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arxiv: 1907.05499 · v1 · pith:NJWJNUFHnew · submitted 2019-07-11 · ❄️ cond-mat.stat-mech · hep-th· math-ph· math.MP

Logarithmic correlation functions for critical dense polymers on the cylinder

Alexi Morin-Duchesne , Jesper Lykke Jacobsen This is my paper

Pith reviewed 2026-05-24 22:29 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech hep-thmath-phmath.MP
keywords critical dense polymerslogarithmic correlation functionscylinderTemperley-Lieb algebraconformal field theorycentral charge -2boundary condition changing fieldsnon-abelian fusion
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The pith

Lattice correlation functions for critical dense polymers on a cylinder match ratios of conformal correlators at central charge c=-2.

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

The paper computes explicit expressions for correlation functions in the critical dense polymer model on a semi-infinite cylinder of perimeter n, defined as ratios of partition functions with modified boundary conditions at two points. These are obtained using the XX spin chain representation of the Temperley-Lieb algebra and yield asymptotics involving powers and logarithms of the scaled distance τ as n goes to infinity. The authors then interpret these as coming from a conformal field theory with c=-2, where boundary condition changing fields have dimensions -1/8 or 0. Solving the resulting differential equations yields perfect agreement with the lattice results, allowing computation of structure constants in the operator product expansions.

Core claim

We find explicit expressions for these correlators for finite n using the representation of the enlarged periodic Temperley-Lieb algebra in the XX spin chain. The resulting asymptotics as n→∞ are expressed as simple integrals that depend on the parameter τ=(x-1)/n ∈(0,1). For small τ, the leading behaviours are proportional to τ^{1/4}, τ^{1/4} log τ, log τ and log²τ. We interpret the lattice results in terms of ratios of conformal correlation functions. We assume that the corresponding boundary changing fields are highest weight states in irreducible, Kac or staggered Virasoro modules, with central charge c=-2 and conformal dimensions Δ=-1/8 or Δ=0. We obtain differential equations satisfied

What carries the argument

Boundary condition changing fields in Virasoro modules with c=-2 and dimensions Δ=-1/8 or 0, whose correlators satisfy differential equations that match the lattice asymptotics.

If this is right

  • The leading small-τ behaviors of the correlators are τ to the power 1/4, τ^{1/4} log τ, log τ, and log squared τ.
  • Structure constants appearing in the operator product expansions can be computed from the solutions of the differential equations.
  • The fusion of the boundary condition changing fields is non-abelian.
  • Ratios of these structure constants are determined explicitly.

Where Pith is reading between the lines

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

  • If the agreement holds, similar lattice-to-CFT mappings could apply to other values of the loop fugacity or different boundary conditions.
  • The non-abelian fusion suggests that the operator algebra may have a richer structure than in rational CFTs, potentially affecting higher-point functions.
  • Extensions to the full cylinder or other topologies might reveal additional logarithmic terms in the correlators.

Load-bearing premise

The boundary changing fields correspond to highest weight states in irreducible, Kac or staggered Virasoro modules with central charge c=-2 and conformal dimensions -1/8 or 0.

What would settle it

A mismatch between the small-τ asymptotic expansions derived from the lattice partition function ratios and those obtained by solving the differential equations for the conformal correlators would falsify the interpretation.

Figures

Figures reproduced from arXiv: 1907.05499 by Alexi Morin-Duchesne, Jesper Lykke Jacobsen.

Figure 1
Figure 1. Figure 1: Loop configurations on the 10 × 8 cylinder, with the boundary conditions corresponding to Z0, Z (x = 6) and Z (x = 7). We denote by Z0 the partition function for the model where the lower segment is identical to the top segment of the cylinder and is therefore exclusively decorated with simple arcs. We consider α ∈ (0,∞) for which Z0 6= 0. Likewise, we denote by Z (x) the partition function wherein the bot… view at source ↗
Figure 2
Figure 2. Figure 2: The correlation function C (x) for φ = 1. The points are the exact values for n = 200. The blue, yellow and purple solid curves are respectively drawn from (3.28), (3.29) and (3.33). where K = 2π Z 1/2 0 dy  cos φ(y − 1 2 )  sin πy − cos(φ 2 ) πy  . (3.34) Recalling from (3.9) that C (x) = C (n + 2 − x), the final results for the asymptotics of these correlation functions for 1 ≪ x ≪ n are C (x) 1≪x≪n −… view at source ↗
Figure 2
Figure 2. Figure 2: The correlation functions C (x) and C (x) for φ = 1. The points are the exact values for n = 200. In the left panel, the blue, orange and purple solid curves are drawn from (4.23), (4.26a) and (4.26b). In the right panel, the blue and orange curves are drawn from (4.28) and (4.29). In the limit τ → 0 +, this yields C (x) [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The map z(x) from the semi-infinite cylinder V to the upper half-plane H. 5.2 Differential equation for the four-point function We consider the correlation functions appearing in (5.1), but defined on H. Using the method of images, the correlation functions on H are equal to correlation functions in the full complex plane: hφ(z1)φ(z2)ψα(z, z∗ )iH = hφ(z1)φ(z2)ψα(z)ψα(z ∗ )iC, hψα(z, z∗ )iH = hψα(z)ψα(z ∗ )… view at source ↗
Figure 4
Figure 4. Figure 4: The Loewy diagrams for the modules V, I, K, M and S. 6.3 Differential equation for the four-point function The correlators on the right sides of (6.1) involve one non-chiral field, which we rewrite as the product of two chiral fields using the method of images. We proceed to compute the correlator G = hω1(z1)ω2(z2)ψα(z3, z4)iH = hω1(z1)ω2(z2)ψα(z3)ψα(z4)iC. (6.7) We work under the hypothesis that ω1(z) and… view at source ↗
read the original abstract

We compute lattice correlation functions for the model of critical dense polymers on a semi-infinite cylinder of perimeter $n$. In the lattice loop model, contractible loops have a vanishing fugacity whereas non-contractible loops have a fugacity $\alpha\in(0,\infty)$. These correlators are defined as ratios $Z(x)/Z_0$ of partition functions, where $Z_0$ is a reference partition function wherein only simple arcs are attached to the boundary of the cylinder. For $Z(x)$, the boundary is also decorated with simple arcs, but it also has two positions $1$ and $x$ where the boundary condition is different. We investigate two such kinds of boundary conditions: (i) there is a single node at each of these points where a long arc is attached, and (ii) there are pairs of adjacent nodes at these points where two long arcs are attached. We find explicit expressions for these correlators for finite $n$ using the representation of the enlarged periodic Temperley-Lieb algebra in the XX spin chain. The resulting asymptotics as $n\to\infty$ are expressed as simple integrals that depend on the parameter $\tau=\frac{x-1}n\in(0,1)$. For small $\tau$, the leading behaviours are proportional to $\tau^{1/4}$, $\tau^{1/4}\log \tau$, $\log\tau$ and $\log^2\tau$. We interpret the lattice results in terms of ratios of conformal correlation functions. We assume that the corresponding boundary changing fields are highest weight states in irreducible, Kac or staggered Virasoro modules, with central charge $c=-2$ and conformal dimensions $\Delta = -\frac18$ or $\Delta=0$. We obtain differential equations satisfied by the conformal correlators, solve these equations, and find a perfect agreement with the lattice results. We compute structure constants and ratios thereof which appear in the operator product expansions of the boundary condition changing fields. The fusion of these fields is found to be non-abelian.

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

0 major / 2 minor

Summary. The manuscript computes explicit expressions for correlation functions in the critical dense polymer model on a cylinder using the XX spin chain representation of the enlarged periodic Temperley-Lieb algebra. The n→∞ asymptotics are given as integrals over τ=(x-1)/n, with small-τ expansions showing behaviors proportional to τ^{1/4}, τ^{1/4} log τ, log τ and log²τ. These are matched to ratios of conformal correlators at c=-2 by assuming the boundary changing fields are highest-weight states in irreducible, Kac or staggered Virasoro modules with Δ=-1/8 or Δ=0, deriving and solving the corresponding null-vector differential equations, and reporting perfect agreement. Structure constants and their ratios are computed from the OPEs, and the fusion of the fields is concluded to be non-abelian.

Significance. If the reported matching holds, the work supplies a non-circular, lattice-derived verification of logarithmic correlators and fusion rules in c=-2 LCFT. The lattice side proceeds from an independent XX-chain representation and explicit integrals without CFT input, while the CFT side solves the differential equations under the stated module assumptions; the agreement therefore constitutes a genuine test of the operator identification. Explicit structure constants are obtained, providing concrete, falsifiable numbers for further checks in polymer models and LCFT.

minor comments (2)
  1. The abstract states that the asymptotics 'are expressed as simple integrals' but does not display the explicit integral expressions; including them in the main text (with the precise measure and limits) would improve reproducibility.
  2. A short table or side-by-side comparison of the numerical coefficients extracted from the lattice integrals and those obtained from the CFT solutions would make the 'perfect agreement' claim easier to inspect at a glance.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the results, and recommendation to accept. We are pleased that the work is viewed as providing a genuine, non-circular test of logarithmic correlators and fusion rules in c=-2 LCFT.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper first derives explicit finite-n correlators and their n→∞ asymptotic integrals directly from the XX spin-chain representation of the enlarged periodic Temperley-Lieb algebra, without reference to CFT or module assumptions. It then separately assumes the boundary fields lie in standard c=-2 Virasoro modules (Kac or staggered) with given Δ values, derives the corresponding null-vector differential equations, solves them, and reports agreement with the already-computed lattice integrals. Because the lattice side is self-contained and independent of the CFT assumptions, the match constitutes an external test rather than a reduction of any claimed derivation to its own inputs. No self-citations are invoked as load-bearing uniqueness theorems, and no fitted parameters are relabeled as predictions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the identification of lattice boundary conditions with specific Virasoro modules; no numerical free parameters are introduced.

axioms (1)
  • domain assumption Boundary changing fields are highest weight states in irreducible, Kac or staggered Virasoro modules with c=-2 and Δ=-1/8 or Δ=0
    Explicitly stated in the abstract as the assumption required to interpret the lattice results in CFT.

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