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arxiv: 2604.15749 · v1 · submitted 2026-04-17 · ❄️ cond-mat.stat-mech

Recognition: unknown

Extracting conformal data from finite-size tensor-network flow in critical two-dimensional classical models

Sing-Hong Chan , Pochung Chen
Authors on Pith no claims yet

Pith reviewed 2026-05-10 08:09 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords conformal datatensor-network renormalizationfinite-size scalingfinite-entanglement scalingcentral chargecritical two-dimensional modelsIsing modelclock model
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The pith

A self-consistent finite-size window below a crossover scale in tensor-network flows extracts central charge, scaling dimensions, and conformal spins from critical two-dimensional lattice models without a unique fixed-point tensor or priorC

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

The paper develops a framework that locates, from transfer-matrix spectra, a finite-size window and an accompanying crossover scale separating finite-size scaling from finite-entanglement scaling caused by bond truncation. Inside this window the central charge, scaling dimensions, and conformal spins become extractable for the Ising and three-state clock models. The estimates remain consistent across three distinct tensor-network renormalization schemes and do not require reaching a unique critical fixed-point tensor or knowing the underlying conformal field theory in advance. The same spectra also supply an operational definition of entanglement scaling and a second route to the central charge.

Core claim

By tracking transfer-matrix spectra under successive tensor-network renormalizations, one identifies a self-consistent finite-size window bounded above by a crossover scale that marks the onset of finite-entanglement effects; within this window the central charge, scaling dimensions, and conformal spins can be read off directly, and the procedure remains insensitive to the particular renormalization scheme employed.

What carries the argument

The self-consistent finite-size window together with the crossover scale that separates the finite-size-scaling regime from the finite-entanglement-scaling regime in the transfer-matrix spectrum.

If this is right

  • Conformal data up to relatively high levels can be obtained for the critical Ising and three-state clock models.
  • Universal behavior below the crossover scale is reproducible across multiple tensor-network renormalization schemes.
  • The spectra furnish both an operational definition of entanglement scaling and a complementary estimator of the central charge.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The window-based separation of scaling regimes offers a systematic way to test conformal invariance in other two-dimensional lattice models whose critical theories are not known in advance.
  • Similar finite-size flow diagnostics could be examined in quantum tensor-network settings to isolate entanglement effects from universal scaling.

Load-bearing premise

A robust crossover scale between finite-size and finite-entanglement regimes always exists and the universal behavior below it does not depend on the choice of tensor-network renormalization scheme.

What would settle it

If the extracted central charge or scaling dimensions shift by more than a few percent when the same model is renormalized with a different scheme, or if no clear crossover scale can be identified from the spectra, the method fails to deliver scheme-independent conformal data.

Figures

Figures reproduced from arXiv: 2604.15749 by Pochung Chen, Sing-Hong Chan.

Figure 1
Figure 1. Figure 1: FIG. 1. Conformal tower of the Ising model. The plot is [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. HOTRG results for the Ising model for [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. HOTRG results of Ising model for [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. HOTRG results for the Ising model for self-consistent spin-resolved subgroups [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. First row [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. First row [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. First row [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. HOTRG results of 3-state clock model. First row: [PITH_FULL_IMAGE:figures/full_fig_p010_12.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. HOTRG results for the 3-state clock model for self [PITH_FULL_IMAGE:figures/full_fig_p010_11.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. HOTRG results of 3-state clock model. First row: [PITH_FULL_IMAGE:figures/full_fig_p011_13.png] view at source ↗
Figure 17
Figure 17. Figure 17: Here the results are obtained with HOTRG at [PITH_FULL_IMAGE:figures/full_fig_p011_17.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Central-charge estimator [PITH_FULL_IMAGE:figures/full_fig_p012_14.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Plots of self-consistent spin-resolved subgroups [PITH_FULL_IMAGE:figures/full_fig_p013_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. Plots of self-consistent spin-resolved subgroups [PITH_FULL_IMAGE:figures/full_fig_p014_17.png] view at source ↗
read the original abstract

We present a general framework for extracting conformal data from critical two-dimensional classical lattice models using finite-size tensor-network flow. The central idea is to identify, from transfer-matrix spectra, a self-consistent finite-size window together with a crossover scale that separates the finite-size-scaling regime from the finite-entanglement-scaling regime induced by bond-dimension truncation. Within this window, the central charge, scaling dimensions, and conformal spins can be estimated without requiring a unique critical fixed-point tensor or detailed prior knowledge of the underlying conformal field theory. We benchmark the framework using three tensor-network renormalization schemes for the critical two-dimensional Ising and three-state clock models. Across schemes, we find robust universal behavior below the crossover scale, enabling accurate extraction of conformal data up to relatively high conformal levels. The analysis also yields a natural operational definition of entanglement scaling for classical tensor-network calculations and, in turn, a complementary estimator of the central charge.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a framework for extracting conformal data (central charge, scaling dimensions, and conformal spins) from finite-size tensor-network renormalization flows in critical 2D classical lattice models. The approach identifies a self-consistent finite-size window below a crossover scale separating finite-size-scaling from finite-entanglement-scaling regimes induced by bond-dimension truncation, enabling extraction without a unique critical fixed-point tensor or detailed prior CFT knowledge. Benchmarks are performed on the Ising and three-state clock models using three distinct tensor-network schemes, reporting robust universal behavior below the crossover and an operational definition of entanglement scaling for central-charge estimation.

Significance. If the central claim holds, the framework provides a practical, scheme-robust method for determining CFT data from tensor-network simulations of unknown critical points, which is valuable for statistical mechanics and condensed-matter applications. The multi-scheme benchmarks on two models and the complementary entanglement-scaling estimator for the central charge are strengths that support broader utility. The self-consistent window procedure, however, requires careful validation to confirm it does not introduce hidden dependence on the identification algorithm itself.

major comments (2)
  1. [§3] §3 (central procedure): The self-consistent definition of the finite-size window and crossover scale is extracted from the same transfer-matrix spectra used to obtain the conformal data. This introduces moderate circularity risk; the manuscript should provide the explicit algorithmic steps (thresholds, fitting routines, or exclusion rules) and a sensitivity test (e.g., variation of identification parameters or synthetic spectra) to demonstrate that the extracted values remain stable and independent of the window-selection details.
  2. [§4] §4 (benchmarks): While consistency across the three renormalization schemes is stated, the results lack quantitative error bars on extracted quantities, explicit data-exclusion criteria for points inside the window, and direct numerical comparisons to exact known values (Ising: c = 1/2, lowest Δ = 1/8; clock model: c = 4/5, etc.). These omissions make it difficult to assess the claimed accuracy up to high conformal levels and weaken the evidence that the window is robustly identifiable without prior CFT input.
minor comments (2)
  1. [Abstract] Abstract: The phrase 'accurate extraction' would be strengthened by including at least one quantitative metric (e.g., relative deviation from known values or typical error size) rather than remaining purely qualitative.
  2. [§4] Figures in §4: Plots of spectra or scaling flows should explicitly mark the identified crossover scale and the boundaries of the self-consistent window to improve readability and reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major point below and will revise the manuscript to incorporate the suggested clarifications and additions.

read point-by-point responses

  1. Referee: [§3] §3 (central procedure): The self-consistent definition of the finite-size window and crossover scale is extracted from the same transfer-matrix spectra used to obtain the conformal data. This introduces moderate circularity risk; the manuscript should provide the explicit algorithmic steps (thresholds, fitting routines, or exclusion rules) and a sensitivity test (e.g., variation of identification parameters or synthetic spectra) to demonstrate that the extracted values remain stable and independent of the window-selection details.

    Authors: We agree that explicit documentation of the window-identification procedure is necessary to address potential concerns about circularity. In the revised manuscript we will add a dedicated subsection to §3 that spells out the algorithmic steps in full: the precise criteria used to locate the crossover scale (deviation thresholds in the log-log scaling of extracted dimensions versus system size), the fitting routines employed to define the finite-size window boundaries, and the exclusion rules applied to individual data points. We will also include sensitivity analyses in which the identification parameters are systematically varied and the procedure is applied to synthetic spectra generated from known CFT data; these tests will be reported to demonstrate that the extracted conformal quantities remain stable within the quoted precision. revision: yes

  2. Referee: [§4] §4 (benchmarks): While consistency across the three renormalization schemes is stated, the results lack quantitative error bars on extracted quantities, explicit data-exclusion criteria for points inside the window, and direct numerical comparisons to exact known values (Ising: c = 1/2, lowest Δ = 1/8; clock model: c = 4/5, etc.). These omissions make it difficult to assess the claimed accuracy up to high conformal levels and weaken the evidence that the window is robustly identifiable without prior CFT input.

    Authors: We accept that the current presentation of the benchmarks is incomplete. In the revised §4 we will supply quantitative error bars derived from the linear fits performed inside each identified window, state the explicit data-exclusion criteria (e.g., points whose scaling deviates by more than a fixed percentage from the expected power-law behavior), and add direct numerical comparisons to the exact CFT values for both models. These comparisons will be presented in tables that list the extracted central charges, scaling dimensions, and spins together with the absolute and relative deviations from the known Ising (c = 1/2, Δ = 1/8, …) and three-state clock (c = 4/5, …) results, thereby allowing readers to evaluate the accuracy up to the higher conformal levels reported. revision: yes

Circularity Check

1 steps flagged

Self-consistent window from spectra introduces moderate dependence on identification procedure

specific steps
  1. self definitional [Abstract]
    "The central idea is to identify, from transfer-matrix spectra, a self-consistent finite-size window together with a crossover scale that separates the finite-size-scaling regime from the finite-entanglement-scaling regime induced by bond-dimension truncation. Within this window, the central charge, scaling dimensions, and conformal spins can be estimated without requiring a unique critical fixed-point tensor or detailed prior knowledge of the underlying conformal field theory."

    The window is defined self-consistently from the spectra, after which the conformal quantities are extracted from the same spectra restricted to that window. The selection criterion and the extracted values are therefore interdependent by construction rather than independently validated.

full rationale

The central framework identifies a self-consistent finite-size window directly from the transfer-matrix spectra and then extracts conformal data (central charge, dimensions, spins) from within that window. This creates a moderate self-referential loop in the analysis pipeline, as the window selection and data extraction draw from the same underlying spectra rather than fully external or independent benchmarks. Benchmarks on Ising and clock models with multiple schemes show consistency, but the procedure is not parameter-free or externally falsifiable in the window choice itself. No load-bearing self-citations, ansatz smuggling, or renaming of known results are present; the derivation remains largely self-contained outside this identification step.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard assumptions of conformal invariance and finite-size scaling in 2D critical systems plus the existence of distinct scaling regimes induced by bond-dimension truncation.

axioms (2)
  • domain assumption Two-dimensional critical lattice models are described by conformal field theories whose spectrum appears in the transfer-matrix eigenvalues.
    Invoked in the central idea of identifying scaling regimes from transfer-matrix spectra.
  • ad hoc to paper A crossover scale separating finite-size and finite-entanglement regimes can be located self-consistently without prior CFT knowledge.
    This is the key operational step introduced by the paper.

pith-pipeline@v0.9.0 · 5457 in / 1362 out tokens · 26687 ms · 2026-05-10T08:09:20.715169+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

156 extracted references · 15 canonical work pages

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    + (1,0), while theS=−1 branch is consistent with (2 5 , 7

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    We therefore identifyϕ 1 1 = ( 7 5 , 2 5),ϕ 2 1 = ( 2 5 , 2 5)+(1,0),ϕ 1 2 = ( 2 5 , 7 5), andϕ 2 2 = ( 2 5 , 2 5)+ (0,1)

    and (2 5 , 2 5)+(0,1). We therefore identifyϕ 1 1 = ( 7 5 , 2 5),ϕ 2 1 = ( 2 5 , 2 5)+(1,0),ϕ 1 2 = ( 2 5 , 7 5), andϕ 2 2 = ( 2 5 , 2 5)+ (0,1). The total degeneracy is reproduced exactly, since d(ϕ1

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    This example therefore shows that the method can extract the common conformal data of multiple operators even when it cannot resolve them individually within the present framework

    = 1, and the relative error atD= 100 andn= 6 is only about 1.8×10 −5. This example therefore shows that the method can extract the common conformal data of multiple operators even when it cannot resolve them individually within the present framework. 2.X Q=1 46,···,53 ≈4.33 Our final case study examines the highest-scaling- dimension group that we can sti...

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    We also list the numerical results of the associated crossover length scaleL ∗ y and the relative error of the scaling dimension RE ∆ in these tables

    Each operator’s degeneracy, scaling dimension, and conformal spin are also listed. We also list the numerical results of the associated crossover length scaleL ∗ y and the relative error of the scaling dimension RE ∆ in these tables. TABLE VII: Ising model,Q= 0 sector h, ¯h I, ¯I d× ¯d∆S L ∗ y Rerr (100,6) (0,0) (0,0) 1×1 0 0 0 0 ( 1 2 , 1

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    (6,0) 3×1 7 +6 144 1.2×10 −3 TABLE VIII: Ising model,Q= 1 sector h, ¯h I, ¯I d× ¯d∆S L ∗ y Rerr (100,6) ( 1 16 , 1

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    (7,0) 5×1 7 1 8 +7 144 1.1×10 −3 Appendix B: Conformal data of 3-state clock model In this appendix, we present exact conformal data, numerical estimates, and plots for the 3-state-clock- model results obtained with HOTRG atD= 100 and 0.5 0.6 0.7 0.8 0.9 10□3 10□2 10□1 100 101 102 103 □4 □2 0 2 4 1.5 1.6 1.7 1.8 1.9 10□3 10□2 10□1 100 101 102 103 □4 □2 0 ...

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    In these two tables, we also list the numerical results of the associated crossover length scaleL ∗ y and the relative error of the scaling dimension RE ∆

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