{"id":"47622a2e-c6ab-48b6-8e88-5f288dac5037","arxiv_id":"2411.12731","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A linear-growth tensor renormalization contraction, PTMRG, is combined with automatic differentiation to optimize periodic PEPS and extract finite-size scaling data for 2D quantum critical models.","lead":"This paper introduces PTMRG, a tensor network renormalization scheme that grows a periodic 2D lattice one row or column at a time, and combines it with automatic differentiation to optimize projected entangled-pair states on a torus. The method produces finite-size ground state energies for 2D spin models close to quantum Monte Carlo results, and is aimed at extracting critical properties of 2D quantum lattice systems.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The PTMRG truncation error is uncontrolled for the PEPS benchmarks; Appendix B shows χ-sensitivity and non-variational energies, but no χ values or convergence checks are reported, so the observed finite-size scaling may be contaminated by truncation.","rationale":"The reader's weakest assumption is the uncontrolled PTMRG truncation and the unverified sweet-spot χ. I agree that this is the most load-bearing weakness. The paper's abstract and conclusion claim an efficient and accurate FSS method; the evidence for accuracy is the comparison to QMC/DMRG and the extraction of Δσ. All of these are computed with the approximate PTMRG contraction, and Appendix B explicitly shows that the approximation can be discontinuous and non-variational at χ=30 for a benchmark model. Since the paper does not report the χ used in the main runs or any χ-convergence study, one cannot distinguish physical FSS from truncation-induced FSS. This is not an internal inconsistency; the method could be perfectly fine at the unpublished χ values. But the burden is on the authors to show convergence, especially because they themselves warn about a 'sweet spot'. The classical Ising test is reassuring for the contraction of a single tensor with χ=112, but the PEPS optimization is a more complex setting where AD propagates through the truncation and where the D^2=9 double-layer bond makes χ-sensitivity more severe. The QMC comparisons at fixed L provide a sanity check, but the scaling exponents and Casimir constants are extracted from the L-dependence, which is precisely what truncation could distort. My proposed test directly checks whether the reported energies are converged in χ at the relevant L; if they are, the central claim stands, and if not, the FSS results need revision. Therefore I do not change the reader's CONDITIONAL verdict.","tokens_in":19732,"tokens_out":4861,"duration_ms":48915,"concrete_test":"Re-optimize the TFIM PEPS at λ=λ_c with D=3 for L=8,9,10 using at least two PTMRG truncation dimensions, e.g., χ=40 and χ=50, with all other settings identical. If any energy in Table I shifts by more than ~1×10^-4 (the size of the D=3 vs D=4 difference), the L^-3 fit and the extrapolated ϵ0=-3.2342 are not converged in χ. A complementary check: for the AF Heisenberg model at D=3, L=8, sweep χ=30,31,40,50 and compare with the QMC value -0.6735; if the energy does not approach QMC monotonically or varies by more than 2×10^-3, the sweet-spot assumption fails for the most challenging benchmark.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that PTMRG combined with AD produces reliable finite-size scaling (FSS) data on the torus. This requires the PTMRG truncation error to be small and, in particular, to have an L-dependence that does not mimic or mask the physical scaling laws used in the fits (L^-3 for energies, L^{3-2Δσ} for magnetization). The paper provides no error bound for the truncation and no proof that the error is benign. Appendix B demonstrates the danger: for the AF Heisenberg model with D=3, χ=30 gives an approximate energy that is discontinuous in the variational parameters and can fall below the exact energy; χ=31 removes the discontinuities locally. The main text states that a 'sweet spot' χ must be found, but it does not report the χ values used for Tables I-II, Fig. 4, Fig. 6, or Table III, nor does it show convergence of any observable with χ. Consequently, the benchmark energies and extracted exponents could be shifted by truncation errors whose size and L-dependence are unknown. The L^-3 energy fits and the Δσ=0.508 extraction are exactly the quantities that would be biased if the truncation error varies with L; the QMC comparisons at fixed L do not rule this out because the scaling analysis, not the individual energies, carries the central claim. The classical Ising benchmark (Fig. 2) shows the contraction is accurate for a single local tensor at χ=112, but it does not address the double-layer PEPS setting with AD where the truncation acts on a D^2=9 index and where the non-variational energy can mislead the optimizer.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript introduces PTMRG, an approximate contraction scheme for two-dimensional tensor networks on a torus whose renormalization step grows the system size linearly (L to L+1) with an O(χ^5) cost per step. The authors combine PTMRG with automatic differentiation to optimize translation-invariant periodic PEPS, and benchmark the resulting finite-size ground-state energies for the transverse-field Ising, XY, and antiferromagnetic Heisenberg models, also extracting a scaling dimension Δσ from magnetization data. The classical Ising transfer-matrix benchmark shows that PTMRG maintains accurate CFT scaling up to larger L than TRG/HOTRG at comparable cost. The PEPS results approach QMC and DMRG values, but the reported PEPS simulations do not specify the crucial PTMRG truncation dimension χ or provide convergence checks, leaving the central finite-size-scaling claims incompletely supported.","tokens_in":20081,"tokens_out":4109,"duration_ms":45911,"significance":"If the performance claims withstand scrutiny, the paper describes a practical tool for finite-size scaling of two-dimensional quantum lattice models on the torus: linearly spaced system sizes and lower asymptotic cost than TRG/HOTRG are attractive, and the fully translation-invariant periodic PEPS parametrization is a natural fit for the geometry. Several strengths are explicit: the O(χ^5) cost analysis in Appendix C is concrete, the classical Ising benchmark provides a machine-checkable accuracy check with many data points, the PEPS energies are compared against independent QMC and DMRG results, and Appendix B honestly documents a known truncation-induced pathology. The principal weakness is that the PEPS benchmarks omit the values of χ used and contain no systematic χ-convergence study, so the L-dependent behavior that underlies the scaling fits could be contaminated by truncation errors whose size and L-dependence are unknown. This is a fixable reporting and validation gap rather than an obviously wrong central idea.","major_comments":[{"comment":"The PEPS benchmarks do not report the PTMRG truncation dimension χ used for any model, bond dimension D, or system size L. The central claim is finite-size scaling, i.e., that the L-dependence of the approximate energies follows the physical L^-3 or L^{3-2Δσ} forms. Appendix B shows that for the Heisenberg model with D=3, χ=30 gives discontinuous, non-variational energies and that χ=31 only removes the discontinuities locally, and the main text (Sec. III) states that a 'sweet spot' χ must be found. Without specifying the χ values and demonstrating convergence of, for example, ϵ0(L) with increasing χ at fixed L, the observed scaling in Figs. 4 and 7 and the extraction Δσ=0.508 in Fig. 6 may be biased by an L-dependent truncation error. This is the load-bearing issue for the paper's main claim, so a table of χ per model/D/L and a convergence check are required.","section":"Sec. IV, Tables I–II, Figs. 4, 6, 7; Sec. III; Appendix B"},{"comment":"The accuracy benchmark on the classical Ising model (Fig. 2) validates PTMRG for a single-layer local tensor with small local bond dimension, but it does not directly validate the double-layer PEPS contraction under automatic differentiation. For the PEPS benchmarks the contracted tensor has bond dimension D^2 (9 for D=3), and Appendix B demonstrates failure of gradient optimization precisely in that double-layer setting at insufficient χ. Consequently, the classical Ising accuracy result cannot by itself support the periodic PEPS finite-size-scaling claims; a PEPS-specific accuracy check is needed, for example comparing PTMRG energies against direct contraction for small L or against an independent boundary-MPS contraction for the same PEPS tensor.","section":"Sec. II.B versus Secs. IV–VI"},{"comment":"The scaling dimension is reported as Δσ=0.508 without an uncertainty or a systematic-error estimate. The fit uses magnetization data at a single longitudinal field hz=10^-3 and for D=2,3, but the paper does not verify that hz is small enough for the linear-order perturbation theory leading to Eq. (15) to hold, nor that irrelevant-operator corrections are negligible over the fitted L range. Since the literature value is Δσ=0.518148806(24), the reported agreement is only qualitative without a quantified error bar; a short study of the fitted exponent versus hz and versus the L-window would make the claim testable.","section":"Sec. V, Eq. (15), Fig. 6"}],"minor_comments":[{"comment":"The text says 'Figure 2(a) shows the computational time required to perform one update step', but the caption of Fig. 2(a) describes the finite-size effect δΔσ(L), while Fig. 2(b) shows the computational time; the section and caption should be brought into agreement.","section":"Sec. II.B and Fig. 2"},{"comment":"The word 'insert' in the caption of Fig. 8 should be 'inset'.","section":"Appendix D"},{"comment":"Reference [82] contains a typo: 'Journal of Marchine Learning Research' should be 'Journal of Machine Learning Research'.","section":"References"},{"comment":"The notion of a 'sweet spot' for χ is informal; since the paper does not define a criterion for choosing χ or for detecting when the energy landscape becomes smooth, a more precise guiding rule would improve reproducibility.","section":"Sec. III"},{"comment":"The QMC comparison would be more informative if the QMC statistical errors were shown on the finite-L values, since the differences between PEPS and QMC at the largest L are of the same order as typical QMC error bars.","section":"Tables I–II and Fig. 7"},{"comment":"Table I reports energies for D=4 up to L=10, and Fig. 4 includes D=4, but Eq. (8) gives only the D=2 and D=3 thermodynamic extrapolations; reporting the D=4 extrapolation would clarify how 'nearly identical' the D=4 results are at the level of the fit.","section":"Sec. IV"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is well within the scope of the journal and the central algorithmic idea is promising. The main revision requirement is to supply the missing χ values and convergence checks for the PEPS benchmarks; this is a feasibility-of-reproduction issue, not a fundamental flaw. The note added about Ref. [115] is appropriate, and the relationship to that work should be clarified in the revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core claim is real: PTMRG contracts periodic 2D tensor networks with linear system-size growth at O(χ^5), and the classical Ising benchmark supports that. The paper also does something genuinely useful by combining this contraction with AD-based optimization of a single uniform torus PEPS, and the benchmarks against QMC, DMRG, and exact Ising data are encouraging. The scaling-dimension extraction (Δσ≈0.508) and the Casimir constants are reasonable, and the O(χ^5) cost analysis in Appendix C is clear.\n\nWhere the paper is soft: the PEPS results never report the truncation dimension χ used for Tables I-II, Figs. 4/6/7, or the scaling-dimension fit, and there is no systematic convergence check in χ. Given that the PTMRG truncation is uncontrolled and Appendix B explicitly shows χ=30 giving discontinuous, non-variational energies for the Heisenberg model, this is a real hole. The L^-3 energy fits and the m_z~L^{3-2Δ} extraction are exactly the quantities that would be biased if truncation error varies with L. The classical Ising benchmark at χ=112 shows the contraction works for a single tensor, but it doesn't address the double-layer PEPS setting where D^2=9 and the optimizer can exploit artifacts. Also, the note added concedes that Ref. 115 already introduced a corner-plus-edge linear coarse-graining scheme with O(χ^4) complexity, so the contraction novelty is mostly in the factorized update and the PEPS-AD application, not the overall idea.\n\nI don't think the paper is fatally flawed. The central method is plausible, the external benchmarks are appropriate, and the authors are honest about the pathology in Appendix B. But the missing χ values and convergence checks are not cosmetic; without them, the finite-size scaling results are not reproducible and the central claim is under-supported. No code or data are provided either, which makes this worse.\n\nIf you work on tensor network methods for 2D quantum systems, this is worth reading and citing, but with caution about the PEPS scaling results until the χ-dependence is reported. I would send it to peer review rather than desk reject, with a request for a convergence table and explicit χ values for every benchmark. The paper is a solid contribution, just not as clean as the abstract suggests.","headline":"Useful method paper, but the novelty is partly undercut by Lan-Evenbly and the PEPS benchmarks omit the χ values needed to trust the finite-size scaling.","tokens_in":20647,"tokens_out":1264,"would_cite":true,"duration_ms":15365,"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":"A contraction step that grows a torus tensor network one site at a time turns periodic PEPS into a variational finite-size scaling tool.","keywords":["projected entangled-pair states","tensor network renormalization","periodic boundary conditions","finite-size scaling","automatic differentiation","transverse-field Ising model","quantum criticality","Casimir energy"],"falsifier":"Run the PTMRG optimization of the antiferromagnetic Heisenberg model with $D=3$ on, say, $L=8$ at $\\chi=30$ and $\\chi=31$, and examine the energy along a gradient direction for discontinuities; if discontinuities persist for all affordable $\\chi$, gradient optimization is not reliable. More directly, reproduce the TFIM critical-point energies of Table I with the $\\chi$ values used in the paper; if any reported energy lies below the exact ground-state energy or shifts non-monotonically when $\\chi$ increases, the approximate energy is not variational and the scaling fit is not trustworthy.","tokens_in":19538,"feed_emoji":"🔄","tokens_out":9378,"duration_ms":85240,"temperature":0.7,"pith_summary":"This paper builds an efficient way to contract two-dimensional tensor networks wrapped on a torus, and uses it to variationally optimize projected entangled-pair states (PEPS) with periodic boundary conditions. The new contraction step grows the system from linear size $L$ to $L+1$ rather than doubling it, so a single sweep collects data at every system size and costs $O(\\chi^5)$ instead of the $O(\\chi^6)$-$O(\\chi^7)$ of standard tensor renormalization schemes. With automatic differentiation supplying gradients, a single translation-invariant PEPS tensor is optimized directly on the torus. The resulting finite-size ground-state energies for the transverse-field Ising, XY, and Heisenberg models agree with quantum Monte Carlo data, and finite-size scaling yields Casimir constants and a scaling dimension $\\Delta_\\sigma = 0.508$ consistent with established values. If correct, this gives the tensor-network community a practical finite-size scaling tool for two-dimensional quantum criticality under periodic boundary conditions.","feed_headline":"PTMRG contracts torus tensor networks in linear system-size steps","feed_subtitle":"One local tensor, grown L to L+1, gives finite-size energies that match Monte Carlo.","key_machinery":"The central object is the set of coarse-grained tensors in PTMRG: a corner tensor $C^{n,m}$ representing an $n \\times m$ patch, and edge tensors $h^n$, $v^n$ representing $n \\times 1$ and $1 \\times m$ chains of the original four-leg tensor $a$. The mechanism is a single horizontal or vertical sweep in which an HOSVD of $C^{n-1,n-1}$ and $h^{n-1}$ yields isometries that absorb the edge tensor into the corner, growing the patch from $(n-1) \\times (n-1)$ to $n \\times n$. This linear growth step is what reduces the asymptotic cost to $O(\\chi^5)$, and when the contraction is differentiated automatically it carries gradients from the energy back to the local PEPS tensor.","core_discovery":"The paper introduces the periodic transfer matrix renormalization group (PTMRG), a contraction scheme for homogeneous $L \\times L$ tensor networks with periodic boundary conditions. Instead of coarse-graining by powers of two, PTMRG keeps a corner tensor $C^{n,m}$ plus horizontal and vertical edge tensors $h^n, v^n$, and applies higher-order-singular-value-decomposition isometries to add one row or column at a time, so the final $C^{L,L}$ is obtained after $L$ horizontal and $L$ vertical sweeps. The paper shows that a single PTMRG step costs $O(\\chi^5)$, that the transfer-matrix spectrum of the critical 2D Ising model follows the conformal prediction with roughly fifty usable data points up to $L \\approx 100$, and that feeding PTMRG contraction through automatic differentiation produces stable quasi-Newton optimization of a single periodic PEPS tensor. Benchmarks on the torus yield energies matching quantum Monte Carlo and scaling behavior $\\epsilon_0(L) \\approx \\epsilon_0 + \\alpha/L^3$ at criticality; from the magnetization in a small longitudinal field the paper extracts $m_z \\propto L^{3-2\\Delta_\\sigma}$ with $\\Delta_\\sigma = 0.508$.","pith_inferences":["Editorial inference: the same $L \\to L+1$ contraction should carry over to twisted boundary conditions through tube algebras, since only the wrapping of the corner tensor legs changes; a cheap check would be comparing PTMRG energies to exact diagonalization on small twisted tori.","Editorial inference: if an adaptive cutoff that detects level crossings, for example through gradient-norm spikes, can be implemented, the method becomes applicable to sign-problematic or frustrated models, where QMC benchmarks are absent and the sweet-spot issue is the main obstacle.","Editorial inference: the finite-size data sets of fifty or more $L$ values could be used to test higher-order corrections to the $\\alpha/L^3$ scaling, potentially extracting subleading Casimir amplitudes that the paper does not analyze.","Editorial inference: the roughly two-percent gap between $\\Delta_\\sigma = 0.508$ and the conformal bootstrap value $0.518148806(24)$ offers a quantitative target; repeating the extraction at $D=4$ and increasing $\\chi$ would show whether the error is controlled by bond dimension or by truncation."],"forward_implications":["With PTMRG, one can evaluate a periodic $L \\times L$ tensor network at every integer $L$, giving $O(L)$ data points for finite-size scaling instead of the logarithmic spacing of TRG and HOTRG.","For the 2D transverse-field Ising model at $\\lambda_c$, optimized PEPS energies follow $\\epsilon_0(L) \\approx \\epsilon_0 + \\alpha/L^3$ and extrapolate to thermodynamic values ($-3.2322$ for $D=2$, $-3.2342$ for $D=3$) matching infinite-PEPS studies.","A longitudinal-field perturbation yields magnetization $m_z \\propto L^{3-2\\Delta_\\sigma}$, from which the paper extracts $\\Delta_\\sigma = 0.508$, within a few percent of conformal bootstrap and fuzzy-sphere values.","For the XY and antiferromagnetic Heisenberg models, $D=3$ and $D=4$ PTMRG energies approach QMC values as $L$ grows, and the extracted Casimir constants $\\alpha_c \\approx 0.813$ (XY) and $1.302$ (Heisenberg) agree with QMC estimates.","The PTMRG contraction reproduces the universal transfer-matrix spectrum of the critical 2D Ising model with roughly fifty reliable data points up to $L \\approx 100$, at equal cost per step where TRG and HOTRG yield only five to eight."],"supporting_citations":[{"why":"Supplies the tensor renormalization group whose $O(\\chi^6)$ cost and logarithmic system-size growth PTMRG improves upon.","marker":"[21]"},{"why":"The higher-order tensor renormalization group baseline whose $O(\\chi^7)$ scaling is compared with PTMRG's $O(\\chi^5)$ step.","marker":"[24]"},{"why":"Corner transfer matrix renormalization ideas that PTMRG combines with TRG for periodic edge and corner tensors.","marker":"[54]"},{"why":"The higher-order singular value decomposition used to build the isometries in each PTMRG sweep.","marker":"[69]"},{"why":"Gradient-based optimization framework for infinite PEPS that the new periodic PEPS optimizer adapts to finite tori.","marker":"[81]"},{"why":"Automatic differentiation through tensor network contractions, the mechanism used to obtain gradients for PEPS optimization.","marker":"[83]"},{"why":"Finite-size scaling relation $\\epsilon_0(L) \\approx \\epsilon_0 + \\alpha/L^3$ used to extrapolate torus ground-state energies.","marker":"[91]"},{"why":"Conformal bootstrap value $\\Delta_\\sigma = 0.518148806(24)$ against which the extracted scaling dimension is checked.","marker":"[100]"},{"why":"Quantum Monte Carlo ground-state energies and Casimir constants for the XY model used as benchmark.","marker":"[102]"},{"why":"Quantum Monte Carlo ground-state energies and Casimir constants for the antiferromagnetic Heisenberg model used as benchmark.","marker":"[103]"}],"fun_headline_variants":["PTMRG: torus tensor networks scale linearly with L","Linear-time torus tensor contraction for PEPS","Torus PEPS: add a row or column, match Monte Carlo","PTMRG: finite-size energies that match QMC on torus","PTMRG for torus PEPS: linear L, QMC-accurate"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method assumes that for each system size there is a cutoff $\\chi$ for the singular-value truncation that is small enough to be affordable but large enough that the approximate energy is smooth and accurate; Appendix B shows this assumption fails at $\\chi=30$ for the Heisenberg model, and the paper does not report the cutoff values used for its headline results.","fun_headline_variants_meta":{"raw":{"variants":["PTMRG: torus tensor networks scale linearly with L","Linear-time torus tensor contraction for PEPS","Torus PEPS: add a row or column, match Monte Carlo","PTMRG: finite-size energies that match QMC on torus","PTMRG for torus PEPS: linear L, QMC-accurate"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001505,"raw_usage":{"total_tokens":6029,"prompt_tokens":935,"completion_tokens":5094,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":551,"completion_tokens_details":{"reasoning_tokens":5002}},"tokens_in":551,"tokens_out":5094,"duration_ms":36591,"temperature":1.0,"reasoning_tokens":5002,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T17:12:20.866253+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the PTMRG optimization of the antiferromagnetic Heisenberg model with $D=3$ on, say, $L=8$ at $\\chi=30$ and $\\chi=31$, and examine the energy along a gradient direction for discontinuities; if discontinuities persist for all affordable $\\chi$, gradient optimization is not reliable. More directly, reproduce the TFIM critical-point energies of Table I with the $\\chi$ values used in the paper; if any reported energy lies below the exact ground-state energy or shifts non-monotonically when $\\chi$ increases, the approximate energy is not variational and the scaling fit is not trustworthy.","supporting_citations":[{"cited_title":"Nishino and K","cited_arxiv_id":null,"evidence_quote":"Corner transfer matrix renormalization ideas that PTMRG combines with TRG for periodic edge and corner tensors."},{"cited_title":"De Lathauwer, B","cited_arxiv_id":null,"evidence_quote":"The higher-order singular value decomposition used to build the isometries in each PTMRG sweep."},{"cited_title":"Vanderstraeten, J","cited_arxiv_id":null,"evidence_quote":"Gradient-based optimization framework for infinite PEPS that the new periodic PEPS optimizer adapts to finite tori."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Finite-size scaling relation $\\epsilon_0(L) \\approx \\epsilon_0 + \\alpha/L^3$ used to extrapolate torus ground-state energies."},{"cited_title":"Henkel and A","cited_arxiv_id":null,"evidence_quote":"Conformal bootstrap value $\\Delta_\\sigma = 0.518148806(24)$ against which the extracted scaling dimension is checked."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Quantum Monte Carlo ground-state energies and Casimir constants for the antiferromagnetic Heisenberg model used as benchmark."}],"review_version":1}