Gauge Theory Picture of an Ordering Transition in a Dimer Model

We study a phase transition in a 3D lattice gauge theory, a “coarse-grained” version of a classical dimer model. Duality arguments indicate that the dimer lattice theory should be dual to a $XY$ model coupled to a gauge field with geometric frustration. The transition between a Coulomb phase with dipolar correlations and a long range ordered columnar phase is understood in terms of a Higgs mechanism. Monte Carlo simulations of the dual model indicate a continuous transition with exponents close but apparently different from those of the 3D $XY$ model. The continuous nature of the transition is confirmed by a flowgram analysis.

Figure 1

(a) Second moment $M_2$ vs $T$ for different system sizes. Inset: Zoom on the peak. (b) Columnar order parameter $C$ and transverse magnetic field correlator $G({\mathbf{p}}_{\min })$ as a function of $T$ for different system sizes.

Figure 2

Binder cumulant of the columnar parameter versus $T$. Left-hand inset: Divergence of the derivative $dB/dT$ close to $T_c$ versus system size (log-log scale). Right-hand inset: Binder cumulant scaling collapse.

Figure 3

$LG$ versus $T$ for different system sizes. Left-hand inset: Scaling of the derivative $LdG/dT$ versus system size $L$ (log-log scale). Right-hand inset: Data collapse of $LG$.

Figure 4

Data collapse of the flowgram of the model. Inset: Flows for different values of $\mathcal{K}$. Note how the flows look more convergent for smaller stiffness.