Asymptotic freedom, lost: Complex conformal field theory in the two-dimensional $O(N>2)$ nonlinear sigma model and its realization in Heisenberg spin chains

Christopher Yang; Thomas Scaffidi

REVIEW 4 major objections 5 minor 2 cited by

Asymptotic freedom, lost: Complex conformal field theory in the two-dimensional $O(N>2)$ nonlinear sigma model and its realization in Heisenberg spin chains

T0 review · 4 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The two-dimensional O(N) nonlinear sigma model acquires a pair of complex-conjugate fixed points when the coupling is allowed to be complex, described by a complex conformal field theory.

desk verdict Serious numerical evidence for a complex O(3) CFT in non-Hermitian spin chains, but the NLSM port rests on a marginally irrelevant operator and two headline exponents are fit targets, not predictions. read the letter →

arxiv 2601.02459 v2 pith:QBRXGIQF submitted 2026-01-05 cond-mat.stat-mech cond-mat.str-elhep-thquant-ph

Christopher Yang , Thomas Scaffidi This is my paper · ORCID

classification cond-mat.stat-mechcond-mat.str-elhep-thquant-ph

keywords complexconformalfieldtheoryO(N)nonlinearsigmamodelasymptoticfreedomnon-HermitianspinchainsloopcentralchargeLindbladdynamicsentanglemententropy

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 asymptotic freedom in the 2D O(N) nonlinear sigma model for N>2 is not the whole story: if the coupling constant is allowed to be a complex number, the theory flows to a pair of complex-conjugate fixed points governed by a complex conformal field theory (CCFT) with a complex central charge. The fixed point is generic, having a single relevant singlet operator, so it should arise in any non-Hermitian model with O(N) symmetry after tuning one complex parameter. The authors confirm this at N=3 by exact diagonalization of two non-Hermitian spin-1 Heisenberg chains and a spin-1/2 ladder, matching the predicted complex central charge and scaling dimensions. They also construct a Lindbladian whose no-click dynamics realize the non-Hermitian Hamiltonian; since the CCFT vacuum is the slowest-decaying eigenstate, dissipative evolution naturally relaxes into the CFT state, offering a route to preparing long-range entangled states.

What carries the argument

The carrying object is the O(N) loop model's Coulomb-gas data, analytically continued to N>2: the fixed-point branches g̃± = 1 ± e(N) with e(N) = (1/π)cos⁻¹(N/2) give closed-form central charge and scaling dimensions that are real for N≤2 and move into a complex-conjugate pair for N>2. The link to the NLSM is provided by the loop-crossing operator, the singlet component of the ℓ=4 watermelon operator, which the paper computes to be irrelevant for all N>2 (Re Δ_{ℓ=4} rises from about 2.04 at N=3 to 2.5 as N→∞). That irrelevance is what makes the loop-model CCFT the generic critical behavior of the non-Hermitian NLSM rather than an artifact of the loop-model truncation.

What would settle it

Measure the finite-size scaling of the loop-crossing coupling in the non-Hermitian spin-1 chain: if Re Δ_{ℓ=4} ≤ 2, the near-marginal operator will grow with system size and the spectrum will drift away from the predicted CCFT; alternatively, a high-precision transfer-matrix calculation of the ℓ=4 watermelon scaling dimension at N=3 would settle whether the loop-model fixed point carries over to the NLSM.

Watch

Extended reading notes

Core claim

The central claim is that the 2D O(N) nonlinear sigma model, for N>2, possesses a pair of complex-conjugate fixed points at complex values of the coupling g, obtained by analytic continuation of two branches of real fixed points that annihilate at N=2. At these points the theory becomes a CCFT with central charge c±(N) = 1 − 6(1−g̃±)²/g̃±, where g̃± = 1 ± (1/π)cos⁻¹(N/2); for N=3 this gives c ≈ 1.51 ± 0.158i. The fixed point is generic: it has a single relevant singlet operator, the energy operator, so it requires tuning only one complex parameter. The port from the loop model to the NLSM rests on the irrelevance of loop crossings, identified with the singlet component of the ℓ=4 watermelon

Load-bearing premise

The argument stands on the claim that loop crossings—the singlet component of the ℓ=4 watermelon operator—are irrelevant for N>2; at N=3 the computed exponent Re Δ_{ℓ=4} ≈ 2.04 is barely above the marginal value of 2, so if this near-marginal operator is actually relevant or exactly marginal, the NLSM would flow elsewhere and the CCFT would not be the generic fixed point.

Editorial extensions

If this is right

Asymptotic freedom is lost in the complex coupling plane: a wide region of initial couplings flows to the CCFT rather than to the trivial g=0 fixed point.
The CCFT is the generic critical point of any non-Hermitian O(N>2)-symmetric model, requiring only a single complex tuning parameter.
Numerical agreement at N=3 in spin-1 chains and a spin-1/2 ladder supports the analytic continuation of the N≤2 real fixed points to complex N>2.
Because the CCFT vacuum is the longest-lived state, no-click dissipative dynamics of the proposed Lindbladian relaxes into a CFT state, providing a route to preparing long-range entangled states through engineered dissipation.
Perturbative beta functions at any fixed loop order should generically have zeros away from the positive real axis; whether these align with the CCFT points is a testable consistency check on perturbation theory.

Reading between the lines

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

Editorial inference: If the complex fixed point exists beyond the one-loop approximation, the NLSM on the real axis may pass near the complex fixed point and exhibit slow 'walking' RG flow or weakly first-order behavior, analogous to other complex-fixed-point phenomena.
Editorial inference: The near-marginal loop-crossing exponent at N=3 (Re Δ=2.04) makes the universality claim delicate; a high-precision transfer-matrix or Monte Carlo calculation of the ℓ=4 watermelon dimension would directly sharpen or refute the port from loop models to the NLSM.
Editorial inference: The ladder model with exact non-invertible symmetry may be a cleaner experimental and numerical platform for observing the CCFT, since it eliminates the slow finite-size convergence caused by loop crossings.
Editorial inference: The dissipative preparation protocol relies on postselecting no-click trajectories; a natural extension, noted as open by the authors, is whether the CCFT can appear as the steady state of the full Lindbladian without postselection.

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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 real step forward. The CCFT itself was already in the loop-model literature, but the paper does three genuinely new things: it argues the O(N>2) NLSM flows to those fixed points in the complex plane, it realizes them in microscopic non-Hermitian Heisenberg chains, and it sketches a Lindblad no-click preparation protocol. The numerical work is extensive and mostly convincing: the spectrum of watermelon operators, current, stress tensor, and the complex central charge from entanglement entropy all agree with the analytic continuation, and the DMRG central charge holds up to L=40. The ladder model with exact non-invertible symmetry is a particularly clean test — no loop crossings, single complex parameter, and it lands on the same CCFT.

The soft spots are real but proportionate. First, the NLSM-to-loop-model port hangs on the singlet component of the l=4 watermelon operator being irrelevant at N=3, with ReDelta approx 2.04. That is one epsilon above marginal. If corrections at the complex fixed point push it below 2, the NLSM flows elsewhere and the 'asymptotic freedom lost' framing collapses. The paper acknowledges this but does not test the crossing RG eigenvalue directly. Second, the spin-1 chain fixed point is located by optimizing residuals against the predicted l=1 and epsilon scaling dimensions (SM Eq. S3), so the 1% agreement of those two operators is not independent evidence. The other operators and central charge are independent, so the paper is not circular overall, but the two headline numbers in the main-text table should be labeled as fit targets. The ladder model's single-parameter tuning is a genuine check of the genericity claim, which helps.

The absence of error bars and the L=14 ED are minor at this stage; the DMRG central charge helps. Citation pattern is fine — Ref. [22] is the same group's earlier loop-model result and is properly credited. The self-citation is legitimate.

Who is this for? Condensed-matter theorists working on non-Hermitian criticality, monitored dynamics, and CFT. It deserves a serious referee. I would send it to review, with the expectation that the loop-crossing relevance and the fit-versus-prediction distinction get highlighted. The central existence claim looks credible; the genericity claim needs a direct finite-size test of the crossing RG eigenvalue.

Referee Report

4 major / 5 minor

Summary. The paper claims that the two-dimensional O(N>2) nonlinear sigma model (NLSM), when the coupling g is allowed to be complex, possesses a pair of complex-conjugate fixed points described by a complex CFT (CCFT) with complex central charge c(N). This is obtained by analytically continuing the known O(N) loop-model fixed points from N≤2 to N>2 and by arguing that loop crossings—the singlet component of the ℓ=4 watermelon operator—are irrelevant for N>2. The authors then locate the N=3 CCFT numerically in non-Hermitian spin-1 Heisenberg chains, in a K1-K2 chain, and in a spin-1/2 ladder with an exact non-invertible symmetry, comparing the complex spectrum and the biorthogonal entanglement entropy with Coulomb-gas predictions. They further propose a Lindbladian whose no-click trajectories realize the non-Hermitian Hamiltonian, so that dissipative evolution can prepare the CCFT vacuum.

Significance. If the central claim holds, it is conceptually significant: it would show that asymptotic freedom is lost in the complex coupling plane, that a generic non-Hermitian O(N>2) model can flow to a CCFT with a single complex relevant coupling, and that dissipative no-click dynamics can prepare a long-range entangled critical state. The numerical work is extensive: several distinct microscopic models are studied, and the non-fitted quantities—central charge, current, stress tensor, and several watermelon multiplets—show reasonable agreement. The Lindblad construction is explicit and physically motivated. However, the NLSM-to-loop-model port rests on the marginal irrelevance of loop crossings, and the main numerical confirmation uses the predicted ϵ and ℓ=1 dimensions as fitting targets. These issues are load-bearing and require additional evidence before the strong claims can be accepted.

major comments (4)

[Predictions for the O(N) NLSM, Eq. (4), Fig. 2(b)] The crossover from the O(N) loop model to the NLSM rests entirely on the loop-crossing operator (singlet component of the ℓ=4 watermelon) being irrelevant at the CCFT. At N=3, Eq. (4) predicts Re∆_lc ≈ 2.04, only 0.04 above marginal. The numerical spectrum in Fig. 3(d) gives Re∆ for D0 as 1.94, actually below 2. Because the critical point was located by fitting the predicted ℓ=1 and ϵ dimensions, this near-marginal/relevant value is not an independent check. Please provide a direct finite-size determination of the loop-crossing RG eigenvalue, e.g. the L-dependence of the D0 gap at fixed (J2,K), and state whether Re∆_lc > 2 in the thermodynamic limit. If it is ≤2, the NLSM does not flow to the CCFT and the central claim fails.
[Supplemental Material Sec. II, Eq. (S3)] The finite-size critical point (J2,K) is found by minimizing the cost function with R1 and R2 set to zero against the predicted ∆ϵ and ∆ℓ=1. Reporting these two operators in the main text as 'good agreement' is circular. The independent confirmation consists of the central charge, J, T, and the ℓ=2,3,4 watermelon operators (with the caveat in Comment 1). Please label ϵ and A as fitting targets, remove them from the confirmation table, and quantify the agreement using only the non-fitted data. This is essential for the non-perturbative claim.
[Microscopic model and numerical results, Eq. (6)] The genericity claim—single relevant singlet and single complex tuning parameter—is in tension with the actual numerical procedure, which tunes two complex parameters (J2,K) or (K1,K2) and uses an optimization weight w in Eq. (S3). The explanation is that loop crossings are weakly irrelevant, but this means the identified point is not shown to lie on the one-complex-parameter critical manifold. Please demonstrate either that a one-complex-parameter family through (J2,K)+ remains critical in the thermodynamic limit, or provide a quantitative estimate of the residual loop-crossing coupling at L=14 and its RG flow. Without this, the 'generic' statement is not supported by the numerics.
[End Matter, Ladder model with exact non-invertible symmetry] The spin-1/2 ladder is an exact realization of the dilute Temperley-Lieb/loop model and therefore excludes loop crossings by construction. It strengthens the loop-model side but cannot test the NLSM port. Please state this limitation explicitly where the ladder is presented as confirmation of the NLSM CCFT, and avoid presenting it as independent evidence for the NLSM-to-CCFT claim.

minor comments (5)

[Fig. 3(d)] The table of identified CCFT states appears to be duplicated in panels (a) and (b)/(c) of Fig. 3; please remove the duplicate and ensure the table is a single panel.
[Eq. (S3)] Define clearly that R2 is complex and that the cost function uses |R2| after the feature-scaling weight w; the notation f(J2,K)=|R1|+w|R2| is only presented in the text and could be made explicit.
[Discussion] The phrase 'unnecessary transition' is used without a precise definition. Please define what is meant in terms of the RG flow or the phase diagram.
[Monitored dynamics, Eq. (8)-(9)] The claim that the system 'naturally relaxes' to the CCFT should be qualified: no-click postselection has a success probability that decays exponentially in time and system size. This is not a fatal issue, but it should be stated to calibrate the proposed state-preparation protocol.
[General] For a numerics-heavy Letter, please include a data/code availability statement, especially since exact diagonalization and DMRG details are only partially described in the Supplemental Material.

Circularity Check

1 steps flagged · score 6.0 of 10

The spin-1 chain 'confirmation' of the ϵ and ℓ=1 scaling dimensions is fitted: SM Eq. S3 uses these predicted values as the optimization target, and Fig. 3(d) then lists the same numbers as predicted-vs-calculated. Central charge and other operators remain independent, so the paper is partially but not fully circular.

fitted input called prediction [Supplemental Material Sec. II, Eqs. (S2)-(S3); main text around Eq. (6) and Fig. 3(d) table]
"We do so by matching the thermal and ℓ= 1 watermelon scaling dimensions in the spin chain spectrum with their predicted values. ... R1 = ∆ℓ=1/∆ϵ − ∆calcℓ=1/∆calcϵ, R2 = ∆ϵ/∆calcϵ −1. ... Energy ϵ 0 0 1.66 − 1.12i 1.67 − 1.1i ... ℓ= 1 WM A 1 π 0.168 + 0.0251i 0.168 + 0.0252i"

The gradient-descent cost function f(J2,K)=|R1|+w|R2| is minimized by construction when the calculated ϵ and ℓ=1 dimensions equal the Coulomb-gas predictions: R2=0 forces ∆calcϵ=∆ϵ, and R1=0 then forces the ratio, hence ∆calcℓ=1=∆ℓ=1. The resulting finite-size point (J2,K) is then used in Fig. 3(d), where the same two operators are listed as 'Predicted' vs 'Calculated' and reported as confirmed. This is a fitted input presented as a prediction. The same optimization is reused for the K1-K2 model ('A similar optimization scheme as the one used in the main text'), so its low-energy ϵ and ℓ=1 agreement is likewise not independent. Other quantities — central charge, ℓ=2,3,4 operators, current, stress tensor — were not in the cost function and remain independent evidence.

full rationale

The paper's central field-theoretic claim — that the O(N>2) NLSM in the complex coupling plane flows to the loop-model CCFT — is not itself circular: it rests on Coulomb-gas formulas that are standard and cited to multiple independent sources (Refs. [22,70-76]), and on the porting argument that loop crossings are irrelevant (Eq. (4), Ref. [80]). The numerical confirmation, however, contains a genuinely circular element for two headline numbers. The SM explicitly states that the optimization searches for the finite-size critical point 'by matching the thermal and ℓ=1 watermelon scaling dimensions in the spin chain spectrum with their predicted values,' via cost functions R1,R2 (Eq. S3). Minimizing these residuals forces the calculated ϵ and ℓ=1 dimensions to coincide with the predicted ones, so the Fig. 3(d) entries for those operators are fitted inputs, not independent predictions. The same procedure is used for the K1-K2 model. By contrast, the central charge (c=1.529−0.161i vs 1.51−0.158i), the ℓ=2,3,4 watermelon multiplets, the current, and the stress tensor are not used in the cost function and are compared with predictions after the fact; they provide genuine, non-circular support. The spin-1/2 ladder analysis also uses a scale-invariance residual (Eq. 13) rather than a predicted-dimension target, so it is independent. The near-marginal loop-crossing exponent Re∆ℓ=4≈2.04 is a fragility of the porting argument, not a circularity. Overall, the central claim has substantial independent content, but at least two of the paper's headline 'confirmed' scaling dimensions are fitted rather than predicted, giving a partial-circularity score of 6.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The central claim rests on prior loop-model CFT results and on the assumed mapping from the NLSM to loop models. The microscopic parameters (J2,K) are fitted to the predictions; no new physical entities are introduced.

free parameters (6)

(J2,K) critical point in spin-1 chain = (0.0660+0.338i, 0.176+0.335i)
Chosen by gradient descent (SM Eq. S3) to make the calculated ϵ and ℓ=1 scaling dimensions equal the loop-model predictions; these two entries in Fig. 3(d) are therefore fitted.
(K1,K2) critical point in K1-K2 chain = (0.385+0.243i, −0.414−1.77i)
Found by the same matching procedure described in SM Sec. II; used to locate the CCFT at L=12.
complex velocity v = 0.796−1.03i (spin-1); 0.578−0.72i (K1-K2); 0.767+0.235i (ladder)
Extracted from energy differences (SM Eq. S1) and used to convert gaps into scaling dimensions; quoted without error bars.
central charge c = 1.529−0.161i (spin-1); 1.565−0.2i (K1-K2); 1.54−0.214i (ladder)
Fitted from entanglement entropy S=(c/3)log[...]+s0; used as evidence but is a fitted output rather than a parameter of the derivation.
optimization weight w = 0.2
Hand-chosen feature scaling weight in the cost function f(J2,K)=|R1|+w|R2| (SM Eq. S3).
bulk energy density e∞ = not quoted
Free parameter in the ground-state energy scaling fit E_g/L = e∞ − πvc/(6L²) (SM Fig. S2).

assumptions (6)

domain assumption Coulomb gas formulas for O(N) loop model central charge and scaling dimensions (Eqs. 2-4) from Refs [22,70-76].
The entire paper tests these analytic-continuation predictions; their validity is assumed from prior loop-model results.
domain assumption Loop crossings correspond to the singlet component of the ℓ=4 watermelon operator and are irrelevant for N>2.
Computed from Eq. (4) plus identification from Ref [80]; load-bearing for porting loop-model CCFTs to the NLSM. The marginal value Re∆≈2.04 at N=3 makes this delicate.
ad hoc to paper Complexifying J2 and K in the spin-1 Heisenberg chain yields an O(3) NLSM with complex coupling g(J2,K).
The standard Haldane mapping from the spin-1 chain to the O(3) NLSM is assumed to survive complexification; this is stated but not derived.
standard math Biorthogonal entanglement entropy for non-Hermitian critical systems obeys S=(c/3)log[(L/π)sin(πl/L)]+s0 with complex c.
Taken from the non-Hermitian CFT literature (Refs [39,55,104-114]); used for the central charge extraction.
standard math No-click Lindblad dynamics are governed by H_eff = H0 − (i/2)Σ_a L_a†L_a.
Standard quantum-trajectory result; assumed for the state-preparation protocol.
domain assumption The dilute Temperley-Lieb algebra and Yang-Baxter point analytically continued to N>2 describe the O(3) loop model and the spin-1/2 ladder.
Basis for the exact non-invertible-symmetry ladder construction (Refs [79,90,91] and [22]).

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Pith. "Pith review of Asymptotic freedom, lost: Complex conformal field theory in the two-dimensional $O(N>2)$ nonlinear sigma model and its realization in Heisenberg spin chains." pith.science (2026). https://pith.science/paper/QBRXGIQF

@misc{pith2026260102459,
  author       = {Pith},
  title        = {Pith review of: Asymptotic freedom, lost: Complex conformal field theory in the two-dimensional $O(N>2)$ nonlinear sigma model and its realization in Heisenberg spin chains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QBRXGIQF}},
  note         = {Machine review of arXiv:2601.02459}
}

abstract

The two-dimensional $O(N)$ nonlinear sigma model (NLSM) is asymptotically free for $N>2$: it exhibits neither a nontrivial fixed point nor spontaneous symmetry-breaking. Here we show that a nontrivial fixed point generically does exist in the complex coupling plane and is described by a complex conformal field theory (CCFT). This CCFT fixed point is generic in the sense that it has a single relevant singlet operator, and is thus expected to arise in any non-Hermitian model with $O(N)$ symmetry upon tuning a single complex parameter. We confirm this prediction numerically by locating the CCFT at $N = 3$ in two non-Hermitian spin-1 antiferromagnetic Heisenberg chains, and in a non-Hermitian spin-$1/2$ ladder, finding good agreement between the complex central charge and scaling dimensions and those obtained by analytic continuation of real fixed points from $N\leq 2$. We further construct a realistic Lindbladian for a spin-1 chain whose no-click dynamics are governed by the non-Hermitian Hamiltonian realizing the CCFT. Since the CCFT vacuum is the eigenstate with the smallest decay rate, the system naturally relaxes under dissipative dynamics toward a CFT state, thus providing a route to preparing long-range entangled states through engineered dissipation.

Figures

Figures reproduced from arXiv: 2601.02459 by the authors.

**Figure 2.** FIG. 2. (a) Schematic two-parameter RG flow for the [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** FIG. 3. Spectrum of the Heisenberg spin-1 chain at the finite-size complex fixed point [Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4. Realization of a CCFT in a spin-1 Heisenberg chain. [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

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Forward citations

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