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 $V_s\approx 2.1$), 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 $z_c\left(V\right)$ 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 $V_{cb}<V\le 0$ $(V_{cb}\approx -1.2)$, 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 $\left|V\right|$. For $V_c<V<V_{cb}$ $(V_c\approx -1.55)$, 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 $z_{\mathrm{AT}}\left(V\right)$. 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 $z$.

• Received 11 November 2020
• Accepted 16 March 2021

DOI:https://doi.org/10.1103/PhysRevE.103.042136