Evidence of the de Almeida-Thouless transition in three-dimensional spin glasses

Abstract

The nature of spin-glass states in a magnetic field remains a major open problem in statistical physics. The existence of the de Almeida-Thouless (dAT) transition for three-dimensional (3D) spin glasses in a field is still debated. We introduce a new computational method to define the spin-glass susceptibility, which is robust against the broad tail in the overlap distribution that undermines conventional analyses. Applying this approach to the Edwards-Anderson spin-glass model in 2D and 3D, and contrasting with the 3D Ising (without disorder) and mean-field spin-glass models, we find a stark difference: the locus of susceptibility maxima bends to the right in the field-temperature plane for the Ising and 2D spin-glass cases, indicating a supercritical crossover line, but bends to the left for the mean-field and 3D spin glasses - a signature of the dAT line. Finite-size scaling further suggests that the peak susceptibility diverges with system size in 3D spin glasses under a field, while saturating in 2D. These results provide direct numerical evidence for the dAT transition in 3D, supporting the replica symmetry breaking scenario.

Figures (9)

• Figure 1: Schematic $H$-$T$ phase diagrams. (a) Ising model; the dashed line is the $L^+$ crossover line. (b) Droplet picture for spin glasses (SG). (c) RSB picture for spin glasses; the solid line is the dAT line.
• Figure 2: Distribution of the overlap order parameter $P(q)$ and various susceptibilities. Data are obtained from simulations for the 3D Ising and spin-glass (SG) models ($L=16$). Error bars represent the standard error of the mean, in all figures.
• Figure 3: Results for the 3D spin glasses. (a-c) Data of susceptibility $\chi_W(T)$: (a) $L= 16$, (b) $h=0$ and (c) $h=0.2$. (d) Peak value $\chi_W^{\rm max}$ as a function of $N=L^3$. The lines represent power-law fitting, $\chi_W^{\rm max} \sim N^{\alpha}$.
• Figure 4: Results of $h(T_{\rm max})$ for four models, determined by the loci of maximum $\chi_W(T)$. In (a), the loci of maximum $\chi_|q|(T)$ are added for comparison.
• Figure 5: Distribution of the overlap order parameter $P(q)$ and various susceptibilities. Data are obtained from simulations for the mean-field (MF) and 2D spin-glass (SG) models ($N=4096$ and $L=46$, respectively).