Dynamical heterogeneities as fingerprints of a backbone structure in Potts models

We investigate slow nonequilibrium dynamical processes in a two-dimensional $q$-state Potts model with both ferromagnetic and $\pm J$ couplings. Dynamical properties are characterized by means of the mean-flipping time distribution. This quantity is known for clearly unveiling dynamical heterogeneities. Using a two-times protocol we characterize the different time scales observed and relate them to growth processes occurring in the system. In particular we target the possible relation between the different time scales and the spatial heterogeneities originated in the ground-state topology, which are associated to the presence of a backbone structure. We perform numerical simulations using an approach based on graphis processing units (GPUs) which permits us to reach large system sizes. We present evidence supporting both the idea of a growing process in the preasymptotic regime of the glassy phases and the existence of a backbone structure behind this process.

Figure 1

(a) Relaxation of the excess energy for different temperatures for the $q=9$ Potts model. (b) Dependence with temperature of the MFTD for the $q=9$ Potts model (colors and symbols for each temperature are conserved from the panel above). All the curves correspond to $\Delta t=t_w=4\times {10}^3$, $L=8192$, $m=100$.

Figure 2

Evolution with the waiting time of the MFTD for the $q=9$ Potts model at $T=0.98T_c$ (within the coarsening regime). The time evolution is followed using a two-times protocol with $\Delta t=t_w$. Averages were taken with $L=8192$, $m=100$. The inset shows the evolution of the relaxation function ${\phi }_E\left(t\right)$ at this temperature. The vertical dashed line indicates approximately the beginning of the power law decay corresponding to the coarsening regime ($t\approx 4\times {10}^3$).

Figure 3

MFTD for the $\pm J$ EA model (equivalently $\pm J$ Potts model with $q=2$) in two dimensions for $T=0.25J$ and $\Delta t=t_w={10}^6$. We show the full MFTD for $L=22$ and 512 using spin-flip dynamics with serial random updates together with the result for $L=512$ using the parallel dynamics described in Sec. 2. The separation of the full MFTD into the contribution from solidary spins (ss) and nonsolidary spins (nss) is also shown for $L=22$. Averages were taken over $M=2000$ samples for $L=22$, $M=100$ samples for $L=512$, and $M=1$ sample for $M=16384$.

Figure 4

Evolution of the MFTD given by the solidary spins in the two-dimensional $\pm J$ EA model. Each curve was obtained using $\Delta t=t_w$. The temperature is $T=0.25J$ in all cases. The arrow emphasizes how the contribution to the fast peak is evolving with increasing $t_w$. Averages were taken within $M=2000$ samples for $L=22$.

Figure 5

MFTD for the $\pm J$ Potts model with different $q$ states, $q=2,3,5,10$, at a fixed temperature $T=0.1J$. Curves correspond to $\Delta t=t_w={10}^7$, $L=4096$, and $M=10$.

Figure 6

MFTD for the $\pm J$ Potts model model with $q=5$, $\Delta t=t_w={10}^7$, and different temperatures as indicated. Curves correspond to $L=2048$, $M=10$.

Figure 7

Arrhenius plot for the characteristic time scale of the second peak (${\tau }_2$) of the MFTD for $q=2,3,5$ ($\Delta t=t_w={10}^7$). The dashed lines are linear fits of the data (symbols), giving slopes of $1.99\pm 0.01$ for the $q=2$ case and $0.99\pm 0.01$ for $q=3$ and 5 cases.