Probing the Almeida-Thouless line away from the mean-field model

Results of Monte Carlo simulations of the one-dimensional long-range Ising spin glass with power-law interactions in the presence of a (random) field are presented. By tuning the exponent of the power-law interactions, we are able to scan the full range of possible behaviors from the infinite-range (Sherrington-Kirkpatrick) model to the short-range model. A finite-size scaling analysis of the correlation length indicates that the Almeida-Thouless line does not occur in the region with non-mean-field critical behavior in zero field. However, there is evidence that an Almeida-Thouless line does occur in the mean-field region.

Figure 1

(Color online) Sample equilibration plot for $\sigma =0.85$, $L=128$, $H_{\mathrm{R}}=0.1$, and $T=0.10$ (the lowest temperature simulated at finite fields). Data for the average energy $U$, and $U(q_l,q)$ defined in Eq. (10), as a function of equilibration time $t_{\mathrm{eq}}$. They approach their common value from opposite directions and, once they agree, do not change on further increasing $t_{\mathrm{eq}}$. The inset shows data for the correlation length divided by system size as a function of equilibration time. The data are independent of $t_{\mathrm{eq}}$ once $U$ and $U(q_l,q)$ agree.

Figure 2

(Color online) Each figure shows data for ${\xi }_L∕L$ vs $T$ for $H_{\mathrm{R}}=0$ for different system sizes, for a particular value of $\sigma$. For all values of $\sigma$, the data cross indicating that there is a spin-glass transition at finite temperature.

Figure 3

(Color online) Each figure shows results for ${\xi }_L∕L$ vs $T$ for $H_{\mathrm{R}}=0.10$ for different system sizes, for a particular value of $\sigma$. For $\sigma =0.55$, the data cross at $T_{\mathrm{c}}=0.96\left(2\right)$ showing that an AT line seems to be present. For $\sigma =0.65$, the data show close to marginal behavior. This may indicate that $\sigma =0.65$ is close to the borderline value for having an AT line. For $\sigma =0.75$ and $\sigma =0.85$, the data do not cross for any temperature down to $T=0.10$, which is considerably smaller than the zero field transition temperatures. This indicates that there is no AT line. Overall the results indicate that, on increasing $\sigma$, the AT line disappears.