  We examine the magnetic correlations in quantum spin models that were derived
recently as effective low-energy theories for electronic correlation effects on
the edge states of graphene nanoribbons. For this purpose, we employ quantum
Monte Carlo simulations to access the large-distance properties, accounting for
quantum fluctuations beyond mean-field-theory approaches to edge magnetism. For
certain chiral nanoribbons, antiferromagnetic inter-edge couplings were
previously found to induce a gapped quantum disordered ground state of the
effective spin model. We find that the extended nature of the intra-edge
couplings in the effective spin model for zigzag nanoribbons leads to a quantum
phase transition at a large, finite value of the inter-edge coupling. This
quantum critical point separates the quantum disordered region from a gapless
phase of stable edge magnetism at weak intra-edge coupling, which includes the
ground states of spin-ladder models for wide zigzag nanoribbons. To study the
quantum critical behavior, the effective spin model can be related to a model
of two antiferromagnetically coupled Haldane-Shastry spin-half chains with
long-ranged ferromagnetic intra-chain couplings. The results for the critical
exponents are compared also to several recent renormalization group
calculations for related long-ranged interacting quantum systems.
