Self-sustained clusters and ergodicity breaking in spin models

Self-sustained spin clusters are analytically linked to ergodicity breaking in fully connected Ising and Sherrington-Kirkpatick (SK) models, relating the less understood spin space to the well understood state space. This correspondence is established through the absence of clusters in the paramagnetic phase, the presence of one dominant cluster in the Ising ferromagnet, and the formation of nontrivial clusters in SK spin glass. Yet unobserved phenomena are also revealed such as a first order phase transition in cluster sizes in the SK ferromagnet. The method could be adapted to investigate other spin models.

Figure 1

Phase diagram of the SK model as a function of coupling mean $J_0$ and temperature $T$. The right-hand side of the dashed line corresponds to the region where a first order phase transition in cluster sizes is observed; i.e., self-sustained clusters of certain size are absent. The blue and orange regions correspond to the negative entropy region of the RS and 1RSB ansatz, respectively; the cross and triangle symbols correspond to the points at which cluster statistics are obtained in Fig. 3a.

Figure 2

The entropy $S\left(r\right)$ of self-sustained clusters of size $r$ at $T=0.5$ and various coupling mean values $J_0$ in the SK model. Curves with open symbols have been obtained with a fine resolution of $\gamma$ at intervals of $0.02$. The black line $J_0\to \infty$ corresponds to $S\left(r\right)$ in an Ising ferromagnet.

Figure 3

(a) Entropy $S\left(r\right)$ in the spin glass case with $J_0=0.5,T=0.2,0.5$, compared to Eq. (12). Inset: The maximum of $S\left(r\right)$ obtained by exhaustive search in the ground state of small SK systems with $J_0=0.01$, compared to the value of $S\left(0.5\right)=\left[\ln C_{N/4}^{N/2}\right]/N=0.346$ (red dashed line); both mean values and standard deviation are shown. (b) The difference $q_{s\sigma s\sigma }-q{m}_{\sigma }^2$ as a function of $r$ for various $J_0$ values.

Figure 4

(a) The self-sustained cluster size $r_{\mathrm{SK}}$ as a function of $\gamma$ at temperature $T=0.5$ with $J_0$ from $1.2$ to $1.8$. (b) In-cluster and out-cluster magnetizations as a function of the cluster size $r$.