Fate of the de Almeida-Thouless line as dimension approaches six
In plain words
Above six dimensions spin-glass theory becomes simple and the field-driven transition certainly exists. One prediction says it disappears exactly as the dimension drops to six; if so, it cannot exist in three dimensions.
Precise statement
For Ising spin glasses on d-dimensional hypercubic lattices, or one-dimensional chains with couplings $J_{ij} \sim r_{ij}^{-\sigma}$ that mimic dimension d, determine whether $h_{\mathrm{AT}}(T)$ is nonzero at $d = 6$ and slightly below, testing the Moore-Bray prediction $h_{\mathrm{AT}}^2/T_c^2 = C (d-6)^4 (1 - T/T_c)^{d/2-1}$ as $d \to 6$ from above. Answer: yes or no for $d = 6$ and for d just below 6.
What would settle it
Simulations of 6D lattices or long-range one-dimensional proxies with controlled finite-size scaling of the replicon susceptibility, or an RG computation that fixes the fate of the AT fixed point.
Status in the literature
Unverified note
Large simulations report a transition in a field in $d = 6$ (arXiv:2306.00569, 2023) and in $d = 5$ (arXiv:2510.14446, 2025), while long-range one-dimensional simulations find $h_{\mathrm{AT}}^{2} \sim (2/3 - \sigma)$, vanishing at the proxy of $d = 6$ (arXiv:2402.03711, 2024), as predicted in 2011 (arXiv:1102.1675).