Lack of Ultrametricity in the Low-Temperature Phase of Three-Dimensional Ising Spin Glasses

We study the low-temperature spin-glass phases of the Sherrington-Kirkpatrick (SK) model and of the 3-dimensional short-range Ising spin-glass (3DISG). By using clustering to focus on the relevant parts of phase space and reduce finite size effects, we found that for the SK model ultrametricity becomes clearer as the system size increases, while for the short-range case our results indicate the opposite, i.e., lack of ultrametricity. Another method, which does not rely on clustering, indicates that the mean-field solution works for the SK model but does not apply in detail to the 3DISG, for which stochastic stability is also violated.

Figure 1

(a) Clustering produces the dendrogram of state clusters, for the equilibrium ensemble of 1000 states, obtained by simulation of one realization of the SK model for 1024 spins, at $T=0.2$. The vertical position of each cluster $\alpha$ is its score $s_{\alpha }$. Logarithmic scale is used for the vertical axis; only clusters with $s>0.04$ are shown. Significant clusters, identified by a long branch above them, have high values of $v$; these clusters reflect the presence of dense regions in the underlying Boltzmann distribution. (b) The distance matrix $d$ of the microstates, that were reordered according to their positions, as the leaves of the dendrogram, on the horizontal axis of (a). Darker colors indicate smaller distances.

Figure 2

The distributions $P(K)$ (see text), calculated for 3DISG and SK systems for various sizes $N$.

Figure 3

The function $\rho (\Delta q)$ for SK and 3DISG systems. The error bars are smaller than the symbols.