Random-diluted triangular plaquette model: Study of phase transitions in a kinetically constrained model

We study how the thermodynamic properties of the triangular plaquette model (TPM) are influenced by the addition of extra interactions. The thermodynamics of the original TPM is trivial, while its dynamics is glassy, as usual in kinetically constrained models. As soon as we generalize the model to include additional interactions, a thermodynamic phase transition appears in the system. The additional interactions we consider are either short ranged, forming a regular lattice in the plane, or long ranged of the small-world kind. In the case of long-range interactions we call the new model the random-diluted TPM. We provide arguments that the model so modified should undergo a thermodynamic phase transition, and that in the long-range case this is a glass transition of the “random first-order” kind. Finally, we give support to our conjectures studying the finite-temperature phase diagram of the random-diluted TPM in the Bethe approximation. This corresponds to the exact calculation on the random regular graph, where free energy and configurational entropy can be computed by means of the cavity equations.

Figure 1

Main: Energy hysteresis cycle for a TPM model with additional long-range plaquettes. We simulated the model on a lattice with the shape of a rhombus (see for instance the representation of Fig. 2) with $L=128$ plaquettes per side and a concentration of additional plaquettes ${\alpha }_L=0.1$. Red circles: Cooling (cooling rate from ${10}^6$ to ${10}^7$ Monte Carlo steps per $\mathrm{\Delta }T=0.04$). Green triangles: Heating. Continuous black line: High-temperature expansion, $e\left(T\right)=-({\alpha }_L+1)tanh(1/T)$. The melting temperature $T_m$ is indicated by the dotted vertical line while the temperature $T_0$ of the entropy crisis of the high-temperature expansion is the filled black square. Inset: Circles show relaxation time in the supercooled liquid phase as a function of $T$, and lines are respectively the two-parameter fit with $aexp(b/T)$ (continuous red line) and the three-parameter fit with $cexp[d/(T-T_K)]$. The critical temperature obtained from the fit is $T_K=0.374$.

Figure 2

Triangular lattice for the generalized TPM with additional short-range plaquettes on a regular sublattice made of equilateral triangles of side 3. The original TPM plaquettes and additional interactions are shown as light gray and dark gray triangles, respectively.

Figure 3

Smallest hyperloop of the high-temperature expansion for the modified TPM with extra short-range plaquettes discussed in Sec. 2b. In the figure can be counted 54 small plaquettes of the original TPM lattice (thick red triangles) and 10 plaquettes of the auxiliary sublattice (thin black triangles), which has as elementary cells equilateral triangles of side 3.

Figure 4

Main: Energy hysteresis cycle for a TPM model with additional short-range plaquettes. We simulated the model on a lattice with the shape of a rhombus (see for instance the representation of Fig. 2) with $L=128$ plaquettes per side and a concentration of additional plaquettes ${\alpha }_L=0.1$: the coupling coefficient of the additional plaquettes is $J_1=1/5$. Red circles: Cooling (cooling rate from ${10}^6$ to ${10}^7$ Monte Carlo steps per $\mathrm{\Delta }T=0.04$). Green triangles: Heating. Continuous black line: High-temperature expansion, $e\left(T\right)=-[{\alpha }_{L}tanh(J_1/T)+tanh(1/T)]$. Inset: Magnetization hysteresis cycle for the same model with the same cooling/heating protocol. Red circles: Cooling. Green triangles: Heating. Continuous line: $m\left(T\right)$ from the high-temperature expansion: $m\left(T\right)={\alpha }_L{[tanh(1/T)]}^{6}tanh(J_1/T)$, with ${\alpha }_L=0.1$ and $J_1=1/5$.

Figure 5

Panel (a): $T=0$ phase diagram of the random-diluted TPM [see Eq. (23)]. Symbols represent numerical data obtained running the leaf-removal algorithm on finite-dimensional geometries, lines the results of analytical calculation on the Bethe lattice. Circles (blue) and the dotted line represent the critical line for the formation of the core, squares (red) and the continuous line are the critical line for the SAT/UNSAT transition. Panel (b): Diamonds, numerical estimate in finite dimensions of the critical value ${\alpha }_L$ for the SAT/UNSAT transition as function of $\sqrt{N}$, with $N$ the size of the system, at fixed ${\alpha }_s=1$; dotted line, linear fit of data. Panel (c): Zoom of the SAT/UNSAT transition line close to the point $({\alpha }_s=1,{\alpha }_L=0)$. Triangles, analytic result on the random graph; squares, numerical data for a system with $N={10}^6$ spins; dotted line, linear fit of analytic prediction.

Figure 6

Panel (a): Circles are the critical values $({\alpha }_s,{\alpha }_L)$ for the formation of the core extrapolated in the thermodynamic limit from numerical data at finite $N$. The continuous line represents a quadratic fit of data $y=a{x}^2$, with prefactor $a=0.877$; see also Eq. (26) in Sec. 3b. Panel (b): Critical values ${\alpha }_L\left(N\right)$ for core formation as a function of the size $N$ of the system for different values of ${\alpha }_s$ (different symbols), from top to bottom: ${\alpha }_s=0.93,0.95,0.97,0.98,0.99$. Panel (c): Concentration of spins in the core as a function of ${\alpha }_L$ for a fixed value of ${\alpha }_s=0.8$, with $N={1024}^2$.

Figure 7

Phase diagram of the random-diluted TPM model on the Bethe lattice in the $({\alpha }_L,T)$ plane for two different values of dilution ${\alpha }_s$: circles (red), ${\alpha }_s=1$; squares (blue), ${\alpha }_s=0.96$; continuous lines, fits of the data with the function ${\alpha }_L\left(T\right)=C_{1}exp(-C_2/T)+{\alpha }_L\left(0\right)$, where $C_1$ and $C_2$ are fit parameters.