Study of the de Almeida–Thouless Line Using Power-Law Diluted One-Dimensional Ising Spin Glasses

We test for the existence of a spin-glass phase transition, the de Almeida–Thouless line, in an externally applied (random) magnetic field by performing Monte Carlo simulations on a power-law diluted one-dimensional Ising spin glass for very large system sizes. We find that a de Almeida–Thouless line occurs only in the mean-field regime, which corresponds, for a short-range spin glass, to dimension $d$ larger than 6.

Figure 1

(a) Finite-size correlation length divided by $L$ as a function of $T$ for different sizes for $H_{\mathrm{R}}=0$ and $\sigma =0.75$ (non-mean-field region). The inset shows ${\chi }_{\mathrm{SG}}/L^{2-\eta }$ using the exact value $\eta =3-2\sigma =1.5$. In both cases, the data cross, indicating a phase transition at zero field. (b) The same as (a) but for $H_{\mathrm{R}}=0.1$. The absence of an intersection down to low $T$ shows that there is no transition in a field [the shaded area corresponds to $T_c(H_{\mathrm{R}}=0)$]. The inset shows data for a bimodal ($\pm J$) distribution of bonds, as used in Ref. 19, for sizes $L=256–1024$ on a linear topology. While Ref. 19 find a finite-temperature transition (shaded area in the inset), we see no sign of it. The absence of a transition is even more clear in (c), where we show data as in (b) but for $\sigma =0.85$, i.e., deeper into the non-mean-field regime. In (d) we show data for the correlation length divided by $L^{\nu /3}(=L^{5/3})$ as a function of $T$ for different sizes for $H_{\mathrm{R}}=0.1$ and $\sigma =0.60$ (in the mean-field region). The inset shows ${\chi }_{\mathrm{SG}}/L^{1/3}$. The intersections show that there is a transition in a field, i.e., an AT line for this value of $\sigma$.

Figure 2

Distribution of the spin overlap $q$ for $\sigma =0.75$, $T=0.71$, and $H_{\mathrm{R}}=0.1$. Even for the largest $L$ studied, there is a tail which extends into the negative-$q$ region.