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

Abstract

We study the fully-packed dimer model on the bilayer square lattice with fugacity equal to $z$ ($1$) for inter-layer (intra-layer) dimers, and intra-layer interaction $V$ between neighbouring 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 $V_s \approx 2.1$), we demonstrate the existence of a {\em bilayer Coulomb phase} with purely dipolar two-point functions, {\em 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 \leq 0$ ($V_{cb} \approx -1.2$), 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 $V_{c} < V < V_{cb}$ ($V_{c} \approx -1.55$), we predict that any small nonzero $z$ immediately gives rise to long-range {\em bilayer columnar order} although the $z=0$ decoupled layers remain power-law correlated in this regime; this implies a non-monotonic $z$ dependence of the columnar order parameter for fixed $V$ in this regime. This bilayer columnar ordered state is separated from the large-$z$ disordered state by a line of Ashkin-Teller transitions $z_{\rm AT}(V)$.