Phase diagram of an extended classical dimer model

We present an extensive numerical study of the critical behavior of dimer models in three dimensions, focusing on the phase transition between Coulomb and crystalline columnar phases. The case of attractive interactions between parallel dimers on a plaquette was shown to undergo a continuous phase transition with critical exponents close to those of the $O\left(N\right)$ tricritical universality class, a situation which is not easily captured by conventional field theories. That the dimer model is exactly fine tuned to a highly symmetric point is a nontrivial statement which needs careful numerical investigation. In this paper, we perform an extensive Monte Carlo study of a generalized dimer model with plaquette and cubic interactions and determine its extended phase diagram. We find that when both interactions favor alignment of the dimers, the phase transition is first order, in almost all cases. On the opposite, when interactions compete, the transition becomes continuous, with a critical exponent $\eta \sim 0.2$. The existence of a tricritical point between the two regimes is confirmed by simulations on very large size systems and a flowgram method. In addition, we find a highly degenerate crystalline phase at very low temperature in the frustrated regime which is separated from the columnar phase by a first-order transition.

Figure 1

(Color online) Phase diagram of the attractive dimer model with cubic interactions. Dots indicate values of interactions where simulations were performed. Dashed lines and green dots represent second order transitions. Solid lines and red dotes correspond to first-order transitions. The white dot represents a point where the nature of the transition is still unclear.

Figure 2

(Color online) The three different possibilities with four parallel dimers parallel on a unit cube. Each pattern contributes $v_4$ to the energy $E_{\mathcal{C}}$ in Eq. (2).

Figure 3

(Color online) Illustration of occurrences of the $v_2$ (top) and $v_4$ (bottom) terms of the model defined in Eq. (2): interacting plaquettes (top) and cube (bottom) are represented with shaded surfaces.

Figure 4

(Color online) Left: evolution of the columnar order parameter $C$ and inverse stiffness $K^{-1}$ as a function of temperature for $v_2=0$ and $v_4=-1$. Right: temperature dependence of the average energy per site $⟨E⟩/N$. Here $L=16$.

Figure 5

(Color online) Energy per site probability $P\left(e\right)$ as a function of energy per site $e=E/N$, for different ratios $v_4/v_2>0$ at the critical temperature separating columnar and Coulomb phases. The size $L$ corresponds to the minimal length for which the histograms start to display a double peak distribution. For $v_4/v_2=0.2$, a deformed single peak is observed in the distribution for $L=140$, the maximal sample size simulated for this parameter.

Figure 6

(Color online) For $v_4=1$ and $v_2=0$. Left: evolution of the stiffness $K^{-1}$ and columnar order $C$ as a function of temperature for $L=16$. The nonzero value reached by $C$ is due to the finite size of the sample. Right: the height of the peak of the specific heat per site does not display any dependence on system size.

Figure 7

(Color online) Evolution of the specific heat per site as a function of temperature for $v_4/v_2=-1$. The first peak at $T_{c_1}\sim 0.5$ is associated with the crystallization of the system in an ordered phase different from the columnar phase, which can be readily seen by the evolution of the columnar order parameter (see upper inset). The second peak, at $T_{c_2}\sim 0.95$, delimitates a region with nonzero columnar order from a region with nonzero inverse stiffness. Upper inset: evolution of the inverse stiffness $K^{-1}$ and of the columnar order parameter $C$ for $L=16$. Lower inset: zoom of the specific heat close to $T_{c_2}$.

Figure 8

(Color online) Three configurations on the $L=6$ cube which maximize the number of parallel plaquettes without having any parallel cubes. Configuration $A$ is built out of a unit pattern containing two planes which repeats itself three times. This unit pattern possesses 42 parallel plaquettes, leading to a number of 7/12 parallel plaquettes per unit site. Configurations $B$ and $B^{\prime }$ both consist of a unit pattern of three planes repeating itself twice. The unit pattern possesses 63 parallel plaquettes, leading also to a number of 7/12 parallel plaquettes per unit site. Configuration $B^{\prime }$ is obtained from configuration $B$ by shifting the dimer pattern on the two lowest planes by one unit cell on the right direction. Mixing $B$ and $B^{\prime }$ patterns, one can create several dimer configurations having the same energy.

Figure 9

(Color online) Top: evolution as a function of temperature of the columnar order parameter $C$ for different ratios $v_4/v_2$ at $L=32$. For $v_4/v_2<-0.8$, $C$ does not converge toward 1 at $T=0$. Inset: evolution of the average energy per site. A discontinuity corresponding to the phase transition can easily be detected. Bottom: evolution of $C$, $⟨N_{\text{plaquette}}⟩/N$ and $⟨N_{\text{cube}}⟩/N$ as a function of $v_4/v_2$ for $T=0.3$ and $L=32$.

Figure 10

(Color online) As a function of temperature, for $v_4/v_2=-10$. Top: specific heat per site $C_v/N$. The dashed line represents a linear interpolation of the regular part of the specific heat. Inset: maximum of $C_v/N$ as a function of system size. Middle: Binder cumulant and inverse stiffness crossings. Insets: Binder cumulant derivative (at $T=0.672$) and derivative $L.d{K}^{-1}/dT$ (at $T=0.6715$) as a function of system size (log-log scale). Bottom: columnar susceptibility $\chi$ per site. Inset: maximum of $\chi$ as a function of system size (log-log scale). All lines in insets denote power-law fits of critical exponents (see text for details).

Figure 11

(Color online) Specific heat per site $C_v/N$ at $v_4/v_2=-1.0$. The dashed line represents a linear interpolation of the regular part of the specific heat. Inset: Maximum of $C_v/N$ as a function of system size.

Figure 12

(Color online) Flowgram (top) and performed collapses (bottom) for the extended dimer model. See text for details of the flowgram procedure.

Figure 13

(Color online) Left: flows at $v_4/v_2=-0.02$ and $v_4/v_2=0.6$ at the rescaled length $L_{\text{eff}}$. Top right: Maximum of the specific heat for $v_4=0$ as a function of the system size. Data are taken from Ref. 16 except for the value at $L=180$. The solid line is the power-law fit, giving rise to the estimate $\alpha /\nu =1.15\left(15\right)$ compatible with the result of Ref. 16 without the knowledge of the $L=180$ point. Bottom right: probability of energy per site at $v_4=0$ and $L=180$.