journal of Mathematical Physics 55, 021902 (2014)

    work page 2014

Show all 11 references
  1. [8]

    Haegeman, C

    J. Haegeman, C. Lubich, I. Oseledets, B. Vandereycken, and F. Verstraete, Unifying time evolution and optimiza- tion with matrix product states, Phys. Rev. B 94, 165116 (2016)

    2016

  2. [9]

    Hubig, J

    C. Hubig, J. Haegeman, and U. Schollw¨ ock, Error esti- mates for extrapolations with matrix-product states, Phys. Rev. B 97, 045125 (2018)

    2018

  3. [11]

    Controlled bond expansion for Density Matrix Renormalization Group ground state search at single-site costs

    S. R. White, Density matrix renormalization group algo- rithms with a single center site, Phys. Rev. B 72, 180403 (2005). Supplemental material: Reply to comment on “Controlled bond expansion for Density Matrix Renormalization Group ground state search at single-site costs” An...

    2005

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.