Ground state of the spin-$1/2$ kagome Heisenberg antiferromagnet
In plain words
The simplest model of spins on the kagome lattice still has no agreed ground state. The two main candidates are a spin liquid with a small energy gap and one with no gap and Dirac-cone excitations (energy growing linearly with momentum).
Precise statement
Model: H = J sum_<ij> S_i . S_j, $S = 1/2$, $J > 0$, nearest neighbors on the kagome lattice. Determine whether the thermodynamic-limit ground state is a gapped $Z2$ spin liquid (finite triplet gap $\Delta_t$, topological entanglement entropy $\ln 2$) or a gapless $U(1)$ Dirac spin liquid ($\Delta_t\sim 1/L$, power-law correlations). An answer gives $\Delta_t/J$ extrapolated to infinite size with error bars, or a proof that it vanishes.
What would settle it
Converged DMRG, PEPS or neural-network variational results on cylinders and tori with circumferences beyond 12 sites that agree on the extrapolated spin gap and entanglement scaling.
Status in the literature
Unverified note
DMRG from 2011 favored a gapped $Z2$ state; DMRG with flux insertion and PEPS studies from 2017 to 2019 favor the $U(1)$ Dirac state; no consensus as of 2026.