{"id":"2403acd9-8a88-4cbc-9d62-f6ef7af54e5a","arxiv_id":"2501.11810","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A delta-function index-shifting algorithm builds locally connected initial tensors for the tensor renormalization group from arbitrary Boltzmann factors, and boundary-style squeezers remove the initial-tensor dependence shown in an Ising benchmark.","lead":"This paper gives a systematic recipe for rewriting a partition function as a tensor network that the tensor renormalization group can process, by inserting delta functions and shifting indices along a tree. It matters because tensor renormalization group methods avoid Monte Carlo sign problems, so a generic way to build their input tensors could widen their use in lattice field theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The index-shift construction is exact, but the claim that squeezers remove initial-tensor dependence rests on a single Ising benchmark; this transfer is the load-bearing soft spot.","rationale":"The reader's weakest assumption correctly identifies the unproven transfer of the squeezer improvement from one 2D Ising test to arbitrary TRG methods. My own reading of Algorithm 1 found no flaw in the exactness of the construction: the delta-insertion and index-shift steps are variable renamings, and the nonlocal example in eq. (10) satisfies the pairwise-index condition after contraction. The real soft spot is the second central claim in the abstract and Section 5, where the general statement that replacing isometries by squeezers removes initial-tensor dependence is supported only by Figure 5 and by references [8,15]. Because the reader already conditioned the verdict on this point, I see no reason to move the verdict; the condition should stand. The proposed test using a second model with different symmetry would settle whether the improvement claim actually generalizes.","tokens_in":13680,"tokens_out":30470,"duration_ms":313134,"concrete_test":"Run boundary-HOTRG with K(delta) and K(exp) on a second model with different tensor symmetry, e.g., the 2D q=3 Potts model or the 2D Z2 gauge theory, computing |1 - F(D)/F_ref| for D = 4, 8, 16, 32 and at several couplings including the critical point. If the two initial-tensor error curves do not agree within a factor of about 2 across D and coupling, the claim that squeezers remove initial-tensor dependence does not transfer beyond the Ising benchmark.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central construction in Algorithm 1 is sound: inserting a delta and shifting it to a neighboring tensor is a variable renaming, and the worked examples in eqs. (7) and (10) are internally consistent. The load-bearing problem instead sits in Section 5. The paper claims that replacing HOTRG isometries by squeezers 'removes the dependence on the form of the initial tensors' and that this holds for any TRG method. The evidence is one benchmark: the 2D Ising free energy at the critical point with V=2^20, comparing K(delta) and K(exp) under HOTRG and b-HOTRG (Fig. 5). The squeezer construction itself is not derived here but imported from [8,15], and no second model, off-critical temperature, different boundary condition, or error bar is given. The abstract's improvement claim and the conclusion that the new initial tensors 'lead to similarly good results as previous, problem-specific methods' depend on this transfer. If the transfer fails for another model or regime, the improvement claim is unsupported even though the exact initial-tensor construction remains valid.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a systematic method for constructing locally connected tensor networks as initial inputs for tensor renormalization group (TRG) methods. Starting from a partition function written as a product of Boltzmann factors, the method introduces auxiliary indices through delta functions and shifts those deltas along a directed tree, so that each index appears on exactly two neighboring tensors. The construction is illustrated on the two-dimensional Ising model, a nonlocal four-spin interaction, and a three-dimensional model. The paper also compares HOTRG and boundary-HOTRG for two different initial tensor choices in the 2D Ising model and argues that boundary-HOTRG's squeezers remove the dependence of TRG accuracy on the initial tensor form.","tokens_in":13845,"tokens_out":4774,"duration_ms":54027,"significance":"The exact initial-tensor construction in Sections 2 and 3 is a genuinely useful contribution: it is parameter-free, involves no expansions or decompositions, and is applicable to arbitrary Boltzmann weights. The worked identities in Eqs. (4)-(8) and (10) are internally consistent, and the use of the Onsager result as an external benchmark for the Ising free energy is appropriate. The Steiner-tree formulation in Section 4 correctly identifies a practical algorithmic cost. However, the improvement claim in Section 5 is supported by only a single numerical example, so the significance of the 'removing initial-tensor dependence' part is presently conditional. If the claim is either strengthened by additional tests or appropriately qualified, the paper would be a solid proceedings contribution.","major_comments":[{"comment":"The abstract and conclusions state that replacing isometries by squeezers 'removes the dependence on the form of the initial tensors', but the evidence is a single benchmark: the 2D Ising free energy at the critical temperature with V=2^20. No off-critical temperature, second model, different boundary condition, or error estimate is provided. Since this is the load-bearing evidence for the improvement claim, either add further tests or qualify the claim to the tested case.","section":"§5, Fig. 5"},{"comment":"The b-HOTRG squeezers P_1 and P_2 are not derived in this manuscript but are imported from Refs. [8,15], and the statement that the construction applies 'in any TRG method' is supported only by references. Because the removal of initial-tensor dependence is a central contribution, the paper should either explain the mechanism by which the squeezers guarantee independence for arbitrary TRG algorithms or explicitly defer the full claim to Ref. [8] with a precise statement of what the present paper demonstrates.","section":"§5, squeezer construction"},{"comment":"The notation after Algorithm 1 is confusing: the text says the new tensor K'(x) depends only on 'a_x and a_{x+|v|}', but the preceding definition and the two-dimensional examples use indices a_v and a_{v+μ}. This should be corrected to a_v and a_{v+μ}, or the convention should be defined precisely, so that the reader can follow the index bookkeeping in Eqs. (7) and (10).","section":"§3, after Algorithm 1"}],"minor_comments":[{"comment":"The delta-function structure in Eq. (10) is not obvious from the short description of the three index shifts; a brief derivation or a labelled intermediate step would help the reader verify the mapping of the a, b, and c indices.","section":"Eq. (10)"},{"comment":"The lower two b-HOTRG curves appear essentially coincident in the printed figure. A short data table or a zoomed inset would make the claimed 'same accuracy' quantitative and would show the D-dependence more clearly.","section":"§5, Fig. 5"},{"comment":"The sentence 'All TRG methods introduce a truncation, which minimizes a cost function' is too broad; some contraction schemes are exact for their respective tensor-network classes. Suggest softening to 'The TRG methods considered here'.","section":"§5, first paragraph"},{"comment":"The conclusion states that the method applies to general observables and inhomogeneous systems, but this is asserted rather than demonstrated. A one-sentence description of how impurity tensors or spatially varying Boltzmann factors enter Algorithm 1 would make the scope claim substantiated.","section":"§6"}],"recommendation":"major_revision","confidential_remarks":"The exact index-shift construction is sound and the numerical test is consistent with it. The requested revisions are modest: either add one or two further numerical checks or qualify the universal 'removes initial-tensor dependence' claim. Given that this is a proceedings contribution, I would not require a full multi-model study, but the current wording overstates the evidence in the manuscript."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the delta-shift construction in Algorithm 1 is exact and as general as advertised. It's an index relabeling, and the paper explains it cleanly. The soft spot is Section 5: the claim that squeezers remove initial-tensor dependence is backed by exactly one 2D Ising test, and the squeezer definition is imported from [8] and [15], not derived here.\n\nWhat's actually new: the Steiner tree framing in Section 4 is the most original bit. Pointing out that the number of arrows in the tree sets the bond dimension, and that minimizing arrows is NP-complete, is a genuinely useful way to think about constructing initial tensors for nonlocal models. The worked examples in Section 3 for a nonlocal interaction and a 3D Boltzmann factor are clear and make the algorithm concrete.\n\nThe paper is also honest that it's built on the authors' PRD [8]; the proceedings doesn't pretend the full derivation lives here. The Ising benchmark against the exact Onsager result is a legitimate external check. No fitting happened; the comparison is exactly what it claims.\n\nWhere it's loose: the opening claim of Section 5—that replacing isometries with squeezers 'removes the dependence on the form of the initial tensors'—is too broad. The evidence is one plot at one volume, one temperature, one model. No error bars. The mechanism for why boundary-HOTRG should generically do this is described, but the proof is left to references. A sharper paper would either add a second model (say, the nonlocal example from Section 3) or soften the statement to 'in the tested case.' Similarly, the conclusion that Steiner tree solutions let you 'estimate the numerical demands in advance' is reasonable but not demonstrated; bond dimension is not the same as truncation error.\n\nNone of this undermines the central construction. If you work with TRG for models where the standard expansion tricks fail, this recipe is useful. The self-citation concern is mild here because the construction is exact and the benchmark is external. I'd send it to a referee, but I'd ask that referee to check whether the general claim on initial-tensor dependence holds beyond the Ising test.","headline":"An exact and clean delta-shift recipe for TRG initial tensors, with an overreaching claim about removing initial-tensor dependence on thin evidence.","tokens_in":14402,"tokens_out":3084,"would_cite":false,"duration_ms":29331,"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":"This paper claims that any partition function written as a product of Boltzmann factors can be converted exactly—without expansions or decompositions—into a locally connected tensor network for the tensor renormalization group, and that a…","keywords":["tensor renormalization group","initial tensor construction","Boltzmann factor","locally connected tensor network","Steiner tree problem","boundary tensor renormalization group","squeezer","exact tensor network representation"],"falsifier":"Contract the tensor network produced by Algorithm 1 for a small lattice, such as a $4\\times4$ Ising model with periodic boundary conditions, and compare with the original Boltzmann sum; exact equality is required by construction, so any discrepancy falsifies the method. For the squeezer claim, run boundary-HOTRG with the delta-initial tensors on a three-dimensional Ising or $\\mathbb{Z}_2$ gauge model and check whether convergence in bond dimension matches the expansion-based initial tensors, which the paper does not test.","tokens_in":13443,"feed_emoji":"🧮","tokens_out":6561,"duration_ms":66737,"temperature":0.7,"pith_summary":"Tensor renormalization group (TRG) methods compute partition functions without Monte Carlo sampling, making them attractive for sign-problem systems, but they require the partition function to be a locally connected tensor network. This paper claims a fully general way to get there: starting from any partition function written as a product of Boltzmann factors, insert delta-function identities to create new indices and shift those deltas along a directed tree connecting the sites, until every index appears on exactly two neighboring tensors. The step uses no Taylor expansion, character expansion, or any special property of the Boltzmann weight, so it is exact and model-agnostic. The paper also demonstrates that truncated TRG algorithms such as higher-order TRG can be sensitive to which initial tensor representation was chosen, and that replacing one-sided isometries with two-sided 'squeezers' from the boundary TRG construction removes this dependence in the 2D Ising benchmark. If these claims hold, the practical entry barrier for applying TRG to a new model drops substantially.","feed_headline":"Every Boltzmann partition function gets an exact tensor network","feed_subtitle":"TRG starts from any Boltzmann weight product, and truncation accuracy stops depending on the initial tensor.","key_machinery":"The central object is Algorithm 1: given a Boltzmann factor $K$ depending on several site variables, insert a delta function $\\delta_{\\sigma,a}$ to duplicate one of its indices, then shift that delta by one lattice spacing and fuse it into a neighboring tensor, repeating along a directed tree that connects all sites of $K$ to a chosen origin. Each arrow of the tree converts a shared index, which would appear in more than two tensors, into two new indices that each appear on exactly two neighboring tensors, producing a locally connected tensor network. The tree-choice optimization is the rectilinear Steiner tree problem, and the number of arrows determines the bond dimension of the initial tensor. For the accuracy claim, the load-bearing object is the 'squeezer' $P$, constructed from both left and right SVD isometries and singular values, which replaces the single-sided isometry used in HOTRG truncation.","core_discovery":"The central claim is that the initial tensor construction does not assume any specific properties of the Boltzmann weights. Given a Boltzmann factor $K_{\\sigma_{\\hat r}\\ldots}$ depending on several site variables, the algorithm inserts a delta function $\\delta_{\\sigma,a}$ that duplicates one index, shifts that delta to a neighboring site, and fuses it into the tensor there; repeating this along a directed tree that connects all sites of $K$ to a chosen origin yields a new tensor whose indices are each shared by exactly two neighboring tensors. This is an exact rewriting of the partition function, so no information is lost before truncation. The choice of tree is the rectilinear Steiner tree problem, and the number of arrows sets the final tensor's bond dimension. The paper further claims that the accuracy of TRG truncations can depend strongly on the initial tensor's symmetry: HOTRG with the delta-constructed asymmetric tensor is less accurate than with the Taylor-constructed symmetric one at fixed bond dimension, but when the coarse-graining step uses squeezers built from both left and right singular-value-decomposition isometries, the two initial tensors give the same accuracy. Replacing isometries by squeezers is stated to remove the dependence on the form of the initial tensors.","pith_inferences":["Editorial extension: because the construction is exact, the only approximation in a TRG run is the truncation, so the delta-initial tensors could serve as a representation-independent benchmark for comparing different TRG algorithms on the same model.","Editorial extension: the paper leaves the Steiner tree step to be solved by hand for small interaction graphs; automating a minimal-tree search for larger Boltzmann weights would be a direct practical follow-up.","Editorial extension: the squeezer claim is demonstrated on one 2D Ising benchmark; applying boundary-type squeezers to non-symmetric initial tensors in three-dimensional or gauge models would test whether the 'removes dependence' statement holds beyond the tested case."],"forward_implications":["Any partition function written as a product of Boltzmann factors, including non-local interactions such as those in eq. (9), can be converted exactly into a TRG-ready tensor network without Taylor, character, or orthogonal-function expansions.","The size of the initial tensor is controlled by the number of arrows in the Steiner tree, so the numerical cost of a TRG calculation can be estimated before coarse-graining begins.","The same construction applies to inhomogeneous systems and to observables with impurity tensors, not just translation-invariant partition functions.","Within the tested 2D Ising benchmark, boundary-type squeezers make HOTRG's free-energy error independent of whether the initial tensor came from the delta construction or from a Taylor expansion, removing a source of algorithm bias.","For lattice gauge theories, the paper reports that the construction already works for the $\\mathbb{Z}_2$ gauge model in a companion work, keeping open the path to sign-problem-free gauge calculations."],"supporting_citations":[{"why":"Introduces the tensor renormalization group that the constructed networks are meant to feed.","marker":"[3]"},{"why":"Companion paper with the full derivation of the construction and the squeezer-based improvement, cited for details.","marker":"[8]"},{"why":"Defines higher-order TRG (HOTRG), the algorithm used for the free-energy benchmarks.","marker":"[12]"},{"why":"Introduces the boundary TRG idea from which the squeezer improvement is taken.","marker":"[14]"},{"why":"Defines the anisotropic TRG squeezers used to remove initial-tensor dependence.","marker":"[15]"},{"why":"Provides the exact Onsager/Kaufman free energy used as the benchmark reference.","marker":"[16]"},{"why":"Background reference for the rectilinear Steiner tree problem that governs the arrow-choice optimization.","marker":"[13]"}],"fun_headline_variants":["Initial tensor recipe makes TRG accuracy independent of setup","Exact tensor networks from arbitrary Boltzmann weights","Squeezers fix the initial tensor bias in TRG","Build initial tensors for TRG without assumptions","Taming initial tensor choice in tensor renormalization"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The claim that squeezers remove the initial-tensor dependence for arbitrary TRG methods is transferred from one 2D Ising benchmark and from a construction summarized from earlier work, so the general improvement rests on that transfer holding beyond the tested case.","fun_headline_variants_meta":{"raw":{"variants":["Initial tensor recipe makes TRG accuracy independent of setup","Exact tensor networks from arbitrary Boltzmann weights","Squeezers fix the initial tensor bias in TRG","Build initial tensors for TRG without assumptions","Taming initial tensor choice in tensor renormalization"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000171,"raw_usage":{"total_tokens":1242,"prompt_tokens":886,"completion_tokens":356,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":502,"completion_tokens_details":{"reasoning_tokens":282}},"tokens_in":502,"tokens_out":356,"duration_ms":4281,"temperature":1.0,"reasoning_tokens":282,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T17:50:24.779450+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Contract the tensor network produced by Algorithm 1 for a small lattice, such as a $4\\times4$ Ising model with periodic boundary conditions, and compare with the original Boltzmann sum; exact equality is required by construction, so any discrepancy falsifies the method. For the squeezer claim, run boundary-HOTRG with the delta-initial tensors on a three-dimensional Ising or $\\mathbb{Z}_2$ gauge model and check whether convergence in bond dimension matches the expansion-based initial tensors, which the paper does not test.","supporting_citations":[{"cited_title":"Nakayama and M","cited_arxiv_id":null,"evidence_quote":"Companion paper with the full derivation of the construction and the squeezer-based improvement, cited for details."},{"cited_title":"Kaufman,Crystal statistics","cited_arxiv_id":null,"evidence_quote":"Provides the exact Onsager/Kaufman free energy used as the benchmark reference."}],"review_version":1}