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sampling-nonergodic — GCNN, spinon pair-density-wave state (chi0, P_Z2 = -1, q =..., -0.4461111 ○
Finding. Published in Phys. Rev. X 15, 011047 (2025) as a ground state 1.78% below the DMRG benchmark. A Comment (arXiv:2605.28861) shows the single-spin-flip update used at this size does not conserve total magnetisation, so the Markov chains freeze and the low energy is a sampling artifact. Under ergodic sampling the same ansatz converges ~3.5% ABOVE DMRG.
Diagnosis. The reported energy is an artifact of non-ergodic Monte Carlo sampling, not a property of the ansatz. The kagome Heisenberg Hamiltonian is SU(2) symmetric and its ground state lies in a fixed total-magnetisation sector, so a single-spin-flip update - which changes S^z_tot - is incompatible with the symmetry. As the network concentrates on the physical S^z_tot = 0 sector the acceptance rate collapses to exactly zero beyond 5000 iterations and the chains freeze, so the reported average is taken over a non-representative set of configurations. With the magnetisation-preserving exchange update the same architecture and optimizer converge stably to E ~ -45.6 against a DMRG value of E ~ -47.3; re-evaluating the spin-flip-optimised PARAMETERS under ergodic sampling raises the energy further, to E ~ -42.6. The published -48.18 lies below all of them.
Ruled out. NOT a variational-principle violation, and that is the point: DMRG at finite bond dimension is itself an upper bound, so an energy below it is not on its own evidence of error - it would ordinarily just be a better state. Nothing about the number alone identifies it as wrong. The refutation had to come from the sampler.
Evidence. arXiv:2605.28861 (Kamal, Kufel, Vu, Laumann & Yao), Fig. 1(a) acceptance-rate collapse and Fig. 1(b) energy convergence; Ðurić et al., Phys. Rev. X 15, 011047 (2025), arXiv:2401.02866, Sec. IV.