Coulomb gas transitions in three-dimensional classical dimer models

Close-packed, classical dimer models on three-dimensional, bipartite lattices harbor a Coulomb phase with power-law correlations at infinite temperature. Here, we discuss the nature of the thermal phase transition out of this Coulomb phase for a variety of dimer models which energetically favor crystalline dimer states with columnar ordering. For a family of these models, we find a direct thermal transition from the Coulomb phase to the dimer crystal. While some systems exhibit (strong) first-order transitions in correspondence with the Landau-Ginzburg-Wilson paradigm, we also find clear numerical evidence for continuous transitions. A second family of models undergoes two consecutive thermal transitions with an intermediate paramagnetic phase separating the Coulomb phase from the dimer crystal. We can describe all of these phase transitions in one unifying framework of candidate field theories with two complex Ginzburg-Landau fields coupled to a $U\left(1\right)$ gauge field. We derive the symmetry-mandated Ginzburg-Landau actions in these field variables for the various dimer models and discuss implications for their respective phase transitions.

Figure 1

(Color online) The six dimer coverings with columnar dimer ordering that maximize the number of parallel dimers on neighboring bonds of the cubic lattice. Our family of classical dimer models energetically favors these ordering patterns with the 6-GS model favoring all six states, the 4-GS model favoring states 3–6, the 2-GS model favoring states 1–2, and the 1-GS model favoring a single state only. Further variations are discussed in the text.

Figure 2

(Color online) Overview of numerical results for the dimer models with one (1-GS), two (2-GS), four (4-GS), and six (6-GS) columnar ground states: (a) the specific heat per site $C_v\left(T\right)/N$, (b) the energy per site $E\left(T\right)/N$, and (c) the monomer confinement length ${\xi }^2\left(T\right)$ defined in the text. All four models undergo a direct transition with clear thermodynamic signatures between the high-temperature Coulomb phase (with deconfined monomers) to the dimer crystal at low temperature (with confined monomers). The sharp, kinklike features in the energy $E\left(T\right)$ and monomer confinement length ${\xi }^2\left(T\right)$ are indicative of a first-order transition for the 2-GS and 4-GS models. The 1-GS and 6-GS models undergo continuous transitions with smooth features. Data are shown for system size $L=48$ for the 1-GS and 6-GS models and $L=32$ for the 2-GS and 4-GS models.

Figure 3

(Color online) Specific heat per site $C_v\left(T\right)/N$ for the 1-GS model as function of temperature $T$ and different system sizes $L$. Inset shows a specific-heat scan over a wide temperature range for a sample $L=48$ system size. Below the peak around $T_c\approx 2.276\pm 0.001$, there is a shoulder which shows no variation with system size.

Figure 4

(Color online) Confinement length ${\xi }^2\left(T\right)$ for the 1-GS model measuring the (squared) average distance between two monomers. Data for different system sizes $L$ are renormalized by the expectation value $\left(L^2+2\right)/4$ for deconfined monomers. The distinct crossing point shown in the inset for temperatures in the vicinity of the transition temperature indicates a continuous transition.

Figure 5

(Color online) Stiffness $\rho$ of the 1-GS model multiplied by $L$ vs temperature $T$ near the transition for different system sizes $L$. The crossing point indicates a continuous transition.

Figure 6

(Color online) Data collapse for the stiffness $\rho$ multiplied by $L$ (top panel) and confinement length ${\xi }^2$ (lower panel) of the 1-GS model as a function of $L^{1/\nu }t$, where $t=\left(T-T_c\right)/T_c$, with $T_c=2.27618$. The critical exponent $\nu =0.6717$ corresponds to the 3D $XY$ universality class.

Figure 7

(Color online) Confinement length ${\xi }^2\left(T\right)$ for the 2-GS model. Data for different system sizes $L$ are renormalized by the expectation value $\left(L^2+2\right)/4$ for deconfined monomers. The absence of a distinct crossing point in the vicinity of the transition temperature indicates a first-order transition.

Figure 8

(Color online) Confinement length ${\xi }^2\left(T\right)$ for the 4-GS model. Data for different system sizes $L$ are renormalized by the expectation value $\left(L^2+2\right)/4$ for deconfined monomers. The absence of a distinct crossing point in the vicinity of the transition temperature indicates a first-order transition.

Figure 9

(Color online) Energies per site for a dimer model which continuously interpolates between the 1-GS $\left(\lambda =0\right)$ and 2-GS $\left(\lambda =1\right)$ models. Data shown are for system size $L=48$.

Figure 10

(Color online) Energy (per site) histograms for the dimer model interpolating between the 1-GS and 2-GS models in the vicinity of the transition temperature of the respective models. Data shown are for system size $L=48$.

Figure 11

(Color online) Energies per site for a dimer model which continuously interpolates between the 4-GS $\left(\lambda =0\right)$ and 6-GS $\left(\lambda =1\right)$ models. Data shown are for system size $L=48$.

Figure 12

(Color online) Energy (per site) histograms for the dimer model interpolating between the 4-GS and 6-GS models in the vicinity of the transition temperature of the respective models. Data shown are for system sizes $L=48$ in the upper panel and $L=96$ in the lower panel.

Figure 13

(Color online) Distance between the two peaks in the bimodal energy histograms shown in Fig. 12. Lines are guides to the eyes. Data for system sizes $L=48$ and $L=96$ are shown.

Figure 14

(Color online) Cubic lattice and its dual. The dual cubic lattice sites sit at the centers of each cube of the cubic lattice.

Figure 15

(Color online) Overview of the $xy$ model that favors one columnar ordering pattern along the $x$ and $y$ lattice directions: (a) specific heat per site $C_v\left(T\right)/N$, (b) energy per site $E\left(T\right)/N$, and (c) confinement length $\xi {\left(T\right)}^2$. This model undergoes an unusual sequence of phases with a paramagnetic phase (shaded area) at intermediate temperature scales. The high-temperature transition out of the Coulomb phase (denoted by an arrow) is a monomer confinement transition, while the dimer crystal forms only at the low-temperature phase transition.

Figure 16

(Color online) Crossing point of the confinement length ${\xi }^2\left(T\right)$ for different system sizes at the high-temperature phase transition out of the Coulomb phase for the $xy$ model.

Figure 17

(Color online) Data collapse for the confinement length ${\xi }^2$ as a function of $L^{1/\nu }t$, where $t=\left(T-T_c\right)/T_c$, with $T_c=2.2475$. The critical exponent $\nu =0.6717$ corresponds to the 3D $XY$ universality class.

Figure 18

(Color online) Bond occupation along the $x$, $y$, and $z$ lattice directions vs temperature for the $xy$ model. At low temperatures, data are shown for a system that spontaneously orders along the $y$ direction (other systems spontaneously order along the $x$ direction). The low-temperature transition between the paramagnet and the dimer crystal is accompanied by a sudden increase of the bond occupation along the preferred lattice direction. The high-temperature transition between the Coulomb gas and the paramagnet where the monomers confine is indicated by the arrow.

Figure 19

(Color online) Energy histogram in the vicinity of the low-temperature phase transition, which we argue to be a thermal analog of a spin-flop transition (see text). Our numerics indicate a weak first-order transition by a double-peak structure in the energy histogram, which emerges for larger system sizes. Here, system size is $L=96$.