Does the random SU(2) Heisenberg magnet freeze at zero temperature?
In plain words
When quantum spins free to point in any direction interact with random couplings of both signs, they may freeze into a glass or stay in a liquid-like state with no frozen moments. For spin $1/2$ with all-to-all random couplings, calculations suggest weak freezing, but its size and the liquid-like behavior above it are unsettled, and short-range versions are open.
Precise statement
Model: $H = (1/\sqrt{N}) \sum_{i<j} J_{ij} S_i . S_j$, $S = 1/2$, $J_{ij}$ independent Gaussian with zero mean and unit variance, and its short-range versions on 2D and 3D lattices. Determine whether the Edwards-Anderson order parameter $q_{\mathrm{EA}}$, the long-time limit of the site- and disorder-averaged $\langle S_i(t) . S_i(0)\rangle$, is nonzero at $T = 0$ in the thermodynamic limit, its value, and the temperature window in which Sachdev-Ye spin-liquid correlations $G(\tau)\sim 1/\tau$ appear. An answer gives $q_{\mathrm{EA}}$ with error bars and the crossover temperature.
What would settle it
Extrapolation in N of exact-diagonalization and matrix-product-state data for the infinite-range model, checked against controlled large-M expansions of its SU(M) generalization, and quantum Monte Carlo or tensor-network data for the lattice versions.
Status in the literature
Unverified note
Christos, Haehl and Sachdev (PRB 2022) described spin-glass order at $T = 0$ with a spin-liquid crossover at higher $T$, and Hosseinabadi, Sachdev and Marino (PRL 2026) studied the crossover to Sachdev-Ye-Kitaev criticality; the short-range lattice versions are open.