Critical behavior of the three- and ten-state short-range Potts glass: A Monte Carlo study

We study the critical behavior of the short-range $p$-state Potts spin glass in three and four dimensions using Monte Carlo simulations. In three dimensions, for $p=3$, a finite-size scaling analysis of the correlation length shows clear evidence of a transition to a spin-glass phase at $T_{\mathrm{c}}\simeq 0.273$ for a Gaussian distribution of interactions and $T_{\mathrm{c}}\simeq 0.377$ for a $\pm J$ (bimodal) distribution. These results indicate that the lower critical dimension of the three-state Potts glass is below 3. By contrast, the correlation length of the ten-state $(p=10)$ Potts glass in three dimensions remains small even at very low temperatures and thus shows no sign of a transition. In four dimensions we find that the $p=3$ Potts glass with Gaussian interactions has a spin-glass transition at $T_{\mathrm{c}}\simeq 0.536$.

Figure 1

(Color online) An equilibration plot showing the left-hand side (upper curve) and right-hand side (lower curve) of Eq. (14). The data are for the three-state Potts glass in $d=3$ for $L=12$ at $T=0.248$ as a function of equilibration time measured in Monte Carlo sweeps, $N_{\mathrm{sw}}$. An equal number of sweeps are also performed for measurement, so the total number of sweeps is $2N_{\mathrm{sw}}$, respectively. The two sets of data merge at long times, as expected, and then remain independent of time. The inset shows the correlation length ${\xi }_L$ for the corresponding number of Monte Carlo sweeps.

Figure 2

(Color online) Finite-size correlation length ${\xi }_L∕L$ vs $T$ for the three-state Potts glass with Gaussian couplings in three dimensions. The curves intersect at $T_{\mathrm{c}}\simeq 0.273$. The inset shows a scaling plot according to Eq. (12) using $T_{\mathrm{c}}=0.273$ and $\nu =1.30$. The solid line is the third-order polynomial used in the fit.

Figure 3

(Color online) Scaling plot of ${\chi }_{\mathrm{SG}}∕L^{2-\eta }$ vs $T$ for the three-state Potts glass with Gaussian interactions in three dimensions according to Eq. (13) with $T_{\mathrm{c}}$ fixed to 0.273 (determined from the data for ${\xi }_L∕L$ shown in Fig. 2). The fit to Eq. (13) then gives $\eta =-0.03$ and $\nu =0.94$.

Figure 4

(Color online) Ferromagnetic susceptibility ${\chi }_{\mathrm{FM}}$ against $T$ for the three-state Potts glass with Gaussian couplings in three dimensions. The upper curves plotted with solid symbols correspond to $J_0=0$ while the lower curves with open symbols correspond to $J_0=-1$, where $J_0$ is the mean of the Gaussian interactions.

Figure 5

(Color online) Graph of ${\xi }_L$ vs $T$ for the three-state Potts glass with Gaussian couplings in three dimensions. The solid symbols correspond to $J_0=0$ while the open symbols correspond to $J_0=-1$.

Figure 6

(Color online) Graph of ${\xi }_L∕L$ vs $T$ for the three-state Potts glass with Gaussian couplings in three dimensions where the mean of the Gaussian distribution is $J_0=-1$. The vertical line is drawn at the transition temperature $T_{\mathrm{c}}=0.273$ of the $J_0=0$ case (see Fig. 2).

Figure 7

(Color online) Graph of ${\xi }_L∕L$ vs $T$ for the three-state Potts glass with a bimodal coupling distribution in three dimensions. The curves intersect at $T_{\mathrm{c}}\simeq 0.377$. The inset shows a scaling plot according to Eq. (12) using $T_{\mathrm{c}}=0.377$ and $\nu =1.18$.

Figure 8

(Color online) Scaling plot of ${\chi }_{\mathrm{SG}}$ according to Eq. (13) with $T_{\mathrm{c}}$ fixed to 0.377 (determined from the data for ${\xi }_L∕L$ shown in Fig. 7) for the three-state Potts glass with bimodal $(\pm J)$ interactions in three dimensions. The fit to Eq. (13) then gives $\eta =0.02$ and $\nu =0.91$.

Figure 9

(Color online) Graph of ${\chi }_{\mathrm{FM}}$ vs $T$ for the three-state Potts glass with Gaussian couplings with $J_0=0$ in four dimensions.

Figure 10

(Color online) Correlation length ${\xi }_L∕L$ vs $T$ for the three-state Potts glass with Gaussian couplings in four dimensions. The curves intersect at $T_{\mathrm{c}}\simeq 0.54$. The inset shows a scaling plot according to Eq. (12) using $T_{\mathrm{c}}=0.536$ and $\nu =0.81$.

Figure 11

(Color online) Scaling plot of ${\chi }_{\mathrm{SG}}$ for the three-state Potts glass with Gaussian interactions in four dimensions according to Eq. (13) with $T_{\mathrm{c}}$ fixed to the value 0.536, which is obtained from the data for ${\xi }_L∕L$ shown in Fig. 10. The fit to Eq. (13) then gives $\eta =-0.08$ and $\nu =0.70$.

Figure 12

(Color online) Ferromagnetic susceptibility ${\chi }_{\mathrm{FM}}$ vs $T$ for the ten-state Potts glass with Gaussian couplings in three dimensions. For $J_0=-1$, represented by the open symbols, ${\chi }_{\mathrm{FM}}$ remains small and size independent. In contrast, ${\chi }_{\mathrm{FM}}$ for $J_0=0$ (solid symbols) becomes very large and shows substantial finite-size effects.

Figure 13

(Color online) Correlation length ${\xi }_L$ (not ${\xi }_L∕L$) vs $T$ for the ten-state Potts glass in three dimensions. The main panel shows the case for $J_0=-1$ where there are hardly any finite-size effects. The inset shows the same for $J_0=0$ showing the size dependence of ${\xi }_L$ in this case.