Tensor renormalization group study of (1+1)-dimensional O(3) nonlinear sigma model with and without finite chemical potential

Xiao Luo; Yoshinobu Kuramashi

REVIEW 4 major objections 6 minor 41 references

Tensor renormalization group study of (1+1)-dimensional O(3) nonlinear sigma model with and without finite chemical potential

T0 review · 4 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The tensor renormalization group can compute the dynamical critical exponent z = 1.96(6) of the (1+1)-dimensional O(3) nonlinear sigma model at finite chemical potential, a regime where Monte Carlo methods face a sign problem.

desk verdict A competent proceedings summary of the authors' own prior work, but the thermodynamic-limit claim in Sec. 3.2 is wrong by orders of magnitude and the 'first z with TRG' claim is already true of their own ref. [33]. read the letter →

arxiv 2412.02995 v1 pith:4CY2D65M submitted 2024-12-04 hep-lat

Xiao Luo , Yoshinobu Kuramashi This is my paper · ORCID

classification hep-lat

keywords tensorrenormalizationgroupO(3)nonlinearsigmamodelfinitechemicalpotentialsignproblementanglemententropycentralchargequantumphasetransitiondynamicalcriticalexponent

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 argues that the tensor renormalization group (TRG) provides a sign-problem-free route into the finite-density physics of the (1+1)-dimensional O(3) nonlinear sigma model, a massive asymptotically free theory used as a testbed for QCD-like gauge theories. At zero chemical potential, it extracts the central charge c = 1.97(9) from the asymptotic scaling of the von Neumann entropy, consistent with the expected c = 2 and with an independent matrix-product-state calculation, and it shows that the Rényi entropies approach the same value only for large Rényi index n. At finite chemical potential, where standard Monte Carlo suffers a sign problem, the paper locates the quantum critical point at μ_c = 0.14512(11), measures the correlation-length exponent ν = 0.512(15), and extracts the dynamical critical exponent z = 1.96(6) from the scaling of the temporal correlation length. These exponents agree with the theoretical expectations ν = 1/2 and z = 2, and the z measurement is presented as the first successful TRG computation of a dynamical critical exponent.

What carries the argument

The machinery is the tensor network representation of the lattice path integral, obtained by discretizing the continuous O(3) spin integration with Gauss-Legendre quadrature and decomposing the resulting four-leg bond tensor by singular value decomposition, then coarse-graining with the higher-order tensor renormalization group (HOTRG) at bond dimension $D_{\rm cut}$. The two observables that carry the finite-density argument are the number density $\langle n \rangle = (1/LN_t)\, \partial \ln Z/\partial \mu$, evaluated by central finite differences of $\ln Z$, and the temporal correlation length $\xi_t = N_t / \ln(\lambda_0/\lambda_1)$, read off from the two largest eigenvalues of the reduced density matrix. These enter the scaling forms $\langle n \rangle \propto \{ \mu - (\mu_c + B_n/D_{\rm cut})\}^\nu$ and $\xi_t \propto |\mu - (\mu_c + B_\xi/D_{\rm cut})|^{-z\nu}$, whose simultaneous fit yields $\mu_c$, $\nu$, and $z$. The theoretical expectations $\nu = 1/2$ and $z = 2$ come from the equivalence at finite density between this field theory and the integer-spin Heisenberg chain in a magnetic field.

What would settle it

Compute the number density $\langle n \rangle$ and the temporal correlation length $\xi_t$ at $\beta = 1.4$ on a $512 \times 512$ or $1024 \times 1024$ lattice at the same bond dimensions $D_{\rm cut} \in \{125, 130, 135\}$, fit the same scaling forms, and check whether $\mu_c$, $\nu$, and $z$ move by more than the quoted errors; any significant shift would falsify the thermodynamic-limit assumption behind the reported exponents.

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Extended reading notes

Core claim

Working with the higher-order tensor renormalization group and taking the bond dimension $D_{\rm cut} \to \infty$ by extrapolation, the authors construct a tensor network representation of the O(3) nonlinear $\sigma$ model partition function at finite chemical potential $\mu$. At $\mu = 0$ they compute both von Neumann and Rényi entanglement entropies on a $128 \times 1024$ subsystem geometry and fit the asymptotic scaling $S_A = (c/3)(2\pi\beta - \ln \beta) + \text{const.}$ to obtain $c = 1.97(9)$ for the von Neumann entropy, with the $n$th-order Rényi entropies yielding $c$ values that converge toward 2 as $n$ grows. For $\mu \ne 0$ at $\beta = 1.4$ they compute the number density by numerical differentiation of $\ln Z$ and the temporal correlation length $\xi_t$ from the leading eigenvalues of the density matrix, on a $225 \times 225$ lattice. Fitting $\langle n \rangle(\mu) = A_n \{ \mu - (\mu_c + B_n/D_{\rm cut})\}^\nu$ and $\ln \xi_t = A_\xi + \alpha \ln|\mu - (\mu_c + B_\xi/D_{\rm cut})|$ with $\alpha = z\nu$ gives $\mu_c = 0.14512(11)$, $\nu = 0.512(15)$, and $z = 1.96(6)$, consistent with the mass gap $m = 0.1449(2)$ from Monte Carlo and with the predictions $\nu = 1/2$, $z = 2$. The consistency of the extracted $\mu_c$ with the independent mass-gap measurement and the agreement of both exponents with theory constitute the paper's central claim.

Load-bearing premise

The paper's thermodynamic-limit claim rests on treating the $225 \times 225$ lattice as effectively zero temperature and infinite volume, supported by the quoted figures $T/m = 2.1 \times 10^{-7}$ and $Lm = 4.9 \times 10^{6}$; those figures appear inconsistent with the stated lattice size and mass gap $m \approx 0.1449$, since they would give $T/m \approx 0.031$ and $Lm \approx 33$, and if the lattice is not truly in the thermodynamic limit the fitted exponents could carry finite-size bias.

Editorial extensions

If this is right

TRG can map the finite-density phase diagram of an asymptotically free (1+1)-dimensional field theory without a sign problem, locating the critical chemical potential consistently with an independent Monte Carlo mass-gap measurement.
The dynamical critical exponent z = 1.96(6), extracted from the anisotropic scaling of the temporal correlation length, matches the Heisenberg-chain prediction z = 2, indicating that TRG captures the anisotropy that the chemical potential induces between space and time directions.
The central charge c = 1.97(9) from the von Neumann entropy, together with the large-n convergence of the Rényi entropies toward c = 2, confirms the expected conformal structure of the mu = 0 critical theory.
The same computational pipeline, numerical differentiation of ln Z plus eigenvalue-based correlation lengths, carries over to other sign-problematic lattice models, including the finite-density gauge and fermion models the paper cites as motivation.

Reading between the lines

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

If z = 2 holds across lattice sizes, the finite-density O(3) NLSM and the integer-spin Heisenberg chain in a magnetic field share not just static but dynamical universality; a stronger test would be to measure the full scaling function of the number density and compare it with the predicted universal curve, not just the exponents.
The paper's thermodynamic-limit figures for T/m and Lm are inconsistent with L = N_t = 225 and m = 0.1449 by orders of magnitude; repeating the fit on 512-by-512 or 1024-by-1024 lattices would show whether the quoted errors already absorb the finite-size bias.
The observed n-dependence of the central charge extracted from Rényi entropies suggests the reduced density matrix has a quickly decaying eigenvalue spectrum; a direct spectral analysis could quantify how many eigenvalues are needed to approximate the von Neumann entropy, which would sharpen the extrapolation method.
The same TRG setup could be applied to finite-density CP(N-1) or SU(2) principal chiral models to ask whether z = 2 is a general feature of massive asymptotically free (1+1)-dimensional theories or specific to the O(3) model.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: this is a Lattice 2024 proceedings that condenses two of the authors' own JHEP papers [32,33]. It adds no new numbers, and the abstract's claim that the dynamical exponent z is extracted 'first' with TRG is already true of their own ref. [33] – so the novelty statement is factually wrong.

What's good: the numerical work is credible. The central charge c = 1.97(9) from von Neumann entropy agrees with MPS, and the consistency check between Rényi and von Neumann entropies is a nice touch. The finite-density results mu_c = 0.14512(11), nu = 0.512(15), z = 1.96(6) match the theoretical expectations from the Heisenberg-chain mapping. The TRG setup is standard and clearly described.

The soft spots: the thermodynamic-limit justification in Sec. 3.2 is off by orders of magnitude. With N_t = L = 225 and m = 0.1449, T/m ≈ 0.031 and Lm ≈ 33, not the quoted 2.1e-7 and 4.9e6. That is a real factual error. It matters because the fits are done for |mu - mu_c| between 0.00063 and 0.0019, corresponding to xi ~ 25–44. That gives L/xi ~ 5–10 and T*xi ~ 0.11–0.20. The system is not cleanly in the zero-temperature thermodynamic limit; finite-size and finite-temperature corrections are expected at the 10–20% level, larger than the quoted errors on nu and z. The agreement with theory is reassuring, but the systematic uncertainty is understated. The paper is also not new relative to its own cited literature – fine for a proceedings summary, but the 'first z' claim needs to be scoped or removed.

Who is this for? People following TRG applications to sign-problematic models. It is a readable summary, but the primary literature is the two JHEP papers.

Recommendation: a serious referee would catch the volume error and the problematic novelty claim, so yes, send it to review – but the bar for acceptance should be that both issues are fixed. If this is a PoS proceedings, the correction of the typo and the scoped claim should be mandatory before publication.

Referee Report

4 major / 6 minor

Summary. This paper applies the higher-order tensor renormalization group (HOTRG) to the (1+1)-dimensional O(3) nonlinear sigma model with chemical potential. At μ=0, the authors compute von Neumann and Rényi entanglement entropies for β=1.4–1.7 on 128×1024 and 1024×1024 lattices, extrapolate in bond dimension, and extract the central charge c=1.97(9) from the von Neumann entropy and c=2.27(16) from the second Rényi entropy, with an n-dependent analysis showing convergence toward c≈2. At μ≠0 they compute the number density and temporal correlation length on a 225×225 lattice with Dcut=125, 130, and 135, fit a critical scaling form with 1/Dcut corrections, and obtain μc=0.14512(11), ν=0.512(15), and, from α=zν with α=1.003(5), z=1.96(6). These results are interpreted as the first TRG determination of a dynamical critical exponent and are consistent with the predictions ν=0.5 and z=2.

Significance. The significance is potentially high: if the finite-density results are correct, the paper demonstrates that TRG can extract critical exponents, including the dynamical exponent z, in a sign-problematic regime, and the μc value agrees with an independent Monte Carlo mass gap. The μ=0 entanglement results provide a useful cross-check of the TRG method against MPS. However, the central finite-density claim rests on a volume that is stated to be 'large enough' on the basis of numerical values (T/m=2.1×10^{-7}, Lm=4.9×10^6) that are internally inconsistent with the stated lattice size and mass gap; the actual T/m and Lm are O(0.03) and O(33). This leaves a 10–20% finite-size systematic in the exponent extraction, comparable to the quoted statistical errors. The paper therefore needs additional analysis or more conservative claims before the central result can be accepted.

major comments (4)

[Sec. 3.2] The statement 'The volume is large enough to be regarded as the thermodynamic limit at zero temperature: T/m = 2.1×10^{-7} and Lm = 4.9×10^6' is inconsistent with the lattice size L=N_t=225 and the mass gap m=1/6.90(1)=0.1449 quoted in the same section. Since T=1/N_t, the actual values are T/m=1/(225×0.1449)=0.0307 and Lm=32.6, smaller than the quoted values by about five orders of magnitude. In the fit window μ∈[0.14575, 0.14700] with μc=0.14512, the reduced distance δ is 0.00063–0.0019, so the correlation length ξ∼δ^{-ν} is approximately 25–44; hence Tξ≈0.11–0.20 and L/ξ≈5–10. The scaling limit requires Tξ≪1, so finite-temperature corrections at the 10–20% level are expected, which is larger than the quoted errors on ν (0.015) and z (0.06). Please either re-run at substantially larger N_t and L, add a finite-temperature/finite-size scaling analysis, or explicitly include this systematic error in the quoted exponents. As written, the agreement with ν=0.5 and z=2 cannot be distinguished from an accidental finite-size effect.
[Sec. 3.2] The dynamical critical exponent is obtained by fixing μc=0.14512 from the density fit and then using ν=0.512(15) from the same density fit to convert the temporal-correlation-length exponent α=1.003(5) into z=α/ν=1.96(6). This sequential procedure does not propagate the correlations among μc, ν, and α and ignores the uncertainty in the fixing of μc. A joint fit of the density and correlation-length data, or at least a fit with μc free in the ξ_t analysis, is needed to determine whether the quoted error on z is realistic. This is load-bearing for the 'first calculation of the dynamical critical exponent with the TRG method' claim.
[Sec. 3.2] The infinite-bond-dimension extrapolation is based on only three closely spaced values, Dcut=125, 130, and 135, with a linear 1/Dcut ansatz for the shift in μc. The paper does not report χ²/dof, the number of data points, or any test of the linear extrapolation (for example, including Dcut=100 or 150). Since the central results μc, ν, and z depend on this extrapolation, the systematic error from the Dcut→∞ limit should be quantified by varying the fit range, including a quadratic term in 1/Dcut, or reporting the fit quality.
[Sec. 3.2] The numerical differentiation used for the number density, ⟨n⟩≈[ln Z(μ+Δμ)-ln Z(μ-Δμ)]/(2Δμ L N_t), does not state the value of Δμ. The closest data points to μc are at δ=0.00063, so if Δμ is not much smaller than this, the derivative will smear the singular behavior and bias ν and μc. Please specify Δμ and verify that the quoted results are stable against reducing it, or provide the raw ln Z values.

minor comments (6)

[Abstract] The abstract says the study is performed 'with the infinite limit of the bond dimension Dcut→∞', but the paper actually uses a linear 1/Dcut extrapolation from Dcut=125, 130, and 135; please rephrase to 'extrapolated to Dcut→∞'.
[Sec. 3.1, Fig. 4] The text states that the plotted entropies are obtained by linear extrapolations in 1/Dcut, but the figure caption says 'with Dcut=130'; please clarify which data are shown.
[Sec. 3.1] The statement that N_t=1024 'is large enough to be regarded as the zero temperature limit' should be quantified by giving T/m for the β values used, since the correlation length varies from about 6.9 to 34.6.
[Sec. 2.2, Eq. (15)] The definition of ξ_t via λ0 and λ1 would be clearer if the paper stated that these are the leading eigenvalues of the temporal transfer matrix and explained how the reduced tensor T* is constructed in the HOTRG step.
[Fig. 8] The horizontal axis combines the fitted shift B_ξ/Dcut with μ; please define the effective variable, for example δ_eff=μ-(μc+B_ξ/Dcut), in the caption so that the reader can see what is plotted.
[Throughout] The manuscript contains many typos and grammatical slips (for example, 'tranasition', 'inital', 'featute', 'nad', 'Futhermore', 'polynominal', 'presisely', 'etropies'); a careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the extracted exponents are fit outputs benchmarked against external MPS, Monte Carlo, and analytic results, not inputs to the fits.

full rationale

The paper's central quantities are obtained by explicit fits to TRG data: c from the entanglement-entropy scaling forms in Eqs. (19)-(20), and μc, ν, and z from the number-density and temporal-correlation-length fits in Sec. 3.2. The target values c=2, ν=0.5, and z=2 are not used as fit inputs; they serve only as external comparisons, together with the MPS central charge [41] and the Monte Carlo mass gap [39]. The citations to the authors' own prior papers [32,33] describe the same TRG method and preliminary results, but the present fits and data are shown, so those self-citations are not the load-bearing justification. The 'first successful calculation of z with TRG' statement is a novelty claim, not a derivation step. The statement in Sec. 3.2 that a 225x225 lattice gives T/m=2.1e-7 and Lm=4.9e6 is numerically inconsistent with N_t=L=225 and m=1/6.90 (one obtains T/m≈0.031 and Lm≈33); I flag this as a substantive correctness and finite-size-systematics concern, but it is not a circular-reasoning step because the claimed exponents are not defined or fitted in terms of that statement.

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

The central results are obtained by fitting data with assumed power-law and logarithmic scaling forms; the nuisance parameters A_n, B_n, A_xi, B_xi, and intercepts are fitted. No new entities are introduced. The main axioms are the standard finite-density formulation of the O(3) NLSM and the scaling hypotheses used to extract critical exponents.

free parameters (5)

A_n (density amplitude) = 0.20(2)
Fit parameter in the assumed scaling form for number density near mu_c (Sec. 3.2).
B_n (finite-D_cut shift in density fit) = 0.068(12)
Corrects the critical chemical potential for finite bond dimension; fit together with A_n, mu_c, and nu.
A_xi (temporal correlation length offset) = -0.030(29)
Constant in the global fit of ln xi_t versus ln|mu - (mu_c + B_xi/D_cut)|.
B_xi (finite-D_cut shift in correlation length fit) = 0.0599(9)
Finite-D_cut shift of the transition point in the correlation length fit.
Intercepts in entropy scaling fits = not quoted
The constants in Eqs. (19)-(20) are fit parameters in the extraction of the central charge.

assumptions (4)

domain assumption The lattice action with chemical potential introduced via the SO(3) twist matrix D(mu,nu) (Eqs. (1)-(3)) yields the correct continuum O(3) NLSM at finite density, and this model is equivalent to the integer-spin Heisenberg chain in a magnetic field.
Used in Sec. 2 to define the model and in Sec. 3.2 to set the theoretical expectations nu = 0.5 and z = 2 (Refs. [34-38]).
domain assumption The asymptotic scaling formula S_A = (c/3) ln xi with xi ~ (1/beta) exp(2*pi*beta) is valid in the fitted beta range (1.4 to 1.7) at mu = 0.
Used to extract the central charge from entanglement entropies via Eqs. (19)-(20) (Refs. [39,40]).
ad hoc to paper The critical scaling forms <n> = A_n (mu - mu_c - B_n/D_cut)^nu and xi_t ~ |mu - (mu_c + B_xi/D_cut)|^(-z nu) correctly describe the data in the fit ranges.
The functional forms and fit ranges (0.14575 <= mu <= 0.14700) are assumed ad hoc; acceptance of the extracted nu and z depends on these forms being correct (Sec. 3.2).
ad hoc to paper Linear extrapolation in 1/D_cut removes finite bond dimension effects for entropies and critical parameters.
Used for central charge (Sec. 3.1) and implicitly for the D_cut corrections B_n/D_cut and B_xi/D_cut in the fits (Sec. 3.2).

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Pith. "Pith review of Tensor renormalization group study of (1+1)-dimensional O(3) nonlinear sigma model with and without finite chemical potential." pith.science (2026). https://pith.science/paper/4CY2D65M

@misc{pith2026241202995,
  author       = {Pith},
  title        = {Pith review of: Tensor renormalization group study of (1+1)-dimensional O(3) nonlinear sigma model with and without finite chemical potential},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4CY2D65M}},
  note         = {Machine review of arXiv:2412.02995}
}

abstract

We study (1+1)-dimensional O(3) nonlinear sigma model using the tensor renormalization group method with the infinite limit of the bond dimension $D_{\rm cut}\rightarrow \infty$. At the vanishing chemical potential $\mu=0$, we investigate the von Neumann and R\'enyi types of entanglement entropies. The central charge is determined to be $c=1.97(9)$ by using the asymptotic scaling properties of the entropies. We also examine the consistency between two entropies. In the finite density region with $\mu\ne 0$, where this model suffers from the sign problem in the standard Monte Carlo approach, we investigate the properties of the quantum phase transition. We determine the transition point $\mu_{\rm c}$ and the critical exponent of the correlation length $\nu$ from the $\mu$ dependence of the number density in the thermodynamic limit. The dynamical critical exponent $z$ is also extracted from the scaling behavior of the temporal correlation length as a function of $\mu$. This is the first successful calculation of the dynamical critical exponent with the TRG method.

Figures

Figures reproduced from arXiv: 2412.02995 by the authors.

**Figure 1.** 𝛽 dependence of internal energy at 𝜇 = 0 on a 1024 × 1024 lattice. Solid curves represent the results of the strong and weak coupling expansions. region we observe that our result show good consistency with the strong coupling expansion up to 𝛽 ∼ 1.2. On the other hand, the result starts to follow the weak coupling expansion curve around 𝛽 ∼ 2.0. The density matrix 𝜌𝐴 is evaluated using HOTRG with the bond dimension… view at source ↗

**Figure 2.** 𝐿 dependence of von Neumann entropy 𝑆𝐴 at 𝑁𝑡 = 1024 in 1.4 ≤ 𝛽 ≤ 1.7 with 𝐷cut = 130. 2 4 8 16 32 64 128 256 512 1024 L 0.0 1.0 2.0 3.0 S A (2) β=1.4 β=1.5 β=1.6 β=1.7 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 4.** 𝛽 dependence of von Neumann entropy 𝑆𝐴 at (𝐿, 𝑁𝑡) = (128, 1024) in 1.4 ≤ 𝛽 ≤ 1.7 with 𝐷cut = 130. 0 1 2 3 4 5 6 7 8 9 10 11 12 n 1.0 1.5 2.0 2.5 3.0 c c w/ entanglement entropy c w/ Renyi entropy c=2 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

Figures from the paper (2 more)

**Figure 6.** Figure 6: 𝑛 dependence of 𝑛th-order Rényi entropy with 𝑁𝑡 = 1024 at 𝛽 = 1.5. Solid symbol at 𝑛 = 1 denotes the von Neumann entropy. All the results are extrapolated values at 𝐷cut → ∞. 3.2 Quantum phase transition with 𝜇 ≠ 0 We evaluate the number density with the numerical diff…

**Figure 7.** Figure 7: 𝜇 dependence of number density ⟨𝑛⟩ at 𝛽 = 1.4 on a 2 25×2 25 lattice with 𝐷cut ∈ [125, 135]. The solid curves represent the fit results at 𝐷cut = 125(red), 130(blue) and 135(green). -7.5 -7.0 -6.5 -6.0 -5.5 ln|µ−(µc +Bξ /Dcut)| 5.0 6.0 7.0 8.0 ln( ξ t ) Dcut=125 Dcut=1…

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