The close-packed dimer model on the square lattice is exactly solved, yet its continuum description is unsettled in a specific, documented way. Local (height) correlations are governed by a Gaussian theory with central charge c=1c=1 — now derived constructively from the fermionic solution — while the transfer-matrix spectrum, the exact torus partition function, and finite-size corrections consistently match a c=−2c=-2 symplectic-fermion (logarithmic) conformal field theory, and no reconciliation has been published. The problem is to establish the conformal data of the scaling limit — identifying which theory describes it, or how both readings coexist.

Background

The close-packed dimer model is one of the few exactly solved models of statistical mechanics. A configuration places dimers (rods covering two neighboring sites) on the bonds of a lattice so that every site is covered exactly once. On the square lattice the number of configurations and all local correlations are known exactly, either by evaluating the Pfaffian (a determinant-like invariant of antisymmetric matrices) of the Kasteleyn matrix — the signed adjacency matrix of the dimer lattice whose Pfaffian gives the partition function — or through a transfer matrix, the operator mapping one row of the lattice to the next, whose trace gives the partition function and whose diagonalization reduces the two-dimensional classical model to free fermions in one dimension (Ferdinand 1967; Wilkins and Powell 2021). The model is the classical counterpart of the quantum dimer model and the reference example of a Coulomb phase (a phase with algebraically decaying correlations and an emergent height field) for a broad class of constrained lattice models. In two dimensions the dimer model admits a height mapping: a height field hh is defined on the dual lattice so that each dimer contributes a fixed step, and the close-packing constraint forces the coarse-grained height to be a zero-divergence (harmonic) field. Blöte and Hilhorst (1982) established the corresponding roughening description, and the long-wavelength theory is a Gaussian free field with stiffness κ\kappa (the coupling in the action (κ/2)∫∣∇h∣2(\kappa/2)\int|\nabla h|^2), equal to κ=π\kappa=\pi for isotropic square-lattice dimers; this yields the algebraic dimer-dimer correlations with their exact known exponents. Kenyon (2001) proved rigorously that the height field converges to the Gaussian free field. A Gaussian free boson is a conformal field theory with central charge c=1c=1; the central charge is a number characterizing a conformal field theory, extractable from finite-size corrections such as the O(1/L)O(1/L) correction to transfer-matrix ground-state energies or the universal O(1)O(1) term in free energies. A second, apparently conflicting line of evidence comes from the transfer-matrix spectrum and the exact torus partition function. On the torus the configuration space splits into topological sectors labelled by the dimer flux (winding numbers) (Φx,Φy)(\Phi_x,\Phi_y), and Ferdinand's exact partition function takes the theta-function form Z∝∑i=24θi2/(2η2)Z\propto\sum_{i=2}^{4}\theta_i^2/(2\eta^2), where the θi\theta_i are Jacobi theta functions and η\eta is the Dedekind eta function, a standard modular form. Sector-resolved spectral analysis (Rasmussen and Ruelle 2012), finite-size corrections (Nigro 2012; Izmailian, Papoyan and Ziff 2019), and the Temperley-Lieb structure (the diagram algebra generated by the transfer-matrix bond operators; Morin-Duchesne, Rasmussen and Ruelle 2016) all match a c=−2c=-2 conformal theory: symplectic fermions, a fermionic ghost system and the standard example of a logarithmic conformal field theory — a theory whose representation modules of the Virasoro algebra (the infinite-dimensional symmetry algebra of two-dimensional conformal field theories) are reducible but indecomposable, so that correlation functions may contain logarithms. In particular, the scaling limit of the transfer matrix generates the integrals of motion (an infinite family of commuting conserved quantities) of the c=−2c=-2 theory, not those of c=1c=1; Morin-Duchesne et al. note explicitly that this does not rule out an alternative transfer-matrix description compatible with c=1c=1. Wilkins and Powell (2021) rederived the exact solution through a modified transfer matrix diagonalized in each flux sector, reproduced the torus partition function and the standard dimer correlations, and computed the central charge from the ground-state finite-size correction, obtaining c=1c=1 — while explicitly recording that the value of the central charge in the dimer model is controversial and that other arguments suggest c=−2c=-2. Allegra (2015) argued that corner free energies become consistent with c=1c=1 once boundary-condition-changing operators (operators at corners that switch the height boundary condition, of dimension 1/32) are included, but this has not been accepted as a settlement: later work (Izmailian et al. 2019) retains the c=−2c=-2 interpretation. The constructive derivation of the height-field theory from the fermionic solution, deferred in Wilkins and Powell (2021), has since been completed by the same authors via bosonization (the exact equivalence mapping the free fermions to the height field; Wilkins and Powell 2023) and via a Grassmann-integral method, an exact rewriting of the partition function as an integral over anticommuting variables, for square and honeycomb lattices (Powell 2026), which also recovers the partition function in Gaussian (tilt-sector) form; neither work addresses the central-charge controversy. Monomers are defects of the close-packing constraint (a site left uncovered); the monomer distribution function is the ratio of the number of configurations with two monomers at prescribed positions to the number of close-packed configurations. Fisher and Stephenson (1963) showed it decays as ∣r∣−1/2|\mathbf r|^{-1/2}, and Hartwig (1966) proved the exact asymptotics for monomers on a diagonal. Wilkins and Powell (2021) derived the exact same-row amplitude E/(2X)E/(2\sqrt X) for odd separation XX, with E=22/3e6ζ′(−1)/(1+α2)1/4E=2^{2/3}e^{6\zeta'(-1)}/(1+\alpha^2)^{1/4} and α\alpha the ratio of horizontal to vertical dimer weights (the anisotropy), by evaluating a Toeplitz determinant — a determinant whose (m,n)(m,n) entry depends only on m−nm-n — with the discontinuous symbol φ(k)=−eike2iθksgn(k)\varphi(k)=-e^{ik}e^{2i\theta_k}\mathrm{sgn}(k), using the Fisher-Hartwig conjecture, a partly conjectural framework for the asymptotics of Toeplitz determinants with discontinuous symbols (Fisher and Hartwig 1969; Basor and Tracy 1991). Their construction covers only monomers on the same row: monomers on different rows generate operators that anticommute with the flux parity (−1)Φy(-1)^{\Phi_y} and require a different sector treatment, though the source presents this as an extension its formalism can accommodate rather than as a posed open question. On the field-theory side the monomer is described by a magnetic vertex operator (an exponential of the height field carrying a localized height charge), with correlations ∼∣r−r′∣−1/2\sim|\mathbf r-\mathbf r'|^{-1/2} for monomers on opposite sublattices (Wilkins and Powell 2023), consistent with the exact lattice results.

Problem Statement

The close-packed dimer model on the square lattice is exactly solved; the objective is to settle its continuum conformal field theory — the conformal data actually characterizing its scaling limit. Establish which conformal field theory describes the scaling limit of the square-lattice dimer model — the Gaussian height theory with c=1c=1, a c=−2c=-2 symplectic-fermion (logarithmic) theory, or a precise relation between the two readings. A complete answer must account for all of the documented conflicting evidence, explaining how each of the following is realized: (i) the exact torus partition function in its theta-function form, (ii) the finite-size corrections of the free energy, including the vanishing corner contribution and its interpretation, (iii) the organization of the transfer-matrix spectrum into conformal towers (families of levels built on a highest-weight state), including the finding that the transfer matrix generates the c=−2c=-2 integrals of motion, and (iv) the rigorously established Gaussian correlations of local observables. Acceptable resolutions include, but are not limited to, the following: a derivation showing that one reading is the correct scaling limit and explaining the other as an artifact (for example of the signed sum over topological sectors, of the sector ground-state analysis, or of neglected boundary-condition-changing operators); the construction of an alternative transfer-matrix description compatible with c=1c=1, whose existence is explicitly not ruled out by Morin-Duchesne, Rasmussen and Ruelle (2016); or an equivalent reconciliation, for example a logarithmic theory whose local bosonic sector is the Gaussian height theory. Two adjacent items from the same source are documented gaps in the same transfer-matrix framework but are not completion requirements of this problem: the exact monomer distribution function for monomers on different rows (requiring operators that anticommute with the flux parity (−1)Φy(-1)^{\Phi_y}, presented in the source as an extension of its formalism to the general case), and the rigorous status of the Fisher-Hartwig Toeplitz asymptotics underlying the same-row monomer amplitude (applied in the source as a conjecture). The analysis of dimer-dimer interactions proposed in the same source is an open-ended program on non-exactly-solved models (advanced in Wilkins and Powell 2023 for the double dimer model) and is not part of the completion boundary here; progress on it does not by itself close this problem.

References

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