Existence of a de Almeida-Thouless line in three dimensions
In plain words
In the solvable infinite-range model a spin glass stays glassy in a magnetic field up to a critical line in the field-temperature plane, the de Almeida-Thouless (AT) line. Whether such a line exists in three dimensions is a direct test between the competing theories.
Precise statement
For the 3D Ising spin glass in a uniform or random field $h$, determine whether a thermodynamic phase transition occurs at nonzero $h_{\mathrm{AT}}(T)$ for $T < T_c$, signaled by a diverging replicon susceptibility and replicon correlation length. Answer: yes or no, and the line $h_{\mathrm{AT}}(T)$ if yes.
What would settle it
Finite-size scaling of the replicon correlation length in equilibrium simulations of the 3D model in a field, at sizes where curves for different L cross at a stable point, or a proof.
Status in the literature
Unverified note
A 2024 analysis argues the AT line vanishes below six dimensions (arXiv:2402.03711), while 2025 simulations in $d = 5$ (arXiv:2510.14446) and a 2026 preprint in $d = 3$ (arXiv:2601.19192) report evidence for it.