Large-scale Monte Carlo simulations of the three-dimensional $XY$ spin glass

We study the XY spin glass by large-scale Monte Carlo simulations for sizes up to ${24}^3$, down to temperatures below the transition temperature found in earlier work. The data for the larger sizes show more marginal behavior than that for the smaller sizes, indicating that the lower critical dimension is close, and possibly equal to three. We find that the spins and chiralities behave in a similar manner. We also address the optimal ratio of “over-relaxation” to “Metropolis” sweeps in the simulation.

Figure 1

(Color online) Equilibration plot testing Eq. (10) for $L=16$ at $T=0.265$. It is seen that the data for $U$ and $U\left(q_l,q_s\right)$, given by Eq. (11), come together when the total number of over-relaxation sweeps, $N_{\text{sweep}}^{\text{OR}}=N_{\text{equil}}^{\text{OR}}+N_{\text{meas}}^{\text{OR}}$, see Table , is equal to about $2\times {10}^5$. These two quantities then stay at their common value, indicating that equilibration has been achieved. It is seen that the energy comes close to its equilibrium value very quickly, whereas $U\left(q_l,q_s\right)$, which depends on the link overlap $q_l$ between two replicas, takes much longer.

Figure 2

(Color online) A plot of the spin-glass and chiral-glass correlation lengths, ${\xi }_L$ and ${\xi }_{L,c}$, divided by $L$ as a function of the total number of sweeps for $L=16$ at $T=0.265$. It is seen that the data flatten off at around $2\times {10}^5$ sweeps, the value where the two sets of data in Fig. 1 start to agree. This indicates that when the data in Fig. 1 agree within high precision, i.e., when Eq. (10) is satisfied, the correlation lengths have reached their equilibrium value.

Figure 3

(Color online) Data for ${\xi }_L/L$, the spin-glass correlation length divided by system size as a function of $T$ for different system sizes.

Figure 4

(Color online) Data for the chiral correlation length (averaged over longitudinal and transverse directions) divided by system size as a function of $T$ for different system sizes.

Figure 5

(Color online) The same data as in Figs. 3, 4, but including only the largest sizes and in a somewhat expanded scale.

Figure 6

(Color online) Data for the ratio of the chiral-glass to the spin-glass correlation lengths for sizes from 8 to 24.

Figure 7

(Color online) Data for the spin-glass susceptibility ${\chi }_{\text{SG}}\equiv {\chi }_{\text{SG}}\left(\mathbf{k}=0\right)$ divided by $L^{2-{\eta }_{\text{SG}}}$, where we took ${\eta }_{\text{SG}}=-0.2$ in order to get the data to intersect [see Eq. (13)], for $T$ around 0.30 since this is roughly where the data for ${\xi }_L/L$ and ${\xi }_{c,L}$ intersect/merge for the largest sizes (see Figs. 3, 4).

Figure 8

(Color online) Similar to Fig. 7 but for the chiral-glass susceptibility ${\chi }_{\text{CG}}\equiv {\chi }_{\text{CG}}\left(\mathbf{k}=0\right)$. Here we took ${\eta }_{\text{CG}}=0.1$.

Figure 9

(Color online) Results for $L=16$, $T=0.265$. The data connected by solid lines is $U\left(q_l,q_s\right)$ in Eq. (11) for different number of over-relaxation (OR) sweeps per Metropolis (MET) sweep as indicated. The horizontal axis is the total number of OR plus MET sweeps. The data connected by the dashed line is the energy $U$, which should equal $U\left(q_l,q_s\right)$ in equilibrium according to Eq. (10). Since the energy equilibrates relatively fast, its value does not depend significantly on the ratio of OR to MET sweeps for the range of sweeps presented. The number of parallel tempering sweeps is the same for all sets of data except for “40 OR,” where it is 1/4 as many.