Boundary conformal data of the dilute $A_2^{(2)}$ loop model at the percolation point: integrable boundary conditions beyond the sine solution and the missing weight $5/96$

Critical site percolation on the triangular lattice is realized exactly by the Yang-Baxter-integrable dilute $A_2^{(2)}$ loop model at crossing parameter $\lambda=\pi/3$, with contractible-loop fugacity $\beta=1$ and central charge $c=0$. Morin-Duchesne, Klümper and Pearce solved this model in the scaling limit by nonlinear integral equations and derived its conformal cylinder and torus partition functions, but only for a restricted strip boundary condition: the sine ('choice 1') solution of the boundary Yang-Baxter equation, taken with equal boundary Boltzmann weights. With this restriction the conformal weights appearing in the boundary partition functions do not exhaust the infinitely extended Kac table; in particular the magnetic weight $\Delta_{1/2,0}=5/96$ never appears. The open problem is to complete the boundary conformal data of this exactly solved model: extend the integrable finite-size analysis to the second (cosine) boundary Yang-Baxter solution in all four left/right combinations, and to further integrable boundary tile choices of the Kac-/Robin type known from the $s\ell(2)$ logarithmic minimal models (the $A_1$-family analogs of this $A_2^{(2)}$ model); compute the resulting conformal weights, boundary characters and strip conformal partition functions; and establish which additional Kac-table weights arise, in particular whether $\Delta_{1/2,0}=5/96$ is realized, thereby completing the classification of integrable boundary conditions and their boundary conformal data. No resolution of this boundary classification is known in the literature to date.

Background

Two-dimensional percolation, in which each site of a lattice is occupied independently with probability $p$, has an exact critical point $p_c=1/2$ on the triangular lattice. Its continuum limit is the archetypal logarithmic conformal field theory (LCFT): a conformal field theory (CFT), i.e. a scale-invariant field theory organized by the Virasoro algebra of local conformal symmetries, whose central charge $c$ vanishes; a rational CFT with $c=0$ would be trivial, so the nontrivial percolation continuum limit is non-rational, with representations that are reducible but indecomposable — the defining feature of logarithmic CFT. Local scaling operators carry conformal weights given by the Kac formula

$$\Delta_{r,s}=\frac{(3r-2s)^2-1}{24}$$

with labels $(r,s)$ ranging over an infinitely extended Kac table that includes half-integer values. The thermal weight, coupling to the deviation of $p$ from $p_c$, is $\Delta_t=\Delta_{2,1}=\Delta_{0,2}=5/8$, and the magnetic weight, coupling to an infinitesimal symmetry-breaking field, is $\Delta_h=\Delta_{1/2,0}=5/96$; all standard percolation critical exponents follow from these two weights.

Site percolation on the triangular lattice admits an exactly solved lattice realization: the dilute $A_2^{(2)}$ loop model, a Yang-Baxter-integrable model in which each face of a square lattice carries one of nine tiles bearing loop segments. 'Yang-Baxter integrable' means the local Boltzmann weights satisfy the Yang-Baxter equation, so that row-to-row transfer matrices at different spectral parameters $u$ commute, yielding infinitely many conserved quantities and, in principle, exact access to the scaling limit. The crossing parameter $\lambda$ specializes to $\lambda=\pi/3$, at which the fugacity of a contractible loop is $\beta=-2\cos 4\lambda=1$, so loops reweight clusters exactly as in site percolation. The transfer matrices are elements of the dilute Temperley-Lieb algebra, the nine-tile diagram algebra generated by the tile operators, whose standard modules are labeled by the number $d$ of defects (lines running through the system). With periodic boundaries the single-row transfer matrix acts in standard modules $\mathcal{W}_{N,d,\omega}$ carrying a twist $\omega=e^{i\gamma}$; on a strip with open boundaries the double-row transfer matrix acts in the standard modules $\mathcal{V}_{N,d}$, labeled only by the defect number.

Finite-size corrections to transfer-matrix eigenvalues have the universal form bulk term plus $\frac{\pi}{24N}(c-24\Delta)$ (with a boundary free-energy term on the strip), so exact diagonalization or Bethe-ansatz solutions in the scaling limit directly yield the conformal data. Morin-Duchesne, Klümper and Pearce (2023) solved the model at $\lambda=\pi/3$ analytically by nonlinear integral equations (NLIEs), the method of Klümper and Pearce that converts the Bethe-ansatz functional relations into integral equations for auxiliary functions. They obtained the strip ground-state weights $\Delta_{1,d+1}=d(d-1)/6$ for $d=0,1,2,\ldots$, the periodic weights $(\Delta,\bar\Delta)=(\Delta_{\gamma/\pi,d/2},\Delta_{\gamma/\pi,-d/2})$, conjectures for the scaling limit of the trace of the transfer matrix in each standard module, and, via the Markov trace (the natural trace on the Temperley-Lieb algebra that weights loops by their fugacities, implementing cylinder and torus geometries), the conformal cylinder and torus partition functions. These coincide exactly with the corresponding results for critical bond percolation on the square lattice, i.e. the dense $A_1^{(1)}$ loop model solved earlier by the same $D_3$ thermodynamic Bethe ansatz, establishing a strong form of universality between the two lattice realizations as logarithmic CFTs.

On the strip, integrable boundary conditions are built from boundary face weights $\kappa_1(u),\kappa_2(u)$ solving the boundary Yang-Baxter (reflection) equation, which guarantees that the strip transfer matrices commute. Two solutions are known from the work of Yung and Batchelor: 'choice 1' with sine weights $\kappa_1^+=\sin(3\lambda/2-u)$, $\kappa_2^+=\sin(3\lambda/2+u)$, and 'choice 2' with cosine weights; at $\lambda=\pi/3$ these reduce to boundary operators proportional to $\cos u$ and $\sin u$ respectively. The two ends of the strip may each take either solution, giving four combinations (SS, CC, SC, CS), and further boundary tile choices are possible in principle. The 2023 analysis was restricted to choice 1 with equal Boltzmann weights, for which the boundary operator is proportional to the identity strand. With this restriction, the conformal weights arising in the strip partition functions do not exhaust the extended Kac table, the ground-state weights lying on the column $r=1$ of Kac labels; in particular the weight $\Delta_{1/2,0}=5/96$ never appears. By analogy with the $s\ell(2)$ logarithmic minimal models (the $A_1$-family counterparts of this model, whose scaling limits are the logarithmic minimal models $\mathcal{LM}(p,p')$), where Kac and Robin boundary conditions (boundary weights beyond the identity strand, realizing half-integer Kac labels) were shown to enlarge the set of boundary characters, the missing weights are expected from the second boundary Yang-Baxter solution and from further integrable boundary tile choices. The companion paper of Boileau, Morin-Duchesne and Saint-Aubin (2023) constructed the fusion hierarchy, T-system and Y-system (a nested family of fused transfer matrices obeying closed functional relations) for all four sine/cosine boundary combinations at generic $\lambda$, and computed the bulk and boundary free energies; but the conformal weights and boundary characters for boundary conditions beyond choice 1 were left open, and are the subject of this problem.

Problem Statement

For the Yang-Baxter-integrable dilute $A_2^{(2)}$ loop model at the percolation point $\lambda=\pi/3$ (contractible-loop fugacity $\beta=1$, central charge $c=0$), the boundary conformal data have been computed only for the strip boundary condition 'choice 1', the sine solution of the boundary Yang-Baxter equation taken with equal boundary Boltzmann weights, for which the boundary face operator is proportional to the identity strand. The problem posed by Morin-Duchesne, Klümper and Pearce is to extend the integrable analysis to the remaining integrable boundary conditions and thereby complete the boundary conformal data of the model:

1. Carry out the finite-size scaling analysis, by functional relations and nonlinear integral equations or an equivalent integrable method, for the second boundary Yang-Baxter solution (the cosine choice), in all four combinations of sine and cosine boundaries on the two ends of the strip (SS, CC, SC, CS), determining the conformal weights of the ground states in each strip standard module $\mathcal{V}_{N,d}$ and, as far as feasible, the excitation structure.

2. Construct and analyse further integrable boundary tile choices of the type known from the $s\ell(2)$ logarithmic minimal models (Kac-, Robin- and boundary-algebra-type boundary weights), which are expected to realize boundary-condition-changing insertions and half-integer Kac labels.

3. Extract, via the Markov trace, the boundary conformal partition functions (boundary characters) for these boundary conditions, as explicit sesquilinear forms in Virasoro characters, in the same sense in which the choice-1 conformal partition functions were obtained.

4. Determine which conformal weights of the infinitely extended Kac table arise, in particular whether the missing magnetic weight $\Delta_{1/2,0}=5/96$ appears and in which boundary conditions and standard modules, and assemble the results into a classification of the integrable boundary conditions of the model together with their boundary conformal data.

These items jointly serve one objective: completing the boundary conformal data of this exactly solved lattice realization of critical site percolation, so that the boundary sector becomes available for comparison with the periodic sector and with critical bond percolation at the same level of completeness as the already-derived torus and cylinder conformal partition functions. Items 1, 2 and 3 are all required; item 4 records their combined outcome, including whether and where $\Delta_{1/2,0}=5/96$ appears.

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