Ultrametric probe of the spin-glass state in a field

We study the ultrametric structure of phase space of one-dimensional Ising spin glasses with random power-law interaction in an external random field. Although in zero field the model in both the mean-field and non-mean-field universality classes shows an ultrametric signature [Phys. Rev. Lett. 102, 037207 (2009)], when a field is applied ultrametricity seems only present in the mean-field regime. The results for the non-mean-field case in an external field agree with data for spin glasses studied within the Migdal-Kadanoff approximation. Our results therefore suggest that the spin-glass state might be fragile to external fields below the upper critical dimension.

Figure 1

Dendrogram obtained by clustering 100 configurations (see text) for a sample system with $\sigma =0.0$ (Sherrington-Kirkpatrick model) and $L=512$ at $T=0.36$, together with the matrix $d_{\alpha \beta }$ (grayscale, distance 0 is black). The order of the states is given by the leaves of the dendrogram (figure rotated clockwise by ${90}^{\circ }$).

Figure 2

Distribution $P\left(K\right)$ for different system sizes (all panels have the same horizontal and vertical scale) and an external random field $H=0.1$. (a) Data for the SK model. The distribution diverges very slightly for $K\to 0$ and $L\to \infty$ thus signaling an UM phase structure. (b) Data for $\sigma =0.60$ (mean-field universality class). There is still a weak hint of a divergence for $K\to 0$. (c) Data for $\sigma =0.75$ (non-mean-field universality class). There is no clear sign of a divergence in $P\left(K\right)$ for $K\to 0$. Note that when $H=0$ data for $\sigma =0.75$ show a clear signature for UM behavior.[12] Error bars are smaller then the symbol size.

Figure 3

Variance $\mathrm{Var}\left(K\right)$ of $P\left(K\right)$ as a function of system size $L$ for different values of $\sigma$. The data can be fit to a power law (dashed lines). In the mean-field regime (SK and $\sigma =0.6$) a fit to a constant is unlikely (see text). The power-law decay of the variance as a function of system size suggests a divergence in $P\left(K\right)$ for $K\to 0$. For $\sigma =0.75$ the data are compatible with a constant (solid line) or a very weak power-law behavior.

Figure 4

Variance $\mathrm{Var}\left(K\right)$ of $P\left(K\right)$ as a function of system size $L$ for spin glasses on MK lattices. The data are compatible with a constant behavior, showing that there is no UM phase-space structure for spin glasses within the MK approximation. The solid line is a guide to the eye.