The $\psi$–$\psi$ sector of height correlations and the local $c=-2$ logarithmic conformal field theory of the two-dimensional Abelian sandpile model

Jeng, Piroux and Ruelle (2006) identified the scaling fields of the four height variables of the two-dimensional Abelian sandpile model (ASM) with a primary field $\phi$ and its logarithmic partner $\psi$ in a $c=-2$ logarithmic conformal field theory (LCFT), reduced the two-site bulk height correlations to three constants $A$, $B$, $C$, and left six explicit open problems. Later literature has resolved part of this program: the conjectured closed forms of the one-site probabilities $P_2$ and $P_3$ (equivalently the integral value $J_2=\tfrac12$ and the mean height $\langle h\rangle=\tfrac{25}8$) are now proved, and the constant $B$ has been computed exactly. What remains open, and is the subject of this problem, is the completion of the exact correlation structure and the identification of the continuum theory: the exact lattice computation of two-site correlations in which both heights are at least 2 — in particular the last undetermined constant $C$ — and of multisite joint probabilities with two or more non-unit heights at macroscopic separations; and the characterization of the local $c=-2$ LCFT realized by the ASM, namely its non-chiral module with indecomposability parameter $\beta=-\tfrac14$ (distinct from the symplectic-fermion value $\tfrac12$), its operator algebra and extended symmetries, and a decision on whether an explicit local field realization exists, together with a microscopic explanation of $\beta$ and of the linear relations that collapse the four height fields onto $\phi$ and $\psi$.

Background

The Abelian sandpile model (ASM) is the archetypal model of self-organized criticality: grains are added one by one on a square grid, any site whose height exceeds 4 topples by sending one grain to each nearest neighbour, and grains leave the system only through open (dissipative) boundary sites, so that the stationary state is critical without any tuned parameter. Stable heights take the values $h\in\{1,2,3,4\}$, and the stationary measure on recurrent (i.e. dynamically reachable) configurations maps, via the burning algorithm (an iterative burning test that characterizes recurrent configurations), onto the uniform measure on spanning trees rooted at the dissipative boundary; the height at a site is determined by how many of its four neighbours precede it on the unique tree path to the root. This mapping made the bulk one-site height probabilities $P_a$ exactly computable: $P_1=2(\pi-2)/\pi^3\approx0.0736$, while $P_2$ and $P_3$ were long expressible only through multiple Brillouin-zone integrals, with the closed forms $P_2=\tfrac14-\tfrac{1}{2\pi}-\tfrac{3}{\pi^2}+\tfrac{12}{\pi^3}\approx0.1739$ and $P_3=\tfrac38+\tfrac1\pi-\tfrac{12}{\pi^3}\approx0.3063$ following once one integral, $J_2$, takes the value $\tfrac12$ — a value verified numerically to $10^{-12}$ but unproven in 2006.

Jeng, Piroux and Ruelle (JPR, 2006) computed the height probabilities in the upper half-plane exactly to order $m^{-2}$ ($m$ being the distance to a boundary that is fully open or fully closed) and matched them to a logarithmic conformal field theory of central charge $c=-2$. A conformal field theory is a two-dimensional critical model described by fields carrying conformal weights $(h,\bar h)$; a primary field transforms covariantly under conformal maps, and its two-point functions decay as a power fixed by the weights. A logarithmic CFT is a non-unitary conformal field theory whose symmetry algebras (two Virasoro copies on the plane) act through reducible but indecomposable representations, so that correlation functions contain logarithms; the canonical local realization at $c=-2$ is the symplectic fermion theory — two free anticommuting scalar fields whose bosonic (vacuum) sector is a local theory, together with its extended chiral triplet algebra. JPR's matching yields the field identification $h_a\to\alpha_a\psi+\beta_a\phi$ with $\alpha_1=0$, $\alpha_2=1$, $\alpha_3=\tfrac{8-\pi}{2(\pi-2)}\approx2.1279$, $\alpha_4=-\tfrac{\pi+4}{2(\pi-2)}\approx-3.1279$, where $\phi$ (the height-1 field) is a primary of weights $(1,1)$ and $\psi$ is its logarithmic partner: the four lattice height fields collapse onto two independent scaling fields (JPR relations (8.13),(8.14)), an observation with no microscopic explanation. Since the infinite plane has no boundary at which sand can dissipate, bulk correlators require the insertion at infinity of a dissipation field $\omega$, the logarithmic partner of the identity, whose lattice meaning is the removal of sand.

Within this framework the two-site height correlations on the plane are predicted to be

$$\sigma_{a,b}(r)=P_{a,b}(r)-P_aP_b=\frac{1}{r^4}\Big[\alpha_a\alpha_b C+(\alpha_a\beta_b+\beta_a\alpha_b)B+\beta_a\beta_b A+\big(2\alpha_a\alpha_b B+(\alpha_a\beta_b+\beta_a\alpha_b)A\big)\log r+\alpha_a\alpha_b A\log^2 r\Big]+\cdots$$

(JPR Eq. (12.15)), with three constants: $A=-P_1^2/2=-2(\pi-2)^2/\pi^6$ is fixed by the height-1 two-point function, while $B$ and $C$ cannot be determined within conformal field theory. The constant $C$ multiplies the $\psi$–$\psi$ sector and can only be measured from joint probabilities in which both sites carry heights $\ge2$, which no lattice method could compute. JPR further showed that the representation carrying $(\phi,\psi)$ is characterized by an indecomposability parameter $\beta$ (the coefficient relating a level-one descendant of a pre-logarithmic field to $\phi$; inequivalent values of $\beta$ label inequivalent Virasoro representations), with the sandpile value $\beta_{\rm ASM}=-\tfrac14$ fixed only by requiring regularity of a four-point function as two boundary-condition-changing fields $\mu$ (chiral fields switching an open boundary segment to a closed one) are sent to infinity — a value inferred, not derived, and different from the symplectic-fermion value $\beta=\tfrac12$, so the sandpile theory is not the symplectic fermion theory.

Two of JPR's open problems have since been settled: the closed forms of $P_2$, $P_3$ and the mean height $\langle h\rangle=\tfrac{25}8$ are now rigorous (2011–2012, via the return probability $\tfrac{5}{16}$ of the loop-erased random walk and via direct analytic integration), and $B$ has been computed exactly (2010). What remains open is the $\psi$–$\psi$ sector — the constant $C$ and the joint probabilities of several non-unit heights at macroscopic separations — and the identification of the local $c=-2$ logarithmic theory realized by the sandpile, including a microscopic account of $\beta=-\tfrac14$ and of the collapse of the four height fields onto $\phi$ and $\psi$.

Problem Statement

Complete the exact solution of the height-variable correlations of the two-dimensional Abelian sandpile model on the square lattice, and identify the logarithmic conformal field theory of its scaling limit. Concretely, of the open problems left by Jeng, Piroux and Ruelle (2006) — after the subsequent rigorous resolution of the one-site closed forms $P_2$, $P_3$, $J_2=\tfrac12$ and the mean height $\tfrac{25}8$, and the exact determination of the constant $B$ — the following remain open and form one linked program.

Both parts (1) and (2) are mandatory, and neither alone resolves the task: determining the constant $C$, or the joint probabilities $P_{a,b}(r)$, without the module characterization of part (2) — or characterizing that module without the lattice correlators of part (1) — is a partial result, not a resolution. Within each nested chain the strongest demand governs: the exact two-site probabilities $P_{a,b}(r)$ of part (1) subsume the single constant $C$, and the multisite probabilities subsume the two-site ones. Part (2) is likewise mandatory in all four of its listed demands — the non-chiral module with all its indecomposability parameters, the local operator algebra and extended symmetries, the decision on whether an explicit local field realization exists, and the microscopic or algebraic derivation of $\beta$ and of the relations (8.13),(8.14) — and either answer to that existence decision resolves it: a negative answer settles that no such realization exists and leaves no construction owed.

(1) Exact lattice computation of the two-site correlations $\sigma_{a,b}(r)=P_{a,b}(r)-P_aP_b$ in which both heights satisfy $a,b\ge2$, at separation $r\gg1$: in particular the determination, in closed form, of the last undetermined constant $C$ in the two-site formula (JPR Eq. (12.15))

$$\sigma_{a,b}(r)=\frac{1}{r^4}\Big[\alpha_a\alpha_b C+(\alpha_a\beta_b+\beta_a\alpha_b)B+\beta_a\beta_b A+\big(2\alpha_a\alpha_b B+(\alpha_a\beta_b+\beta_a\alpha_b)A\big)\log r+\alpha_a\alpha_b A\log^2 r\Big]+\cdots,$$

together with the joint probabilities $P_{a,b}(r)$ themselves (including subleading orders); and, beyond two sites, the exact computation of multisite joint probabilities involving two or more non-unit heights at macroscopic separations, established as exact lattice results rather than conformal-field-theory predictions or numerical fits, with no specific method prescribed.

(2) Characterization of the local $c=-2$ logarithmic conformal field theory realized by the ASM: the identification of the non-chiral module containing the identity, the dissipation field $\omega$ and the Jordan pair $(\phi,\psi)$, including all its indecomposability parameters (among them the parameter $\beta$, with the sandpile value $-\tfrac14$ in the JPR normalization, distinct from the symplectic-fermion value $\tfrac12$); the local operator algebra and any extended symmetries; a decision whether an explicit local field realization exists distinct from the symplectic-fermion/triplet construction; and a microscopic or algebraic derivation — rather than an inference from the regularity of a four-point function — of the value of $\beta$ and of the linear relations by which the four lattice height fields reduce to the two independent scaling fields $\phi$ and $\psi$ (JPR relations (8.13),(8.14)), which are expected to hold in any geometry and for any boundary condition.

The two parts are scientifically linked: the missing lattice correlators of part (1) are precisely the data needed to pin down the non-chiral module of part (2), and the joint objective is a complete match between all exact height correlations of the sandpile and a local logarithmic conformal theory.

References

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