The close-packed dimer model on the square lattice is exactly solved, and its finite-size free-energy corrections depend non-locally on the boundary conditions and on the parity of the lattice dimensions; Izmailian, Priezzhev and Ruelle (SIGMA 3 (2007) 001) showed that these effects are consistently explained by the c=−2c=-2 logarithmic conformal field theory of symplectic fermions once a change of parity is understood as a change of boundary condition in the spanning-tree formulation. The same authors stressed, however, that this does not prove that c=−2c=-2 is forced: a c=1c=1 Gaussian (height-function) description also exists, a boundary-condition and parity analysis within the c=1c=1 framework had not been carried out, the embedding of the c=−2c=-2 subtheory into the c=1c=1 theory was left unclear, and the spanning-tree account cannot treat monomers in generic positions. Yet the precise relationship between the two continuum theories, the dictionary of observables selecting each, and the extension to the monomer-dimer ensemble remain open.

Background

The dimer model asks for the statistics of coverings of a lattice by dimers (dominoes) - bonds occupying two adjacent sites - such that every site is covered exactly once (the close-packed limit). On the square lattice the number of coverings of an M×NM\times N rectangle was computed exactly in 1961 by Kasteleyn, and independently by Temperley and Fisher, by writing the partition function as a Pfaffian (a signed square root of a determinant) of a signed adjacency matrix now called the Kasteleyn matrix. In the infinite-volume limit the model is critical - correlations decay algebraically for any nonzero ratio of horizontal to vertical dimer weights - so its scaling limit is expected to be a two-dimensional conformal field theory (CFT). A CFT is parametrized in part by its central charge cc, which controls the universal finite-size correction to the free energy: for a critical system on a strip or cylinder of width NN, the free energy per unit length expands as FN=fbulkN+2fsurf+A/N+⋯F_N=f_{\rm bulk}N+2f_{\rm surf}+A/N+\cdots, and the universal coefficient AA is fixed by cc together with the smallest conformal weight hmin⁡h_{\min} (the ground-state scaling dimension in the sector selected by the boundary conditions) through the effective central charge ceff=c−24hmin⁡c_{\rm eff}=c-24h_{\min}.

Ferdinand's 1967 evaluation of the dimer partition function on a torus already showed that the finite-size corrections depend on the parities of MM and NN - a non-local dependence, since changing the parity of a lattice dimension is not a local modification, and several authors read off parity-dependent central charges from it. Izmailian, Priezzhev and Ruelle (2007) resolved the strip and cylinder cases: using the Temperley bijection between close-packed dimer coverings and spanning trees (subgraphs connecting all vertices without cycles) on a coarser sublattice - equivalently, recurrent configurations of the Abelian sandpile model (a lattice model of self-organized criticality whose stable configurations also biject with spanning trees) or alternating-sign matrices (a combinatorial class of ±1\pm1/0 matrices in bijection with such trees on special domains) - they showed that changing the parity of the width NN genuinely changes the boundary conditions of the associated tree/sandpile problem, whose boundary sides are called closed (non-dissipative) or open (dissipative). With this parity-induced boundary-condition change taken into account, the exact leading coefficients - π/12\pi/12 and −π/24-\pi/24 for free-boundary strips of odd and even width, π/12\pi/12 and −π/6-\pi/6 for cylinders of odd and even perimeter - are all reproduced by a single theory: the logarithmic CFT of symplectic fermions, with c=−2c=-2 and hmin⁡=−1/8h_{\min}=-1/8 or 00 depending on the sector. A logarithmic CFT is one whose representations are indecomposable (the dilatation operator L0L_0 acts with non-trivial Jordan blocks that cannot be diagonalized, producing logarithms in correlation functions); symplectic fermions - two anticommuting, Grassmann-valued scalar fields - form the simplest non-trivial example, with c=−2c=-2.

However, an apparently unrelated description also exists. Every dimer configuration on a planar bipartite lattice can be encoded in an integer-valued height function on the faces (its discrete gradients are fixed by the dimer positions), and Kenyon proved that this height converges in the scaling limit to the Gaussian free field, i.e. the massless free boson with c=1c=1 whose zero mode is compactified - it takes values on a circle rather than on the real line (developed further by Kenyon, Okounkov and Sheffield). Because the effective central charge fixes only the combination c−24hmin⁡c-24h_{\min}, finite-size data alone cannot decide whether the dimer model 'is' a c=−2c=-2 or a c=1c=1 theory. The 2007 paper argued that the c=−2c=-2 theory should be viewed as a subtheory of the c=1c=1 theory: adequate for the pure dimer (spanning-tree) degrees of freedom, but not for monomer insertions (vacancies at fixed lattice sites), whose correlations were computed by Fisher and Stephenson and fit the Gaussian picture. What remains unresolved is the precise relationship between the two continuum descriptions - how they map onto each other, which lattice observables distinguish them - how the parity-dependent finite-size effects are organized within the c=1c=1 framework, and how far either description reaches over the monomer-dimer ensemble.

Problem Statement

The close-packed square-lattice dimer model admits two internally consistent continuum descriptions: a c=−2c=-2 logarithmic conformal field theory (symplectic fermions), naturally reached through the spanning-tree/alternating-sign-matrix mapping, and the c=1c=1 Gaussian theory of the height field, i.e. the compactified free boson (the massless free scalar whose zero mode takes values on a circle rather than on the real line). The open problem is to determine the precise relationship between these two descriptions and the reach of each over the dimer and monomer-dimer ensembles. Specifically: (a) construct the explicit relation - embedding, map, or other precise structural statement - by which the c=−2c=-2 logarithmic theory sits inside the c=1c=1 Gaussian theory of the same lattice model, and identify which lattice observables and degrees of freedom each description captures (partition-function finite-size sectors and spanning-tree/dimer observables versus height correlations and monomer insertions); (b) carry out the boundary-condition and lattice-parity analysis within the c=1c=1 framework that reproduces the known non-local finite-size effects of the close-packed model - in particular the exact leading coefficients π/12\pi/12 and −π/24-\pi/24 for free-boundary strips of odd and even width NN, π/12\pi/12 and −π/6-\pi/6 for cylinders of odd and even perimeter, and the sectors of the torus partition function labelled by the boundary conditions and by the parities of the lattice dimensions - which the c=−2c=-2 framework accounts for through parity-induced changes of boundary condition; and (c) extend this accounting to dimer configurations with monomers in generic positions, i.e. the monomer-dimer ensemble, where the spanning-tree mapping (defined on a single sublattice) breaks down. These parts jointly serve one objective, namely to establish which conformal theory (or theories) describes the dimer and monomer-dimer models and how the candidate descriptions relate: (a) fixes the relation, (b) validates it against the exactly known finite-size data within the Gaussian framework, and (c) tests its reach beyond pure dimers.

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