Fractonic plaquette-dimer liquid beyond renormalization

We consider close-packed tiling models of geometric objects—a mixture of hardcore dimers and plaquettes—as a generalization of the familiar dimer models. Specifically, on an anisotropic cubic lattice, we demand that each site be covered by either a dimer on a $z$ link or a plaquette in the $x\text{−}y$ plane. The space of such fully packed tilings has an extensive degeneracy. This maps onto a fracton-type “higher-rank electrostatics,” which can exhibit a plaquette-dimer liquid and an ordered phase. We analyze this theory in detail, using height representations and T-duality to demonstrate that the concomitant phase transition occurs due to the proliferation of dipoles formed by defect pairs. The resultant critical theory can be considered as a fracton version of the Kosterlitz-Thouless transition. A significant new element is its UV-IR mixing, where the low-energy behavior of the liquid phase and the transition out of it is dominated by local (short-wavelength) fluctuations, rendering the critical phenomenon beyond the renormalization group paradigm.

Figure 1

Close-packed tilings of the cubic lattice. Each site is either part of a plaquettes (red) in the x-y plane or a $z$ dimer (green).

Figure 2

The electric fields $E_{xy},E_z$(red dot) live at the center of the x-y plaquette or at the center of the $z$ link. The height field $h$ (green) lives at the center of each cube on the dual lattice.

Figure 3

(a) The topological sector $m_z(z,y)=\int dx{E}_z$ counts the winding number of dimers along a specific $x$ row. (b) The view of the 3D lattice from the top onto an x-y plane. If the quantity $m_{xy}$ is fixed on all the shaded squares on a row and column, then all $m_{xy}$ are determined on the whole lattice. (c) The topological sector $m_{xy}(x,y)=\int dz{E}_{xy}$ counts the winding number of plaquettes along a specific $z$ row.

Figure 4

The vison operator $B$ living at the center of each cube flips four dimers into two plaquettes.

Figure 5

The rotor field $\theta$ is defined on the sites of the cube lattice, while the dual height field $h$ lives on the dual lattice at the center of each cube. Red dots: $m_{xy}\left(m_z\right)$ live at the center of an x-y plaquette (at the midpoint of a $z$ link).

Figure 6

T-Duality table of the classical rotor model.

Figure 7

(a) The layered triangular lattice. (b) A local resonance between two distinct patterns. (c) Five possible trimer-dimer closed-pack patterns adjacent to a site.

Figure 8

(a) Sign factors ${\eta }_t,{\eta }_q$ on even layers. Red denotes positive while blue denotes negative. Odd layers have exactly opposite sign factors. (b) Illustration of the sign factor ${\eta }_t,{\eta }_q,{\eta }_z$ near each site.