Large-scale Monte Carlo simulations of the isotropic three-dimensional Heisenberg spin glass

We study the Heisenberg spin glass by large-scale Monte Carlo simulations for sizes up to ${32}^3$, down to temperatures below the transition temperature claimed in earlier work. The data for the larger sizes show more marginal behavior than that for the smaller sizes, indicating the lower critical dimension is close to, and possibly equal to, 3. We find that the spins and chiralities behave in a quite similar manner.

Figure 1

${\mathbf{H}}_i$ is the local field on site $i$ due to its neighbors. The spin at $i$ is initially in direction ${\mathbf{S}}_i$. In a microcanonical (over-relaxation) move, the spin is reflected about ${\mathbf{H}}_i$ according to Eq. (9), and so ends up in direction ${\mathbf{S}}_i^{\prime }$.

Figure 2

(Color online) Equilibration plot, testing Eq. (20) for $L=24$ at the $T=0.144$. It is seen that the data for $2TU∕z$ come together at about $3\times {10}^5$ total sweeps (equilibration plus measurement) and then stay at their common value indicating that equilibration has been achieved. The lines are guides to the eye. The figure shows that the energy comes close to its equilibrium value very quickly, whereas $q_l-q_s$ takes much longer.

Figure 3

(Color online) A plot of ${\xi }_L∕L$ as a function of the total number of sweeps for $L=24$ at $T=0.144$. The figure shows that the data flattens off at around $3\times {10}^5$ sweeps, the value where the two sets of data in Fig. 2 start to agree. This indicates that when the data in Fig. 2 agree within high precision, i.e., when Eq. (20) is satisfied, the correlation length has reached its equilibrium value. The line is a guide to the eye.

Figure 4

(Color online) A plot of ${\xi }_{c,L}^{\Vert }∕L$ as a function of the total number of sweeps for $L=24$ at $T=0.144$. As in Fig. 3, this figure shows that the data flattens off at around $3\times {10}^5$ sweeps, the value where the two sets of data in Fig. 2 start to agree. The line is a guide to the eye.

Figure 5

(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. Note that there are very many data points for the larger sizes. This is because a large number of temperatures are needed for the parallel tempering algorithm, as discussed in Sec. 2.

Figure 6

(Color online) Enlarged view of a region of Fig. 5.

Figure 7

(Color online) Data for ${\xi }_{c,L}^{\Vert }∕L$, the “parallel” chiral-glass correlation length divided by system size, as a function of $T$ for different system sizes.

Figure 8

(Color online) Enlarged view of a region of Fig. 7.