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arxiv: 2601.02681 · v2 · submitted 2026-01-06 · ✦ hep-lat · cond-mat.stat-mech· hep-th

Recognition: 2 theorem links

· Lean Theorem

Tensor renormalization group approach to critical phenomena via symmetry-twisted partition functions

Shinichiro Akiyama , Raghav G. Jha , Jun Maeda , Yuya Tanizaki , Judah Unmuth-Yockey
Authors on Pith no claims yet

Pith reviewed 2026-05-16 17:32 UTC · model grok-4.3

classification ✦ hep-lat cond-mat.stat-mechhep-th
keywords tensor renormalization groupsymmetry-twisted partition functionscritical phenomenaIsing modelO(2) modelBerezinskii-Kosterlitz-Thouless transitionfinite-size scalingspontaneous symmetry breaking
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The pith

Symmetry-twisted partition functions computed via tensor renormalization group locate critical points and extract exponents in classical spin models.

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

The paper establishes that locality constraints on field theories make symmetry-twisted partition functions effective order parameters for detecting spontaneous symmetry breaking. The tensor renormalization group supplies an efficient computational route to these twisted functions, allowing the authors to identify critical points directly from their behavior under finite-size scaling. For the three-dimensional O(2) model this yields a critical temperature of 2.2017(2) together with the exponent nu equal to 0.663(33). The same framework extracts the Berezinskii-Kosterlitz-Thouless temperature of the two-dimensional O(2) model as 0.8928(2) by computing the helicity modulus from the twisted sectors. A sympathetic reader would care because the method bypasses the need for conventional order parameters and works uniformly for both discrete and continuous symmetries.

Core claim

The tensor renormalization group offers an efficient framework to compute the symmetry-twisted partition functions, which enables detection of the symmetry-breaking transition and study of associated critical phenomena. Critical points are identified solely from the twisted partition function, and finite-size scaling extracts the critical temperature Tc = 2.2017(2) with exponent nu = 0.663(33) for the 3D O(2) model together with the BKT temperature TBKT = 0.8928(2) for the 2D O(2) model.

What carries the argument

Symmetry-twisted partition function, which encodes the response of the system to twisted boundary conditions and serves as an order parameter for spontaneous symmetry breaking through its low-energy scaling.

If this is right

  • Critical temperatures and exponents can be determined solely from the scaling of twisted partition functions without reference to conventional order parameters.
  • The helicity modulus for BKT transitions becomes directly accessible from the same twisted sectors.
  • The approach applies uniformly to both discrete-symmetry models such as the Ising model and continuous-symmetry models such as O(2).
  • Finite-size scaling analysis on TRG data yields quantitative estimates for Tc and nu that match known benchmarks within reported errors.

Where Pith is reading between the lines

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

  • The same TRG construction could be applied to quantum lattice models by replacing the classical tensor network with a quantum one while retaining the twisted-boundary construction.
  • Because twisted sectors are sensitive to long-distance symmetry realizations, the method may help classify phases protected by higher-form or subsystem symmetries.
  • Numerical cost scales with bond dimension rather than volume, so the technique could reach system sizes where conventional Monte Carlo sampling of twisted boundaries becomes impractical.

Load-bearing premise

The bond-dimension truncation in the tensor renormalization group is large enough to capture the long-distance physics of the symmetry-twisted sectors with accuracy sufficient for reliable finite-size scaling.

What would settle it

A systematic discrepancy between the critical temperature and exponent extracted from the twisted partition functions and the values obtained from conventional Monte Carlo simulations on the same models would falsify the claim that the TRG twisted sectors are sufficient.

Figures

Figures reproduced from arXiv: 2601.02681 by Judah Unmuth-Yockey, Jun Maeda, Raghav G. Jha, Shinichiro Akiyama, Yuya Tanizaki.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Finite-size scaling analysis for [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: presents the ratio of the twisted partition functions Zα=π/Zα=0 for various system sizes with L × L × L. 5 Here, we have focused on relatively smaller system sizes because the large volume results could be affected by artifacts from the finite bond dimension truncation [66], and Appendix B is devoted to discussing the finite bond-dimension effects, particularly in the thermodynamic limit. The behavior of Z… view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Finite-size scaling analysis for [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: shows the helicity modulus Υα(L × L) evaluated at several twist angles, α = π, π/2, π/4, π/6, for fixed L = 32. We observe that the twist-angle dependence is quite tiny for Υα=π/2,π/4,π/6, which is consistent with the theoretical formula (IV.10) as its α-dependence appears only in the subleading higher-winding contributions O(e−#L1/L2 ). Furthermore, Υα=π is significantly lower than the others on the low-t… view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Helicity modulus at [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Critical temperatures determined from [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
read the original abstract

The locality of field theories strongly constrains the possible behaviors of symmetry-twisted partition functions, and thus they serve as order parameters to detect low-energy realizations of global symmetries, such as spontaneous symmetry breaking (SSB). We demonstrate that the tensor renormalization group (TRG) offers an efficient framework to compute the symmetry-twisted partition functions, which enables us to detect the symmetry-breaking transition and also to study associated critical phenomena. As concrete examples of SSB, we investigate the two-dimensional (2D) classical Ising model and the three-dimensional (3D) classical $O(2)$ nonlinear sigma model, and we identify their critical points solely from the twisted partition function. By employing the finite-size scaling argument, we find the critical temperature $T_c=2.2017(2)$ with the critical exponent $\nu = 0.663(33)$ for the 3D $O(2)$ model. In addition, we also study the Berezinskii-Kosterlitz-Thouless (BKT) criticality of the 2D classical $O(2)$ model by extracting the helicity modulus from the twisted partition functions, and we obtain the BKT transition temperature, $T_{\mathrm{BKT}}=0.8928(2)$.

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

1 major / 1 minor

Summary. The manuscript introduces a TRG-based method to compute symmetry-twisted partition functions, which serve as order parameters for spontaneous symmetry breaking due to locality constraints in field theories. It applies this to the 2D Ising model, extracts Tc=2.2017(2) and ν=0.663(33) for the 3D O(2) nonlinear sigma model via finite-size scaling of the twisted partition function, and determines TBKT=0.8928(2) for the 2D O(2) model from the helicity modulus extracted from twisted sectors.

Significance. If the numerical extractions prove robust against truncation artifacts, the approach provides an efficient tensor-network route to critical parameters that relies only on twisted partition functions rather than direct order-parameter measurements, with potential extension to other models exhibiting SSB or topological transitions.

major comments (1)
  1. Results for the 3D O(2) model: The quoted Tc=2.2017(2) and ν=0.663(33) are obtained from finite-size scaling fits to the twisted partition function, yet no data or plots are provided showing stability of these quantities (or of the underlying twisted free energies) under successive increases in TRG bond dimension D. Because the truncation directly affects the entanglement spectrum in the twisted sectors, this omission leaves open a possible D-dependent systematic bias in the extracted critical parameters.
minor comments (1)
  1. The abstract states that critical points are identified 'solely from the twisted partition function,' but the implementation section should explicitly describe how the phase factors or boundary conditions for the twisted sectors are incorporated into the TRG coarse-graining steps.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comment. We address the major point below and will revise the manuscript to incorporate additional checks as suggested.

read point-by-point responses

  1. Referee: Results for the 3D O(2) model: The quoted Tc=2.2017(2) and ν=0.663(33) are obtained from finite-size scaling fits to the twisted partition function, yet no data or plots are provided showing stability of these quantities (or of the underlying twisted free energies) under successive increases in TRG bond dimension D. Because the truncation directly affects the entanglement spectrum in the twisted sectors, this omission leaves open a possible D-dependent systematic bias in the extracted critical parameters.

    Authors: We agree that explicit demonstration of stability under increasing bond dimension D is important for establishing the reliability of the results, particularly given the truncation effects on the entanglement spectrum in twisted sectors. The manuscript presents results obtained at bond dimensions where our internal convergence checks indicated stability within the reported statistical errors, but we did not include the supporting D-dependence data or plots. In the revised version we will add a new figure (or table) showing the twisted free energies and the fitted values of Tc and ν for successive D (e.g., D = 16, 32, 64), confirming that the quoted central values remain stable within the stated uncertainties. revision: yes

Circularity Check

0 steps flagged

No circularity: critical parameters extracted numerically via TRG and finite-size scaling

full rationale

The derivation computes symmetry-twisted partition functions using the standard TRG coarse-graining algorithm applied to the lattice models. Critical temperatures and exponents are then obtained by fitting finite-size scaling forms to the numerically generated data for the twisted free energies and helicity modulus. These steps are independent of the target values; the reported Tc=2.2017(2), ν=0.663(33), and TBKT=0.8928(2) emerge from the scaling analysis rather than being inserted by definition or reduced to prior fitted parameters. No self-definitional relations, fitted-input predictions, or load-bearing self-citations that collapse the central claim are present. The TRG bond-dimension truncation is an approximation whose convergence is external to the logical chain and does not create equivalence by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach rests on the domain assumption that locality constrains twisted partition functions to serve as order parameters, plus the standard assumption that TRG provides a controlled approximation to the partition function.

free parameters (1)
  • TRG bond dimension
    Truncation parameter controlling the accuracy of the tensor network coarse-graining; its value is chosen for convergence but not reported in the abstract.
axioms (1)
  • domain assumption Locality of field theories strongly constrains the possible behaviors of symmetry-twisted partition functions
    Stated explicitly in the abstract as the reason these functions detect low-energy symmetry realizations.

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Reference graph

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