Classical dimer model with anisotropic interactions on the square lattice

Bilayer Coulomb phase of two-dimensional dimer models: Absence of power-law columnar order

Using renormalization group (RG) analyses and Monte Carlo (MC) simulations, we study the fully packed dimer model on the bilayer square lattice with fugacity equal to z (1) for interlayer (intralayer) dimers, and intralayer interaction V between neighboring parallel dimers on any elementary plaquette in either layer. For a range of not-too-large z>0 and repulsive interactions 0<V<V_{s} (with V_{s}≈2.1), we demonstrate the existence of a bilayer Coulomb phase with purely dipolar two-point functions, i.e., without the power-law columnar order that characterizes the usual Coulomb phase of square and honeycomb lattice dimer models. The transition line z_{c}(V) separating this bilayer Coulomb phase from a large-z disordered phase is argued to be in the inverted Kosterlitz-Thouless universality class. Additionally, we argue for the possibility of a tricritical point at which the bilayer Coulomb phase, the large-z disordered phase and the large-V staggered phase meet in the large-z, large-V part of the phase diagram. In contrast, for the attractive case with V_{cb}<V≤0 (V_{cb}≈-1.2), we argue that any z>0 destroys the power-law correlations of the z=0 decoupled layers, and leads immediately to a short-range correlated state, albeit with a slow crossover for small |V|. For V_{c}<V<V_{cb} (V_{c}≈-1.55), we predict that any small nonzero z immediately gives rise to long-range bilayer columnar order although the z=0 decoupled layers remain power-law correlated in this regime; this implies a nonmonotonic z dependence of the columnar order parameter for fixed V in this regime. Further, our RG arguments predict that this bilayer columnar ordered state is separated from the large-z disordered state by a line of Ashkin-Teller transitions z_{AT}(V). Finally, for V<V_{c}, the z=0 decoupled layers are already characterized by long-range columnar order, and a small nonzero z leads immediately to a locking of the order parameters of the two layers, giving rise to the same bilayer columnar ordered state for small nonzero z.

Emergence of cooperative dynamics in fully packed classical dimers

The classical dimer model on the square lattice is a paradigmatic example of a system subject to strong local constraints. We study its behavior under local stochastic dynamics, by means of Monte Carlo simulations and theoretical arguments. We observe clear signatures of correlated dynamics in both global and local observables and over a broad range of time scales, indicating a breakdown of the simple continuum description that approximates well the statics. We show that this collective dynamics can be understood in terms of one-dimensional "strings" of high mobility, which govern both local and long-wavelength dynamical properties. We introduce a coarse-grained description of the strings, based on the Edwards-Wilkinson model, which leads to exact results in the limit of low string density and provides a detailed qualitative understanding of the dynamics in all flux sectors. We discuss the implications of our results for the dynamics of constrained systems more generally.

Interacting fermions picture for dimer models

Recent numerical results on classical dimers with weak aligning interactions have been theoretically justified via a Coulomb gas representation of the height random variable. Here, we propose a completely different representation, the interacting fermions picture, which avoids some difficulties of the Coulomb gas approach and provides a better account of the numerical findings. Besides, we observe that the Peierls argument explains the behavior of the system in the strong interaction case.

Monomer-dimer mixture on a honeycomb lattice

We study a monomer-dimer mixture defined on a honeycomb lattice as a toy model for the spin-ice system in a magnetic field. In a low-doping region of monomers, the effective description of this system is given by the dual sine-Gordon model. In intermediate- and strong-doping regions, the Potts lattice gas theory can be employed. Synthesizing these results, we construct a renormalization-group flow diagram, which includes the stable and unstable fixed points corresponding to M5 and M6 in the minimal models of the conformal field theory. We perform numerical transfer-matrix calculations to determine a global phase diagram and also to proffer evidence to check our prediction.