Critical properties of the four-state commutative random permutation glassy Potts model in three and four dimensions

We investigate the critical properties of the four-state commutative random permutation glassy Potts model in three and four dimensions by means of Monte Carlo simulations and a finite-size scaling analysis. By using a field programmable gate array, we have been able to thermalize a large number of samples of systems with large volume. This has allowed us to observe a spin-glass ordered phase in $d=4$ and to study the critical properties of the transition. In $d=3$, our results are consistent with the presence of a Kosterlitz-Thouless transition, but also with different scenarios: transient effects due to a value of the lower critical dimension slightly below 3 could be very important.

Figure 1

(Color online) Logarithmic data binning analysis (see text) of the nonconnected susceptibility for the $d=4$ model, $L=16$, $\beta =1.41$, and $\beta =1.5$. Notice that the large time region appears on the left in the figure.

Figure 2

(Color online) Correlation length in units of the linear size $L$ as a function of $\beta$ for $d=3$ systems of different volumes.

Figure 3

(Color online) The cumulant $U_4$ as defined in Eq. (8) as a function of $\beta$ for $d=3$ systems of different volumes.

Figure 4

(Color online) Distribution of the overlap in the $d=3$, $L=16$ system at several temperatures.

Figure 5

(Color online) Distribution of the overlap in the $d=4$ model at low temperature $\left(\beta =1.44\right)$ for two different lattice sizes.

Figure 6

(Color online) Evolution of the overlap of a representative sample of the $d=4$ model, $L=16$ system. Here $\beta =1.5$.

Figure 7

(Color online) Correlation length in units of $L$ as a function of $\beta$ in the $d=4$ model.

Figure 8

(Color online) Close-up of the data of Fig. 7 close to the estimated critical point.