Ground-state phase-space structures of two-dimensional $\pm J$ spin glasses: A network approach

We illustrate a complex-network approach to study the phase spaces of spin glasses. By mapping the whole ground-state phase spaces of two-dimensional Edwards-Anderson bimodal $(\pm J)$ spin glasses exactly into networks for analysis, we discovered various phase-space properties. The Gaussian connectivity distribution of the phase-space networks demonstrates that both the number of free spins and the visiting frequency of all microstates follow the Gaussian distribution. The spectra of phase-space networks are Gaussian, which is proven to be exact when the system is infinitely large. The phase-space networks exhibit community structures. By coarse graining to the community level, we constructed a network representing the entropy landscape of the ground state and discovered its scale-free property. The phase-space networks exhibit fractal structures, as a result of the rugged entropy landscape. Moreover, we show that the connectivity distribution, community structures, and fractal structures change drastically at the ferromagnetic-to-glass phase transition. These quantitative measurements of the ground states provide new insight into the study of spin glasses. The phase-space networks of spin glasses share a number of common features with those of lattice gases and geometrically frustrated spin systems and form a new class of complex networks with unique topology.

Figure 1

(a) The frustration in a square. The square has three ferromagnetic (solid lines) bonds ($J_{ij}=1$) and one antiferromagnetic (dashed line) bond ($J_{ij}=-1$). Therefore at least one bond (indicated by an additional red line nearby) is frustrated ($J_{ij}{s}_i{s}_j=-1$) for different arrangements of Ising spins (black and white circles). All four real-space configurations have the lowest possible energy. In general, frustration exists if, for any closed loop of bonds, the number of antiferromagnetic bonds is odd [19]. (b) The real-space configurations of two ground states of a $3\times 3$ sample correspond to nodes 2 and 3 in the phase-space network. Flipping all spins of an arbitrary state $i$ results in state $i^{\prime }$ in the phase-space network. (c) Another $3\times 3$ sample with the same fraction of antiferromagnetic bonds ($p=6/18$) but a different bond distribution. The phase-space network differs from that in (b). (d) The frustrated bonds (bold lines) of the eight ground states in (b). Note that states $i$ and $i^{\prime }$ share the same configuration of frustrated bonds. The joint point of two frustrated bonds is a free spin (filled circle). Flipping a free spin does not change the number of frustrated bonds (energy) and may create (e.g., $\mathrm{①}\to \mathrm{②}$) or annihilate (e.g., $\mathrm{③}\to \mathrm{④}$) a free spin nearby.

Figure 2

The log-normal distributions of the ground-state degeneracy of 2000 $7\times 7$ samples in (a) and 2000 $8\times 8$ samples in (b) with $p=0.5$. The curves are the Gaussian fit $0.44exp[-0.59{(x-2.23)}^2]$ and $0.38exp[-0.43{(x-2.75)}^2]$, where $x={log}_{10}{N}_{\mathrm{node}}$.

Figure 3

The ground-state networks of four $5\times 5$ samples with different number of ground-state clusters.

Figure 4

(a) The connectivity distributions of six $8\times 8$ samples with $p=0.5000$, 0.4375, 0.3750, 0.3125, 0.2500, and 0.1875 and $N_{\mathrm{node}}=96396,63050,33082,15864,6492$, and 3394, respectively. The curves are the Gaussian fits. (b) The mean connectivity of 2000 $8\times 8$ samples with $p=0.5$. (c) The fitted value of $\alpha$ in Eq. (2) for various sample sizes and $p$ values. (d) $\langle {\sigma }_k^2/\overline{k}\rangle$ as a function of $p$, where $\langle \rangle$ denotes the ensemble average over 200 samples.

Figure 5

The ensemble-averaged fraction of frustrated bonds $\langle n\rangle$ as a function of $p$ for $8\times 8$ samples. $\langle n\rangle$ increases rapidly around the glass transition point $p_c=0.103$.

Figure 6

The eigenvalue distributions, i.e., the spectra, of the adjacency matrices for six $8\times 8$ samples with various $p$ and $N_{\mathrm{node}}$. Variances are scaled to 1 by ${\lambda }^{\prime }=\lambda /{\overline{k}}^{-1/2}$ so all curves collapse into the standard Gaussian distribution.

Figure 7

[(a) and (b)] The phase-space networks of two $6\times 6$ samples with $p=0.5$ and $N_{\mathrm{node}}=1312,996$ respectively. (a) Contains two clusters and 14 communities in different colors. (b) Contains four clusters and 16 communities. (c) $\langle Q_{\mathrm{m}}\rangle$ increases rapidly around the glass transition $p_c=0.103$. Each data point is averaged over 200 samples. (d) Ensemble-averaged numbers of communities and clusters at $p=0.5$. Lines are fitted by $\langle n_c\rangle \propto exp\left(\beta L^2\right)$ with $\beta =0.0231\pm 0.0008$ for clusters and $0.0428\pm 0.0006$ for communities. (e) The entropy-landscape network coarse grained from the largest cluster with $16620$ nodes of an $8\times 8$ sample with $p=0.5$. A node in the coarse-grained network corresponds to a community in the phase-space network or an entropy valley in the entropy landscape. The area of a node is proportional to the number of inner-community nodes, representing the area of the entropy valleys in the entropy landscape. The density of inner-community edges, representing the depth of the entropy valley, is color coded. The thickness of an edge is proportional to the number of intercommunity edges, representing the inverse of the entropy-barrier height. (f) The connectivity distribution of the coarse-grained network averaged over 2000 $6\times 6$ samples and over 2000 $8\times 8$ samples with $p=0.5$. The fitted $p\left(k_c\right)\sim k_{\mathrm{c}}^{-5.97\pm 0.07}$ for a $6\times 6$ sample and $p\left(k_c\right)\sim k_{\mathrm{c}}^{-6.00\pm 0.06}$ for $8\times 8$ samples reveal their scale-free structures. Inset: The size of the entropy valley, i.e., the number of nodes in a community, as a function of its connectivity $k_{\mathrm{c}}$ for the coarse-grained networks of $8\times 8$ samples at $p=0.5$. $\langle \rangle$ denotes the ensemble average. The data reveal that a large entropy valley has more neighbors.

Figure 8

The ensemble-averaged number of communities and clusters $\langle n_c\rangle$ as a function of $p$ for $8\times 8$ samples. $\langle n_c\rangle$ for both clusters and communities begin to increase around the glass transition point $p_c=0.103$.

Figure 9

(a) The size distributions of communities and clusters for 2000 $8\times 8$ samples with $p=0.5$. (b) The fraction of $N_{\mathrm{node}}$ communities or clusters in the whole phase space $f\left(N_{\mathrm{node}}\right)\propto N_{\mathrm{node}}\times$ (its probability). The curves show that median-sized clusters and communities make up the highest fraction of the phase space and thus are most important. The clusters with size $N_{\mathrm{node}}\simeq 16384$ and the communities with size $N_{\mathrm{node}}\simeq 4096$ are most influential for $8\times 8$ samples. (c) The box-covering analysis fitted with Eq. (4) for the three largest clusters of the phase space of an $8\times 8$ sample at $p=0.5$. Their fractal dimensions are $2.89$, $3.25$, and $3.59$. (d) The box-covering analysis shows that 12-, 14-, and 16-dimensional hypercubes are not fractals.

Figure 10

The relative fluctuation of network connectivity ${\sigma }_k/\overline{k}$ of 2000 $8\times 8$ samples at $p=0.5$ follows the curve $1/\sqrt{{log}_2{N}_{\mathrm{node}}}$, suggesting that ${\sigma }_k/\overline{k}$ approaches zero for large networks.