Number of thermodynamic states in the three-dimensional Edwards-Anderson spin glass

The question of the number of thermodynamic states present in the low-temperature phase of the three-dimensional Edwards-Anderson Ising spin glass is addressed by studying spin- and link-overlap distributions using population annealing Monte Carlo simulations. We consider overlaps between systems with the same boundary condition, which are the usual quantities measured, and also overlaps between systems with different boundary conditions, both for the full systems and also within a smaller window within the system. Our results appear to be fully compatible with a single pair of pure states such as in the droplet-scaling picture. However, our results for whether or not domain walls induced by changing boundary conditions are space filling or not are also compatible with scenarios having many thermodynamic states, such as the chaotic-pairs picture and the replica-symmetry-breaking picture. The differing results for spin overlaps in the same and different boundary conditions suggest that finite-size effects are very large for the system sizes currently accessible in low-temperature simulations.

Figure 1

Spin-overlap distributions $P_{\mathcal{J}}\left(q\right)$ for three typical disorder realizations for $L=8$ at $T=0.42$. Each panel represents a different disorder realization, and within each panel, the overlap is shown for the boundary condition pairs $(\pi ,\pi )$ (red dashed line), $(\overline{\pi },\overline{\pi })$ (blue dotted line), and $(\pi ,\overline{\pi })$ (black solid line). The peaks in $P_{\mathcal{J}}\left(q\right)$ for the off-diagonal pair are shifted away from $q_{\mathrm{EA}}$ due to the relative domain wall between the periodic and antiperiodic boundary conditions.

Figure 2

Disorder-averaged spin-overlap distributions $P\left(q\right)$ for the full lattice for sizes $L=4,6,8,10$, and 12 at $T=0.42$. The set of curves with peaks at the finite-size value of $\pm q_{\mathrm{EA}}$ and bimodal features corresponds to the diagonal overlap distributions, while the set of curves with broad maxima at the center corresponds to the off-diagonal overlap distributions.

Figure 3

Four statistics of the off-diagonal overlap distribution: $P\left(0\right),\langle q^2\rangle ,I\left(0.2\right)$, and $f_q$ (see text for details) vs inverse system size $1/L$. The expected thermodynamic behavior, i.e., $1/L\to 0$, is described in Eqs. (4) and (5).

Figure 4

Disorder-averaged spin-overlap distributions $P\left(q\right)$ in a $W=4$ observation window within a system of size $L=4,6,8,10$, and 12 at $T=0.42$. The sets of curves that display large peaks near $\pm q_{\mathrm{EA}}$ are the diagonal overlaps. The off-diagonal overlap curves have much lower weight near $\pm q_{\mathrm{EA}}$, but that weight increases slowly with the system size $L$.

Figure 5

Scaling of ${\mathrm{\Gamma }}_{\mathrm{s}}$ (within a window) and ${\mathrm{\Gamma }}_{\ell }$ (defined below, for the full systems) as a function of system size $L$ for $T=0.20$ and $T=0.42$ together with power-law fits (straight lines).

Figure 6

Comparison of the fits of ${\mathrm{\Gamma }}_{\mathrm{s}}$ assuming droplet scaling (red curved line, power law) or a RSB (blue straight line, constant and finite-size corrections) at $T=0.42$. Both functional forms fit the data comparably well; that is, ${\mathrm{\Gamma }}_{\mathrm{s}}$ does not distinguish the two pictures at these length scales. See text for details of fits.

Figure 7

Average of the link overlap $\langle q_{\ell }\rangle$ as a function of system size $L$ for the three pairs of boundary conditions and temperatures $T=0.42$ (left panel) and $T=0.2$ (right panel). Note that as $L$ increases, the off-diagonal link overlap increases toward the diagonal link overlap.

Figure 8

Comparison of the fits of ${\mathrm{\Gamma }}_{\ell }$ assuming droplet scaling (red lighter curve, power law) or a RSB (blue darker curve, constant and finite-size corrections) at $T=0.2$. Both functional forms fit the data comparably well; that is, ${\mathrm{\Gamma }}_{\ell }$ does not distinguish the two pictures at these length scales. See text for details of fits.