Exact lower critical dimension of the Ising spin glass
In plain words
Below a certain space dimension, magnets with random-sign interactions never freeze into a glass at any nonzero temperature. That dimension lies between two and three, and a conjecture says it is exactly two and a half.
Precise statement
For the Edwards-Anderson Ising model with symmetric couplings in zero field, continued to non-integer dimension d (hierarchical lattices, long prisms, or long-range one-dimensional proxies), determine the lower critical dimension $d_l$ below which $T_c = 0$, defined by the vanishing of the domain-wall stiffness exponent $\theta(d)$. Test the conjecture $d_l = 5/2$ exactly, which follows from replica-symmetry-breaking interface arguments (Franz, Parisi, Virasoro 1994). An answer is $d_l$ with error below 0.02, or a derivation of its exact value.
What would settle it
A computation of $\theta(d)$ on a continuous family of geometries interpolating between $d = 2$ and $d = 3$ with controlled finite-size corrections, or an RG derivation of $d_l$.
Status in the literature
Unverified note
Domain-wall stiffness fits give $d_{\ell}$ close to $5/2$ (Boettcher, Phys. Rev. Lett. 95, 197205, 2005) and 2026 long-prism simulations agree with the replica prediction (arXiv:2601.07926), while a 2026 rigorous result finds no spin-glass order on Migdal-Kadanoff hierarchical lattices of fractal dimension just below 3 (arXiv:2607.15676).