Intermediate phase of the square-lattice $J1-J2$ Heisenberg model
In plain words
On the square lattice, a second-neighbor coupling competes with the nearest-neighbor one and destroys magnetic order over a window of ratios. What replaces it, a spin liquid, a pattern of spin singlets or both in sequence, has been debated for decades.
Precise statement
H = J1 sum_<ij> S_i . S_j + J2 sum_<<ij>> S_i . S_j, $S = 1/2$, square lattice, roughly $0.45 < J2/J1 < 0.62$. Determine the phases between Neel and stripe order (gapless spin liquid, plaquette or columnar VBS, or a sequence) with boundaries and the nature of each transition. An answer is the phase diagram with extrapolated order parameters and gaps.
What would settle it
Agreement among DMRG, PEPS, neural-network and variational Monte Carlo on the extrapolated VBS order and spin gap across the window.
Status in the literature
Unverified note
DMRG finds a narrow possible spin liquid between Neel order and a plaquette VBS (Gong et al., PRL 2014), and 2024 DMRG confirms plaquette rather than columnar VBS order (arXiv:2406.17417); a 2025 cluster-embedding study instead finds plaquette and columnar VBS phases split at $J2/J1 = 0.55$ with no spin liquid (arXiv:2511.06423).