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Logarithmic operators in $c=0$ bulk CFTs

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Pith's one-line read This paper argues that zero-norm 'energy' operators in percolation and self-avoiding walk CFTs can have non-vanishing long-range four-point correlations at c=0, via a rank-3 logarithmic Jordan block.

desk verdict A technically rich and genuinely new construction of logarithmic multiplets at c=0, whose central non-vanishing energy four-point function is real but rests on an unproven cluster-decomposition assumption that the paper itself flags. read the letter →

arxiv 2411.18696 v2 pith:GQMJGT66 submitted 2024-11-27 hep-th cond-mat.stat-mechmath-phmath.MP

Yifei He This is my paper · ORCID
classification hep-thcond-mat.stat-mechmath-phmath.MP
keywords c=0conformalfieldtheorylogarithmicoperatorsJordanblocksKacpercolationself-avoidingwalkclusterdecompositionbootstrap
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

Percolation and self-avoiding walks are described by c=0 conformal field theories, whose 'energy' Kac operators have zero norm and were previously thought to have no long-range multipoint correlations. This paper establishes the contrary: after fixing the operators' normalizations by requiring real finite three-point constants at generic c and passing to c=0, the energy operators become bottom fields of logarithmic Jordan blocks, and the four-point function assembled by cluster decomposition is nonzero. The nonvanishing term is carried by a rank-3 Jordan block built around the second energy operator. If the argument holds, zero-norm states in c=0 bulk CFTs build long-range correlations through logarithmic structures, and the operator algebra of these geometrical critical points is richer than previously believed.

What carries the argument

The machinery is the logarithmic (Jordan) multiplet formed when a Kac operator's norm vanishes at $c=0$. The paper defines properly normalized Kac operators $\hat{\Phi}_{r,s}$ at generic $c$, requires real and finite three-point constants, and takes the $c\to 0$ limit. A first-order zero in the norm produces a rank-2 Jordan block $(\tilde O,O)$; a second-order zero produces a rank-3 block $(\Psi_2,\Psi_1,\Psi_0)$. The top field is built as a combination $\phi/B_\phi(c)+\psi/B_\psi(c)$ (and for rank 3, plus $O^{(2;2)}/\gamma_O(c)$), with logarithmic couplings fixed by conformal Ward identities, for example $\gamma = B'_\phi/(h'_\phi-h'_\psi)$ and $a = \gamma'_O/(2h'_O-h'_\phi-h'_\psi)$. The four-point function is then assembled by cluster decomposition over intermediate states, and the load-bearing identity is the nonzero three-point coupling $C^{\mathrm{perco}}_{\varepsilon\varepsilon\Psi_1}$ to the middle field of the rank-3 block.

What would settle it

Compute the connected four-point correlation of the local bond-occupation energy operator on critical percolation clusters in the scaling limit; in the s-channel it should behave as (z \bar z)^{3/4} times -1875/16384 with no logarithm at leading order, rather than vanish.

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Extended reading notes

Core claim

The paper's central discovery is that the four-point function of the bulk energy operator ε at c=0 does not vanish. For percolation, the s-channel limit is given by eq. (5.55): $$\langle \varepsilon(\infty)\varepsilon(1)\varepsilon(z,\bar z)\varepsilon(0)\rangle = (z\bar z)^{-$2h^{{\mathrm{perco}}$}_\varepsilon}\left(\frac{($C^{{\mathrm{perco}}$}_{\varepsilon\varepsilon\Psi_1})^2}{a}(z\bar z)^2+\cdots\right)$$ with $C^{\mathrm{perco}}_{\varepsilon\varepsilon\Psi_1}=-125/512$ and $a=-25/48$. The same construction applies to self-avoiding walks. The nonzero term comes from the middle field $\Psi_1$ of the rank-3 Jordan block $(\Psi_2,\Psi_1,\Psi_0)$ associated with the second energy operator $\varepsilon'\sim\Phi_{3,1}$, rather than from the naive $c\to 0$ limit of the generic-$c$ BPZ correlation, which vanishes. Cluster decomposition using the exact $c=0$ logarithmic conformal data is what reveals the surviving correlation.

Load-bearing premise

The argument assumes the four-point function can be assembled by cluster decomposition at c=0 from three-point functions, inserting only the identified intermediate states (5.51); if cluster decomposition fails or extra states contribute, the nonvanishing result could be modified.

Editorial extensions

If this is right

  • The energy-operator OPE at c=0 is nonsingular after proper normalization: the bottom-field couplings with T and Ψ0 vanish, while couplings to the top fields are nonzero, with C^{perco}_{εεt}=-25/8 and C^{perco}_{εεΨ1}=-125/512.
  • The s-channel four-energy correlation in percolation and self-avoiding walks is nonzero and, at leading order, has no logarithm, so its scaling is a power law, (z z̄)^{3/4} in percolation.
  • Higher Kac operators acquire higher-order zero norms at c=0, and comparison with c<1 Liouville CFT suggests rank-4, rank-5, and possibly arbitrarily high Jordan blocks, though these are invisible in the spin OPE because the corresponding C_{σσΦ_{r,1}} vanish.
  • The hull spin OPE only probes bottom fields, so its four-point function contains no logarithm, meaning the same Jordan blocks act differently in different sectors of the c=0 theories.

Reading between the lines

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

  • If the claim is right, a direct lattice measurement of the connected four-energy correlation in critical percolation should see a nonzero long-range term, with the sharp target coefficient -1875/16384 in the leading s-channel term, something not previously looked for.
  • The mechanism suggests a general principle for c=0 bulk CFTs: any zero-norm Kac operator that logarithmically mixes with a hull-type operator can acquire long-range higher-point correlations through higher-rank Jordan blocks, even when its two- and three-point bottom couplings vanish.
  • The Liouville comparison implies that the order of zeros in Kac-operator norms may be a universal function of c; if so, the proposed rank-4 and rank-5 blocks could be tested independently by loop-model transfer-matrix or bootstrap methods, without waiting for lattice four-energy data.
  • A probabilistic construction of percolation correlations could give a geometric explanation of why the energy four-point function is non-logarithmic at leading order and why it is built through the rank-3 block rather than directly through bond correlations.
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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

4 major / 5 minor

Summary. The paper studies logarithmic operators in c=0 bulk CFTs describing percolation and self-avoiding walks. It first revisits the known rank-2 and rank-3 Jordan blocks involving T and T-bar-T, then proposes proper normalizations for Kac operators in cluster and loop models at generic c, using analytic bootstrap amplitudes and the requirement that three-point constants be real and finite. At c=0 the Kac operators acquire zero norms, and the paper constructs logarithmic partners and computes c=0 conformal data, including the logarithmic couplings b=-5 and a=-25/48, the three-point constant C^perco_εεΨ1=-125/512, and related OPE coefficients. The central claim, stated in Section 5.3 and summarized in the abstract, is that the four-point function of the bulk energy operator does not vanish at c=0, with the leading s-channel term given in eq. (5.55) as (C^perco_εεΨ1)^2/a (z zbar)^2, built through coupling to the middle field Ψ1 of the rank-3 Jordan block. The paper closes with a discussion of the puzzling physical interpretation and suggestions for future checks.

Significance. If the central claim is correct, the paper establishes an intriguing mechanism by which zero-norm Kac operators can generate long-range higher-point correlations through logarithmic Jordan-block structures, overturning the earlier belief that such correlators vanish at c=0. The work contains substantial strengths: the operator normalizations are deduced from parameter-free amplitude recursions of the analytic bootstrap; the c=0 data are obtained by well-defined c→0 limits with many internal consistency checks; the known values b=-5 and a=-25/48 are recovered; and the results satisfy conformal Ward identities as summarized in Appendix D. The comparison with c<1 Liouville CFT normalizations in Section 3.4 is also suggestive and well presented. However, the headline four-point result is explicitly conditional on an unproven cluster-decomposition assumption at c=0, and it is in direct tension with the vanishing O(1) term of the direct c→0 limit of the BPZ four-point function.

major comments (4)
  1. [5.3 (eqs. (5.50)-(5.55))] The central non-vanishing four-point function is assembled by inserting a complete set of intermediate states {ψ} in (5.51) and inverting the Gram matrix. This requires cluster decomposition and completeness of that specific set at c=0, but the only justification offered is the sentence before (5.50) that all conformal dimensions are positive. Positivity of dimensions does not imply cluster decomposition in a non-unitary logarithmic theory with zero-norm states and Jordan blocks, and the finite list in (5.51) is asserted rather than derived; other states with dimensions up to (2,2) could contribute to the O(z zbar)^2 term. The load-bearing character of this assumption is underscored by the direct c→0 limit of the BPZ four-point function, eqs. (5.42)-(5.49), which vanishes at O(1) and leaves only O(c) logarithmic terms. The nonzero O(1) result (5.55) is therefore not obtained as a limit of the generic-c correlator; it depends on an additional assumption about the exact c=0 theory. Please either prove cluster decomposition and the completeness of (5.51) in this logarithmic setting, or provide an independent lattice or probabilistic computation of the four-energy correlator or of the coupling ⟨εεΨ1⟩ that confirms (5.55).
  2. [4.2 (eq. (4.13))] The central coefficient C^perco_εεΨ1=-125/512 and hence the four-point amplitude (5.55) depend on the normalization choice B'_ε=1/2 in eq. (4.13). This choice is motivated by analogy with the stress tensor, but the energy operator is not the stress tensor and no symmetry forces this value. The comparison with [6] in Section 4.2.1 fixes some product of normalization constants, but the paper should state explicitly that the non-vanishing four-point function is contingent on the lattice-derived normalization of the energy operator. A different positive normalization would rescale the four-point function, and a negative B'_ε, which is not excluded by the reality arguments in Section 3.1, could change its sign and potentially the conclusion. Please derive the normalization from an independent physical definition of the energy operator in the cluster/loop model, or clearly delimit the dependence of (5.55) on this choice.
  3. [5.2 (eqs. (5.10)-(5.11))] The derivation of the three-point coupling C^perco_εεΨ1 uses the asymptotic expansion C_{Φ2,1Φ2,1Φ3,1}(c) ≃ h_ε^2 c/2 + ... (footnote 18) and similar singularity-cancellation conditions in Appendix B.2. These expansion coefficients are load-bearing inputs: if the O(c) term or the subleading coefficient differs, the finite three-point constant (5.11) changes and the four-point result (5.55) changes with it. The paper states only that these conditions 'can be checked,' without displaying the expansions or their derivations. Please present the explicit c-expansions and the verification of conditions (B.20)-(B.21) in the main text or appendix.
  4. [5.3 (eqs. (5.56)-(5.57))] The reconciliation with the BPZ limit is interpretive rather than derivational. The paper argues that the generic-c Kac operator 'only knows about' the intermediate rank-2 block (Θ, Φ̂3,1) and not about Ψ1, but if the exact c=0 theory is a limit of the generic-c family, the four-point function should equal the limit (5.49) unless the c→0 limit and the cluster-decomposition sum fail to commute. The paper does not prove such non-commutation, nor does it define the exact c=0 theory by an independent set of axioms that would make the enlarged intermediate-state space (5.51) self-consistent. Please make this logical step explicit and provide a concrete definition of the c=0 theory under which (5.50) is the correct expansion.
minor comments (5)
  1. [Section 6] There is a typo in the Conclusions: 'energy operataor' should be 'energy operator'; the manuscript would benefit from a careful proofread.
  2. [Eq. (5.27)] The displayed expression for C^{SAW}_{εεε} contains the ambiguous factor 'Γ(1/6)^{7/2}'; please clarify the placement of the exponent and the parentheses.
  3. [Eqs. (2.38), (4.47), (5.49)] The notation B^{(2)}_{Φ3,1}, B^{(3)}_{Φ3,1} is used without a uniform definition; please define these as the coefficients of (c-c*)^2 and (c-c*)^3 in the expansion of the two-point constant B_{Φ3,1}(c).
  4. [After eq. (2.48)] The sentence says 'there are five fields with dimensions (2,2)' and lists ∂^2 t̄, ∂̄^2 t, Ψ2, Ψ1, Ψ0, but the preceding discussion refers to four fields in the rank-3 block; the counting deserves a clearer statement.
  5. [Section 3.4] The comparison with c<1 Liouville CFT is used to suggest arbitrarily high-rank Jordan blocks, but the preceding analysis in Section 3.2.1 leaves the order of zeros for Φ4,1 and Φ5,1 unresolved; please state explicitly which higher-rank conclusions are conjectural and which are established.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the c=0 conformal data are obtained by parameter-free c→0 limits of analytic bootstrap data, and the four-point function is assembled from independently computed three-point data under an explicit cluster-decomposition assumption.

full rationale

The paper's normalizations of Kac operators are fixed at generic c from the analytic amplitude recursions (eqs. 3.12, 3.13, 3.27, 3.28) together with the reality condition on three-point constants; they are not fitted to the target four-point function. The rank-3 Jordan block construction in §2.2.2 derives the logarithmic couplings b and a from conformal Ward identities and Kac dimensions (eqs. 2.21, 2.40), with the singularity cancellation conditions checked in Appendix B. The three-point constant C^{perco}_{εεΨ1} in eq. (5.11) follows from the generic-c OPE coefficient h²_{2,1} of TT̄ and the 1/c term in the definition of Ψ1 (eqs. 2.41, 5.60); it is not chosen to reproduce eq. (5.55). The final four-point expression (5.55) is the standard cluster-decomposition sum over intermediate states (5.50)–(5.54), and its coefficient (C^{perco}_{εεΨ1})²/a is a consequence of the previously fixed three-point data, not an input. The only non-derivative premise is the explicit assumption that cluster decomposition survives at c=0, stated in §5.3 and in the Conclusions ('we assume that cluster decomposition holds at c=0, also reasonable since all conformal dimensions are positive'); an assumption of this kind is a physical input and does not make the derivation circular, though it is a legitimate correctness risk. Citations to the author's earlier work [15,20,21] supply analytic bootstrap data and the earlier discovery of the rank-3 Jordan block, but the present paper re-derives the Jordan block construction and checks the required cancellation conditions, so the citation chain is not load-bearing in a circular way. No step reduces to an equivalent equation by construction, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the normalization scheme, which introduces a small number of hand-chosen constants, and on the assumed validity of cluster decomposition and the real-CFT principle. No new particles or forces are postulated. The rank-2 and rank-3 fields are constructed from existing operators, not invented entities.

free parameters (3)
  • B'_epsilon (coefficient of linear term in energy operator two-point function near c=0) = 1/2
    Chosen by hand in eq. (4.13), motivated by the chiral stress-tensor normalization. Fixes the overall scale of the energy operator; affects specific values of computed three-point constants but not the qualitative non-vanishing of the four-point function.
  • mu (scale in logarithmic operators) = 1
    The logarithm ln(mu^2 z z-bar) is set to mu=1 in eqs. (2.17), (4.3), and footnote 13. Conventional choice affecting only basis-dependent constants such as theta and theta_1, not the intrinsic couplings b, a, C_eps_eps_t, C_eps_eps_Psi1.
  • B_Phi0,2 normalization constant = 5 sqrt(3)/(8 pi) (Q-1)(Q-2)
    Tentative guess in eq. (4.34) to satisfy the cancellation condition (4.30); not used for the central non-vanishing four-point result.
assumptions (4)
  • domain assumption Cluster and loop model CFTs are real CFTs, so three-point constants are real and C^2 >= 0.
    Invoked in Section 3.1 to fix the signs and zeros of operator norms from the known amplitudes; based on the microscopic real positive measure of the lattice models.
  • domain assumption Cluster decomposition holds at c=0.
    Stated in Section 5.3 before eq. (5.50); used to assemble the four-point function from three-point data. All operator dimensions are positive, so the argument is plausible but not proven.
  • ad hoc to paper The c->0 limit of properly normalized generic-c operators gives the complete c=0 logarithmic CFT spectrum.
    Assumed throughout Sections 2 and 4; this is the proposed resolution scheme for the c->0 catastrophe, not independently established.
  • standard math Analytic amplitude recursions and conformal data from prior bootstrap work (e.g., refs. [15,16,17,18]) are correct.
    Used as input without re-derivation; these are published analytic results based on Virasoro degeneracy and crossing symmetry.

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Cite this review

Pith. "Pith review of Logarithmic operators in $c=0$ bulk CFTs." pith.science (2026). https://pith.science/paper/GQMJGT66

@misc{pith2026241118696,
  author       = {Pith},
  title        = {Pith review of: Logarithmic operators in $c=0$ bulk CFTs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GQMJGT66}},
  note         = {Machine review of arXiv:2411.18696}
}
abstract

We study Kac operators (e.g. energy operator) in percolation and self-avoiding walk bulk CFTs with central charge $c=0$. The proper normalizations of these operators can be deduced at generic $c$ by requiring the finiteness and reality of the three-point constants in cluster and loop model CFTs. At $c=0$, Kac operators become zero-norm states and the bottom fields of logarithmic multiplets, and comparison with $c<1$ Liouville CFT suggests the potential existence of arbitrarily high rank Jordan blocks. We give a generic construction of logarithmic operators based on Kac operators and focus on the rank-2 pair of the energy operator mixing with the hull operator. By taking the $c\to 0$ limit, we compute some of their conformal data and use this to investigate the operator algebra at $c=0$. Based on cluster decomposition, we find that, contrary to previous belief, the four-point correlation function of the bulk energy operator does not vanish at $c=0$, and a crucial role is played by its coupling to the rank-3 Jordan block associated with the second energy operator. This reveals the intriguing way zero-norm operators build long-range higher-point correlations through the intricate logarithmic structures in $c=0$ bulk CFTs.

Figures

Figures reproduced from arXiv: 2411.18696 by the authors.

Figure 1
Figure 1. The geometrical configurations described by four-spin correlator in cluster and loop models where [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Illustrations of the 2-hull or four-leg operator in cluster model (left) and hull or two-leg operator [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. The amplitude A σ,cluster Φ2,1 of the energy operator Φ2,1 in cluster model four-spin correlator ⟨σσσσ⟩ cluster as a function of 0 ≤ β 2 ≤ 1. The amplitude develops a singularity at β 2 = 2 3 and changes sign. by examining the crossing equations of several four-point functions that involve Φ2,1 or Φ1,2 [22, 23, 24, 14]. We briefly summarize some of these results in appendix A. In the special case where the O is the … view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: The deduced norm (eq. (3.18a)) of the operator [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 5
Figure 5. Figure 5: The deduced norm (eq. (3.18b)) of the operator [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: The deduced norms (eqs. (3.35) and (3.37)) of the energy operator [PITH_FULL_IMAGE:figures/full_fig_p023_6.png]
Figure 7
Figure 7. Figure 7: Normalizations BL(P) of diagonal operators in c < 1 Liouville CFTs with momenta P2,1, P3,1, P1,3, P1,5 – to be compared with the normalizations of Kac operators Φˆ 2,1, Φˆ 3,1, Φˆ 1,3, Φˆ 1,5 in cluster and loop models. 4 Constructing logarithmic operators We have seen…
Figure 8
Figure 8. Figure 8: Normalization of operator with momentum P4,1 in c < 1 Liouville CFT. The norm has a third order zero at β 2 = 2 3 , i.e. c = 0. with B ′ ϕ = −B ′ ψ , (4.2) where B′ indicates the derivative of the norm B(c) evaluated at c = c∗. Here we have used P ∗ 2 to denote the exp…
Figure 9
Figure 9. Figure 9: Normalization of operator with momentum P5,1 in c < 1 Liouville CFT. The norm has a fourth order zero at β 2 = 2 3 , i.e. c = 0. Note that the logarithmic coupling is completely determined by the normalizations and the conformal dimensions as functions of the parameter…
Figure 10
Figure 10. Figure 10: The amplitudes of the 2-hull operator Φ0,2 in the connectivities Paabb, Paaaa, Pabab. Plots are extracted from [15]. and the s-channel limit of the percolation cluster four-spin correlator:15 ⟨σ(∞)σ(1)σ(z, z¯)σ(0)⟩ perco =(zz¯) −2hσ  1 + (zz¯) hε C 2 σσε γ 2 [PITH_F…
Figure 11
Figure 11. Figure 11: The deduced norm (4.34) and the resulting three-point constant squared for the 2-hull operator [PITH_FULL_IMAGE:figures/full_fig_p030_11.png]

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Reviewed August 12, 2026 · model on record in the stance chip above.