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REVIEW 3 major objections 6 minor 115 references

Finite-size scaling on the torus with periodic projected entangled-pair states

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A contraction step that grows a torus tensor network one site at a time turns periodic PEPS into a variational finite-size scaling tool.

desk verdict 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. read the letter →

arxiv 2411.12731 v2 pith:3RT3E6Z6 submitted 2024-11-19 cond-mat.str-el

Gleb Fedorovich , Lukas Devos , Jutho Haegeman , Laurens Vanderstraeten , Frank Verstraete , Atsushi Ueda This is my paper · ORCID
classification cond-mat.str-el
keywords projectedentangled-pairstatestensornetworkrenormalizationperiodicboundaryconditionsfinite-sizescalingautomaticdifferentiationtransverse-fieldIsingmodelquantumcriticalityCasimirenergy
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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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 $\Delta_\sigma = 0.508$.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

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.

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 (3)
  1. [Sec. IV, Tables I–II, Figs. 4, 6, 7; Sec. III; Appendix B] 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.
  2. [Sec. II.B versus Secs. IV–VI] 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.
  3. [Sec. V, Eq. (15), Fig. 6] 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.
minor comments (6)
  1. [Sec. II.B and Fig. 2] 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.
  2. [Appendix D] The word 'insert' in the caption of Fig. 8 should be 'inset'.
  3. [References] Reference [82] contains a typo: 'Journal of Marchine Learning Research' should be 'Journal of Machine Learning Research'.
  4. [Sec. III] 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.
  5. [Tables I–II and Fig. 7] 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.
  6. [Sec. IV] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: PTMRG benchmarks and finite-size scaling analyses are checked against external QMC, DMRG, conformal-bootstrap, and exact Ising data.

full rationale

Walked the derivation chain. PTMRG is an algorithmic construction whose O(χ^5) per-step cost is counted in Appendix C; the 'linear L to L+1' update rule is the literal recursive definition of the coarse-graining, not a quantity fitted from the target. The TFIM energy extrapolation uses Eq. (6), a standard CFT finite-size scaling form, and is compared with independent DMRG and infinite-PEPS results. The Δσ = 0.508 extraction uses Eq. (15), derived by first-order perturbation theory, and the exponent is fit to the authors' own mz data; it is then compared with conformal-bootstrap and fuzzy-sphere values, so the target is not an input. The classical Ising benchmark uses exact Tc and Δσ = 1/8 as external references. The Casimir constants use literature velocities and are compared with QMC values. Several citations are to co-authors' prior work (e.g., Ref. [72] for error diagnostics and Ref. [81] for iPEPS energies), but none is load-bearing: each is background or a comparison target, and the central claims do not reduce to those citations. Appendix B explicitly discloses uncontrolled truncation, discontinuities, and the need for a 'sweet spot' χ; this is an accuracy and convergence limitation, not a circularity. The note-added acknowledgment of similarity to Ref. [115] affects novelty, not circular derivation. No definitional circularity or fitted-input-called-prediction was found.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The method introduces no new physical entities. Its load-bearing choices are numerical: the truncation dimension χ, the PEPS bond dimension D, and the perturbation strength hz. The scaling-law fits add fitted coefficients that carry the quantitative conclusions. The axioms are standard tensor-network and CFT assumptions, with the uncontrolled PTMRG truncation being the least secure.

free parameters (4)
  • PTMRG truncation dimension χ = not stated for main benchmarks; χ=30 and 31 in Appendix B; χ=112 for Ising partition function in Fig. 2
    Controls the accuracy of every contraction. The paper notes a balance must be found between accuracy and cost, and Appendix B shows results depend strongly on this choice.
  • PEPS bond dimension D = D=2, 3, 4 depending on benchmark
    Variational ansatz size. D=2 is insufficient for XY and Heisenberg models, and the TFIM extrapolation changes from -3.2322 (D=2) to -3.2342 (D=3).
  • Longitudinal field hz = 10^-2 for Fig. 5 and 10^-3 for Fig. 6
    Chosen to break the Z2 symmetry in finite systems. No convergence study over hz is reported, and the scaling-dimension fit assumes the field is small enough to be a linear perturbation.
  • Finite-size fit coefficients (ε0, α, Δσ, αc) = ε0(L→∞) = -3.2322 (D=2) and -3.2342 (D=3) for TFIM; Δσ = 0.508; αc = 0.375, 0.813, 1.302
    These are regression outputs from the assumed scaling laws in Eqs. (6), (15), and (9). They are results rather than input assumptions, but the paper's quantitative claims rest on them.
assumptions (5)
  • domain assumption The ground states of the studied 2D critical spin models on a torus are well represented by a uniform PEPS with bond dimension D=3 or 4 for the observables considered.
    The whole variational optimization in Section III assumes this. D=2 is shown to fail for XY and Heisenberg models, so the method depends on D being large enough.
  • domain assumption HOSVD truncation in PTMRG yields a controlled approximation of the exact contraction for a sufficiently large χ.
    No error bound is given. Appendix B shows the approximation is uncontrolled at small χ and can introduce non-variational energy lowering.
  • domain assumption The finite-size scaling formulas ε0(L) ≈ ε0 + α/L^3 (Eq. 6) and mz ∝ L^{3-2Δσ} (Eq. 15) apply to these models at the chosen critical parameters.
    The extrapolations and scaling-dimension extraction are based on these formulas. The paper cites Ref. [91] and CFT arguments but does not verify them independently.
  • domain assumption The transfer matrix spectrum of the critical 2D classical Ising model is given by Λ_n = A exp(-2πΔ_n), with Δ_σ = 1/8.
    Used in Section II B to benchmark PTMRG errors. This is a standard CFT result, not proven in the paper.
  • domain assumption Automatic differentiation through the approximate PTMRG contraction gives gradients that are useful for optimization when χ is large enough.
    The algorithm relies on this. Appendix B demonstrates a failure mode at χ=30 for the Heisenberg model, so the assumption is not always satisfied.

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Pith. "Pith review of Finite-size scaling on the torus with periodic projected entangled-pair states." pith.science (2026). https://pith.science/paper/3RT3E6Z6

@misc{pith2026241112731,
  author       = {Pith},
  title        = {Pith review of: Finite-size scaling on the torus with periodic projected entangled-pair states},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3RT3E6Z6}},
  note         = {Machine review of arXiv:2411.12731}
}
abstract

An efficient algorithm is constructed for contracting two-dimensional tensor networks under periodic boundary conditions. The central ingredient is a novel renormalization step that scales linearly with system size, i.e. from $L \to L+1$. The numerical accuracy is comparable to state-of-the-art tensor network methods, while giving access to much more data points, and at a lower computational cost. Combining this contraction routine with the use of automatic differentiation, we arrive at an efficient algorithm for optimizing fully translation invariant projected entangled-pair states on the torus. Our benchmarks show that this method yields finite-size energy results that are comparable to those from quantum Monte Carlo simulations. When combined with field-theoretical scaling techniques, our approach enables accurate estimates of critical properties for two-dimensional quantum lattice systems.

Figures

Figures reproduced from arXiv: 2411.12731 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic illustrations of the coarse-graining procedure in PTMRG. The coarse-grained tensors consist of four types [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. (a) shows the computational time required to perform one update step for TRG, HOTRG, and PTMRG 1 . When using the same bond dimension χ, PTMRG is faster by one order of magnitude. Conse￾quently, more data points can be collected in the same computational time. In addition to its computational efficiency, PTMRG also demonstrates high accuracy. TRG/TNR algorithms are known to introduce numerical errors due to the bond… view at source ↗
Figure 3
Figure 3. FIG. 3. Schematic picture of the periodic PEPS algorithm. [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: shows that ϵ0 indeed scales as L −3 following the CFT argument presented above. A linear fit provides the following ground state energies in the thermodynamic limit: ϵ0(L → ∞) = ( −3.2322, D = 2, −3.2342, D = 3, (8) agreeing with infinite PEPS studies presented in [81,…
Figure 6
Figure 6. Figure 6: FIG. 6. Magnetization per site [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: shows the optimized ground state energy as a function of the system size L both for the (a) XY and (b) AF Heisenberg models. As expected from 3-D CFT, the energy density scales as L −3 and approaches the QMC results with increasing PEPS bond dimension D. The Casimir co…
Figure 8
Figure 8. Figure 8: FIG. 8. Ground state energy [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 10
Figure 10. Figure 10: FIG. 10. An example of a single PTMRG step that updates [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Ground state energy [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]

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